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+SEQUENT: Towards Traceable Quantum Machine Learning using
+Sequential Quantum Enhanced Training ∗
+Philipp Altmann1, Leo S¨unkel1, Jonas Stein1, Tobias M¨uller2,
+Christoph Roch1 and Claudia Linnhoff-Popien1
+1LMU Munich
+2SAP SE, Walldorf, Germany
+philipp.altmann@ifi.lmu.de
+Keywords:
+Quantum Machine Learning, Transfer Learning, Supervised Learning, Hybrid Quantum Computing.
+Abstract:
+Applying new computing paradigms like quantum computing to the field of machine learning has recently
+gained attention. However, as high-dimensional real-world applications are not yet feasible to be solved us-
+ing purely quantum hardware, hybrid methods using both classical and quantum machine learning paradigms
+have been proposed. For instance, transfer learning methods have been shown to be successfully applicable
+to hybrid image classification tasks. Nevertheless, beneficial circuit architectures still need to be explored.
+Therefore, tracing the impact of the chosen circuit architecture and parameterization is crucial for the devel-
+opment of beneficially applicable hybrid methods. However, current methods include processes where both
+parts are trained concurrently, therefore not allowing for a strict separability of classical and quantum impact.
+Thus, those architectures might produce models that yield a superior prediction accuracy whilst employing the
+least possible quantum impact. To tackle this issue, we propose Sequential Quantum Enhanced Training (SE-
+QUENT) an improved architecture and training process for the traceable application of quantum computing
+methods to hybrid machine learning. Furthermore, we provide formal evidence for the disadvantage of current
+methods and preliminary experimental results as a proof-of-concept for the applicability of SEQUENT.
+1
+INTRODUCTION
+With classical computation evolving towards per-
+formance saturation, new computing paradigms like
+quantum computing arise, promising superior perfor-
+mance in complex problem domains. However, cur-
+rent architectures merely reach numbers of 100 quan-
+tum bits (qubits), prone to noise, and classical com-
+puters run out of resources simulating similar sized
+systems (Preskill, 2018). Thus, most real world appli-
+cations are not yet feasible solely relying on quantum
+compute. Especially in the field of machine learn-
+ing, where parameter spaces sized upwards of 50 mil-
+lion are required for tasks like image classification,
+the resources of current quantum hardware or simula-
+tors is not yet sufficient for pure quantum approaches
+(He et al., 2016). Therefore, hybrid approaches have
+been proposed, where the power of both classical and
+quantum computation are united for improved results
+(Bergholm et al., 2018). By this, it is possible to lever-
+age the advantages of quantum computing for tasks
+with parameter spaces that cannot be computed solely
+*accepted for publication at ICAART 2023
+by quantum computers due to hardware and simula-
+tion limitations. Within those hybrid algorithms the
+quantum part is, analogue to the classical deep neu-
+ral networks (DNNs), represented by so called vari-
+ational quantum circuits (VQCs), which are param-
+eterized and can be trained in a supervised manner
+using labeled data (Cerezo et al., 2021). For hybrid
+machine learning, we will from hereon refer to VQCs
+as quantum parts and to DNNs as classical parts.
+To solve large-scale real-world tasks, like image
+classification, the concept of transfer learning has
+been applied for training such hybrid models (Gir-
+shick et al., 2014; Pan and Yang, 2010). Given a com-
+plex model, with high-dimensional input- and param-
+eter spaces, the term transfer leaning classically refers
+to the two-step procedures of first pre-training using a
+large but generic dataset and secondly fine-tuning us-
+ing a smaller but more specific dataset (Torrey and
+Shavlik, 2010).
+Usually, a subset of the model’s
+weights are frozen for the fine-tuning to compensate
+for insufficient amounts of fine-tuning data.
+Applied to hybrid quantum machine learning
+(QML), the pre-trained model is used as a feature ex-
+arXiv:2301.02601v1  [quant-ph]  6 Jan 2023
+
+tractor and the dense classifier is replaced by a hybrid
+model referred to as dressed quantum circuit (DQC)
+including classical pre- and post-processing layers,
+and the central VQC (Mari et al., 2020). This archi-
+tecture results in concurrent updates to both classical
+and quantum weights. Even though, this produces up-
+dates towards overall optimal classification results, it
+does not allow for tracing the advantageousness of the
+quantum part of the architecture. Thus, besides pro-
+viding competitive classification results, such hybrid
+approaches do not allow for valid judgment whether
+the chosen quantum circuit benefits the classification.
+The only arguable result is that it does not harm the
+overall performance or that the introduced inaccura-
+cies may be compensated by the classical layers in the
+end. However, as we currently are still only exploring
+VQCs, this verdict, i.e. traceability of the impact of
+both the quantum and the classical part, is crucial to
+infer the architecture quality from common metrics.
+Overall, with current approaches we find a mismatch
+between the goal of exploring viable architectures and
+the process applied.
+We therefore propose the application of Sequen-
+tial Quantum Enhanced Training (SEQUENT), an
+adapted architecture and training procedure for hybrid
+quantum transfer learning, where the effect of both
+classical and quantum parts are separably assessable.
+We see our contributions as follows:
+• We provide formal evidence that current quantum
+transfer learning architectures might result in an
+optimal network configuration (perfect classifica-
+tion / regression results) with the least-most quan-
+tum impact, i.e., a solution equivalent to a purely
+classical one.
+• We propose SEQUENT, a two-step procedure of
+classical pre-training and quantum fine-tuning us-
+ing an adapted architecture to reduce the number
+of features classically extracted to the number of
+features manageable by the VQC producing the
+final classification.
+• We show competitive results with a traceable im-
+pact of the chosen VQC on the overall perfor-
+mance using preliminary benchmark datasets.
+2
+BACKGROUND
+To delimit SEQUENT, the following section pro-
+vides a brief general introduction to the related fields
+of quantum computation, quantum machine learning,
+deep learning and transfer learning.
+2.1
+Quantum Computing
+Quantum Computation
+works fundamentally dif-
+ferent than classical computation, since QC uses
+qubits instead of classical bits. Where classical bit
+can be in the state 0 or 1, the corresponding state of
+a qubit is described in Dirac notation as | 0⟩ and | 1⟩.
+However, more importantly, qubits can be in a super-
+position, i.e., a linear combination of both:
+| ψ⟩ = α | 0⟩+β | 1⟩
+(1)
+To alter this state, a set of reversible unitary op-
+erations like rotations can be applied sequentially to
+individual target qubits or in conjunction with a con-
+trol qubit. Upon measurement, the superposition col-
+lapses and the qubit takes on either the state | 0⟩ or
+| 1⟩ according to a probability. Note that α and β in
+(1) are complex numbers where | α |2 and | β |2 give
+the probability of measuring the qubit in state | 0⟩ or
+| 1⟩ respectively. Note that | α |2 + | β |2= 1, i.e., the
+probabilities sum up to 1. (Nielsen and Chuang, 2010)
+Quantum algorithms like Grover (Grover, 1996)
+or Shor (Shor, 1994) provide a theoretical speedup
+compared to classical algorithms. Moreover, in 2019
+quantum supremacy was claimed (Arute et al., 2019),
+and the race to find more algorithms providing a quan-
+tum advantage is currently underway. However, the
+current state of quantum computing is often referred
+to as the noisy-intermediate-scale quantum (NISQ)
+era (Preskill, 2018), a period when relatively small
+and noisy quantum computers are available, however,
+still no error-correction to mitigate them, limiting the
+execution to small quantum circuits.
+Furthermore,
+current quantum computers are not yet capable to ex-
+ecute algorithms that provide any quantum advantage
+in a practically useful setting.
+Thus, much research has recently been put into the
+investigation of hybrid-classical-quantum algorithms.
+That is, algorithms that consist of quantum and clas-
+sical parts, each responsible for a distinct task. In this
+regard, quantum machine learning has been gaining
+in popularity.
+Quantum
+Machine
+Learning
+algorithms
+have
+been proposed in several varieties over the last years
+(Farhi et al., 2014; Dong et al., 2008; Biamonte et al.,
+2017).
+Besides quantum kernel methods (Schuld
+and Killoran, 2019) variational quantum algorithms
+(VQAs) seem to be the most relevant in the current
+NISQ-era for various reasons (Cerezo et al., 2021).
+VQAs generally are comprised of multiple com-
+ponents, but the central part is the structure of the ap-
+plied circuit or Ansatz. Furthermore, a VQA Ansatz
+is intrinsically parameterized in order to use it as a
+
+predictive model by optimizing the parameterization
+towards a given objective, i.e. to minimize a given
+loss. Overall, given a set of data and targets, a param-
+eterized circuit and an objective, an approximation
+of the generator underlying the data can be learned.
+Applying methods like gradient descent, this model
+can be trained to predict the label of unseen data
+(Cerezo et al., 2021; Mitarai et al., 2018). For the
+field of QML, various circuit architectures have been
+proposed (Biamonte et al., 2017; Khairy et al., 2020;
+Schuld et al., 2020).
+For the remainder of this paper, we consider the
+following simple φ-parameterized variational quan-
+tum circuit (VQC) for η qubits:
+VQCφ(z) = meassureσ ◦entangleφδ ◦···◦
+◦entangleφ1 ◦embedη(z)
+(2)
+with the depth δ, and the output dimension σ given
+the input z = (z1,...,zη), where embedη loads the
+data-points z into η balanced qubits in superposi-
+tion via z-rotations, entangleφ applies controlled
+not gates to entangle neighboring qubits followed by
+φ-parameterized z rotations, and measureσ applies
+the Pauli-Z operator and measures the first σ qubits
+(Schuld and Killoran, 2019; Mitarai et al., 2018).
+This architecture has also been shown to be di-
+rectly applicable to classification tasks, using the
+measurement expectation value as a one-hot encoded
+prediction of the target (Schuld et al., 2020).
+Overall, VQAs have been shown to be applica-
+ble to a wide variety of classification tasks (Abo-
+hashima et al., 2020) and successfully utilized by
+Mari et al. (2020), using the simple architecture de-
+fined in (2). Thus, to provide a proof-of-concept for
+SEQUENT, we will focus on said architecture for
+classification tasks and leave the optimization of em-
+beddings (LaRose and Coyle, 2020) and architectures
+(Khairy et al., 2020) to future research.
+2.2
+Deep Learning
+Deep Neural Networks (DNNs)
+refer to parame-
+terized networks consisting of a set of fully-connected
+layers.
+A layer comprises a set of distinct neu-
+rons, whereas each neuron takes a vector of inputs
+x = (x1,x2,...xn), which is multiplied with the cor-
+responding weight vector w j = (w j1,w j2,...w jn). A
+bias b j is added before being passed into an activa-
+tion function ϕ. Therefore, the output of neuron z
+at position j takes the following form (Bishop and
+Nasrabadi, 2006):
+z j = ϕ
+�
+n
+∑
+i=1
+w jixi +bj
+�
+(3)
+Given a target function f(x) : X �→ y, we can de-
+fine the approximate
+ˆfθ(x) : X �→ ˆy = Lhd→o ◦···◦Ln→h1
+(4)
+as a composition of multiple layers L with multiple
+neurons z parameterized by θ, d − 1 h-dimensional
+hidden layers, and the respective input and target di-
+mensions n and o. Using the prediction error J =
+(y − ˆfθ(x))2, ˆfθ can be optimized by propagating the
+error backwards through the network using the gradi-
+ent ∇θJ (Bishop and Nasrabadi, 2006).
+Those feed forward models have been shown ca-
+pable of approximating arbitrary functions, given a
+sufficient amount of data and either a sufficient depth
+(i.e. number of hidden layers) or width (i.e. size of
+hidden state) (Leshno et al., 1993).
+Deep neural networks for image classification
+tasks are comprised of two parts: A feature extrac-
+tor containing a composite of convolutional layers to
+extract a υ-sized vector of features FE : X �→ υ, and
+a composite of fully connected layers to classify the
+extracted feature vector FC : υ �→ ˆy. Thus, the overall
+model is defined as ˆf : X �→ ˆy = FCθ ◦FEθ(x). Those
+models have been successfully applied to a wide vari-
+ety of real-world classification tasks (He et al., 2016;
+Krizhevsky et al., 2012). However, to find a parame-
+terization that optimally separates the given dataset, a
+large amount of training data is required.
+Transfer Learning
+aims to solve the problem of in-
+sufficient training data by transferring already learned
+knowledge (weights, biases) from a task Ts of a source
+domain Ds to a related target task Tt of a target do-
+main Dt. More specifically, a domain D = X,P(x)
+comprises a feature space X and the probability dis-
+tribution P(x) where x = (x1,x2,...,xn) ∈ X. The cor-
+responding task T is given by T = {y, f(x)} with la-
+bel space y and target function f(x) (Zhuang et al.,
+2021). A deep transfer learning task is defined by
+⟨Ds,Ts,Dt,Tt, ˆft(·)⟩, where ˆft(·) is defined according
+to Equation 4 (Tan et al., 2018).
+Generally, transfer learning is a two-stage process.
+Initially, a source model is trained according to a spe-
+cific task Ts in the source domain Ds. Consequently,
+transfer learning aims to enhance the performance of
+the target predictive function ˆft(·) for the target learn-
+ing task Tt in target domain Dt by transferring la-
+tent knowledge from Ts in Ds, where Ds ̸= Dt and/or
+Ts ̸= Tt. Usually, the size of Ds >> Dt (Tan et al.,
+2018). The knowledge transfer and learning step is
+commonly achieved via feature extraction and/or fine-
+tuning.
+
+The feature extraction process freezes the source
+model and adds a new classifier to the output of the
+pre-trained model. Thereby, the feature maps learned
+from Ts in Ds can be repurposed and the newly-added
+classifier is trained according to the target task Tt
+(Donahue et al., 2014). The fine-tuning process ad-
+ditionally unfreezes top layers from the source model
+and jointly trains the unfreezed feature representa-
+tions from the source model with the added classifier.
+By this, the time and space complexity for the tar-
+get task Tt can be reduced by transferring and/or fine-
+tuning the already learned features of a pre-trained
+source model to a target model (Girshick et al., 2014).
+3
+RELATED WORK
+In the context of machine learning, VQAs are of-
+ten applied to the problem of classification (Schuld
+et al., 2020; Mitarai et al., 2018; Havl´ıˇcek et al., 2019;
+Schuld and Killoran, 2019), although other applica-
+tion areas exist. Different techniques, e.g. embed-
+ding (Lloyd et al., 2020; LaRose and Coyle, 2020),
+or problems, e.g. barren plateaus (McClean et al.,
+2018), have been widely discussed in the QML liter-
+ature. However, we focus on hybrid quantum transfer
+learning (Mari et al., 2020) in this paper.
+Classical Transfer Learning is widely applied in
+present-day machine learning algorithms (Torrey and
+Shavlik, 2010; Pan and Yang, 2010; Pratt, 1992) and
+can be extended with concepts of the emerging quan-
+tum computing technology (Zen et al., 2020). Mari
+et al. (2020) propose various hybrid transfer learning
+architectures ranging from classical to quantum (CQ),
+quantum to classical (QC) and quantum to quantum
+(QQ). The authors focus on the former CQ architec-
+ture, which which comprises the previously explained
+DQC. In the current era of intermediate-scale quan-
+tum technology the DQC transfer learning approach
+is the most widely investigated and applied one, as
+it allows to some extend optimally pre-process high-
+dimensional data and afterwards load the most rele-
+vant features into a quantum computer. Gokhale et al.
+(2020) used this architecture to classify and detect im-
+age splicing forgeries, while Acar and Yilmaz (2021)
+applied it to detect COVID-19 from CT images. Also,
+Mari et al. (2020) assess their approach exemplary on
+image classification tasks. Although the results are
+quite promising it is not clear from the evaluation,
+whether the dressed quantum circuit is advantageous
+over a fully classical approach.
+4
+DQC QUANTUM IMPACT
+We argue that within certain problem instance DQCs
+may yield accurate results while not making active
+use of any quantum effects in the VQC. This possi-
+bility exists especially for easy to solve problem in-
+stances, when all purely classical layers are sufficient
+to yield accurate results and the quantum layer rep-
+resents the identity.
+This can be seen by realizing
+that the classical pre-processing layer acts as a hid-
+den layer with a non-polynomial activation function,
+hence being capable of approximating arbitrary con-
+tinuous functions depending on the number of hidden
+units by the universal approximation theorem (Leshno
+et al., 1993). Therefore, the overall DQC architecture
+is portrayed in Figure 1.
+The central VQC is defined according to sec-
+tion 2.1 as introduced above.
+Both pre- and post-
+processing layers are implemented by fully connected
+layers of neurons with a non-linear activation function
+according to subsection 2.2. Formally, the DQC for η
+qubits can thus be depicted as:
+DQC = Lη→σ ◦VQCφ ◦Ln→η
+(5)
+where Ln→η and Lη→σ are the fully connected clas-
+sical dressing layers according to Equation 3, map-
+ping from the input size n to the number of qubits η
+and from the number of qubits η to the target size σ
+respectively, and VQCφ is the actual variational quan-
+tum circuit according to Equation 2 with η qubits and
+σ = η measured outputs.
+Now let us consider a parameterization φ, where
+VQCφ(z) = id(z) = z resembles the identity function.
+Consequently (5) collapses into the following purely
+classical, 2-layer feed-forward network with the hid-
+den dimension η:
+DQC = Lη→σ ◦id ◦Ln→η = Lη→σ ◦Ln→η
+(6)
+By the universal function approximation theorem,
+this allows DQC to approximate any polynomial func-
+tion f : Rn → Ro of degree 1 arbitrarily well, even if
+the VQC is not affecting the prediction at all.
+�
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+�
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+��������
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+������
+������
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+Pre-
+processing
+Post-
+processing
+������
+Figure 1: Dressed Quantum Circuit Architecture
+
+Consequently, one has to be careful in the selec-
+tion of suitable problem instances, as they must not
+be too easy in order to ensure that the VQC is even
+needed to yield the desired results. This becomes es-
+pecially difficult as current quantum hardware is quite
+limited, typically restricting the choice to fairly easy
+problem instances. On top of this, no necessity to use
+a post-processing layer seems apparent, as it has been
+shown in various publications (Schuld et al., 2020;
+Schuld and Killoran, 2019) that variational quantum
+classifiers, i.e, VQCs can successfully complete clas-
+sification tasks without any post-processing. Overall,
+whilst conveying a proof-of-concept, that the com-
+bination of classical neural networks and variational
+quantum circuits in the dressed quantum circuit hy-
+brid architecture is able to produce competitive re-
+sults, this architecture is neither able to convey the
+advantageousness of the chosen quantum circuit nor
+exclude the possibility of the classical part just being
+able to compensate for quantum in-steadiness.
+5
+SEQUENT
+To improve the traceability of quantum impact in hy-
+brid architectures, we propose Sequential Quantum
+Enhanced Training. SEQUENT improves upon the
+dressed quantum circuit architecture by introducing
+two adaptations to it: First, we omit the classical
+post-processing layer and use the variational quan-
+tum circuit output directly as the classification result.
+Therefore we reduce the measured outputs σ from the
+number of qubits η (cf. Figure 1) to the dimension of
+the target ˆy (cf. Figure 2).
+The direct use of VQCs as a classifier has been
+frequently proposed and shown equally applicable as
+classical counterparts (Schuld et al., 2020). By this,
+the overall quality of the chosen circuit and parame-
+terization are directly assessable by the classification
+result, thus the final accuracy. Moreover, a parame-
+ter setting of universal approximation capabilities (cf.
+Equation 6) with the least (identitary) quantum con-
+tribution is mathematically precluded by the removal
+of the hidden state (compare Equation 5).
+Concurrently omitting the pre-processing or com-
+pression layer however would increase the number of
+at least required qubits to the number of output fea-
+tures of the problem domain, or, when applied to im-
+age classification, the chosen feature extractor (e.g.
+512 for Resnet-18). However, both current quantum
+hardware and simulators do not allow for arbitrate
+sized circuits, especially maxing out at around 100
+qubits.
+������
+�
+�
+�
+�
+�
+��������
+��������
+��������
+������
+������
+������
+��������
+��������
+��������
+���������
+������������
+������������
+������
+Figure 2: SEQUENT Architecture: Sequential Quantum
+Enhanced Training comprised of a classical compression
+layer (CCL) parameterized by θ and a variational quantum
+circuit (VQC) parameterized by φ with separate phases for
+classical (blue) and quantum (green) training for variable
+sets of input data X, prediction targets ˆy and VQCs with η
+qubits and δ entangling layers.
+We therefore secondly propose to maintain the
+classical compression layer to provide a map-
+ping/compression X �→ η and, in order to fully clas-
+sically pre-train the compression layer, add a surro-
+gate classical classification layer η �→ ˆy. Replacing
+this surrogate classical classification layer with the
+chosen variational quantum circuit to be assessed and
+freezing the pre-trained weights of the classical com-
+pression layer then allows for a second, purely quan-
+tum training phase and yield the following sequential
+training procedure depicted in Figure 3:
+1. Pre-train SEQUENT: ˆf : X �→ η �→ ˆy = CCLθ(x)◦
+CCLθ(z) containing a classical compression layer
+and a surrogate classification layer by optimizing
+the classical weights θ
+2. Freeze the classical weights θ, replace the sur-
+rogate classical classification layer by the vari-
+ational quantum classification circutit VQCφ(cf.
+Equation 2) and optimize the quantum weights φ.
+This two-step procedure can be seen as an applica-
+tion of transfer learning on its own, transferring from
+classical to quantum weights in a hybrid architecture.
+Overall, the SEQUENT architecture displayed in
+Figure 2 can be formalized as:
+SEQUENTθ,φ : X �→ η �→ ˆy = VQCφ(z)◦CCLθ(x)
+(7)
+CCLθ(x) : X �→ η = Ln→η
+(cf. Equation 3)
+VQCφ(z) : η �→ ˆy
+(cf. Equation 2)
+�������������������
+�������������������
+�����������������
+�������������������
+�����������������
+�����������������������������
+���������������������������
+�������������������
+���
+���
+Figure 3:
+SEQUENT Training Process consisting of
+a classical (blue) pre-training phase (1) and a quantum
+(green) fine-tuning phase (2).
+
+To be used for the classification of high-
+dimensional data, like images, the input x needs to
+be replaced by the intermediate output of an image
+recognition model z (cf. subsection 2.2). Combining
+both two-step transfer learning procedures, the fol-
+lowing three-step procedure is yielded:
+1. Classically pre-train a full classification model
+(e.g. Resnet (He et al., 2016)) ˆf : X �→ υ �→ ˆy =
+FCθ(z) ◦ FEθ(x) to a large generic dataset (com-
+pare subsection 2.2)
+2. Freeze convolutional feature extraction layers FE
+and fine-tune fully-connected layers consisting of
+a compression layer and a surrogate classification
+layer FE : υ �→ η �→ ˆy = CCLθ(z)◦CCLθ(x).
+3. Freeze classical weights and replace surrogate
+classification layer with VQC to train the quan-
+tum weights φ of the hybrid model:
+ˆfθ,φ : X �→ υ �→ η �→ ˆy = VQCφ(z)◦CCLθ(x)◦FE
+For a classification task with n classes, at least η ≥ n
+qubits are required. Whilst we use the simple Ansatz
+introduced in Equation 2 with η = 6 qubits and a
+circuit depth of δ = 10 to validate our approach in
+the following, any VQC architecture yielding a direct
+classification result would be conceivable.
+6
+EVALUATION
+We evaluate SEQUENT by comparing its perfor-
+mance to its predecessor, the DQC, and a purely clas-
+sical feed forward neural network. All models were
+trained on 2000 datapoints of the moons and spirals
+(Lang and Witbrock, 1988) benchmark dataset for
+two and four epochs of sequential, hybrid and clas-
+sical training respectively. To guarantee comparabil-
+ity, we set the size of the hidden state of the classical
+model to h = η = 6. The code for all experiments
+is available here1. The classification results are vi-
+sualized in Figure 4. Looking at the result for the
+moons dataset, all compared models are able to de-
+pict the shape underlying data. Note, that even the
+considerably simpler classical model is perfectly able
+to separate the given classes. Hence, these experi-
+mental results support the concerns about the impact
+of the VQC to the overall DQC’s performance (cf.
+section 4).
+With a final test accuracy of 95%, the
+DQC performs even worse than the purely classical
+model reaching 96%. Looking at the SEQUENT re-
+sults however, these concerns are eliminated, as the
+performance and final accuracy of 97%, besides out-
+performing both compared models, can certainly be
+1https://github.com/philippaltmann/SEQUENT
+Figure 4: Classification Results of SEQUENT, DQC and
+Classical Feed Forward Neural Network for moons (left)
+and spirals (right) benchmark datasets
+denoted to VQC, due to the applied training process
+and the used architecture. Similar results show for the
+second benchmark dataset of intertwined spirals on
+the right side of Figure 4. The overall best accuracy
+of 86% however suggests, that further adjustments to
+the VQC could be beneficial. This result also depicts
+the application of SEQUENT we imagine for bench-
+marking and optimizing VQC architectures.
+7
+CONCLUSIONS
+We proposed Sequential Quantum Enhanced Train-
+ing (SEQUENT), a two-step transfer learning proce-
+dure applied to training hybrid QML algorithms com-
+bined with an adapted hybrid architecture to allow
+for tracing both the classical and quantum impact on
+the overall performance.
+Furthermore, we showed
+the need for said adaptions by formally pointing out
+weaknesses of the DQC, the current state-of-the-art
+approach to this regard. Finally, we showed that SE-
+QUENT yields competitive results for two representa-
+
+SEQUENT
+SEQUENT
+.97
+.86
+DQC
+DQC
+.95
+.81
+Classical
+Classical
+.96
+.79tive benchmark datasets compared to DQCs and clas-
+sical neural networks. Thus, we a provided proof-
+of-concept for both the proposed reduced architecture
+and the adapted transfer learning training procedure.
+However, whilst SEQUENT theoretically is appli-
+cable to any kind of VQC, we only considered the
+simple architecture with fixed angle embeddings and
+δ entangling layers as proposed by (Mari et al., 2020).
+Furthermore, we only supplied preliminary experi-
+mental implications and did not yet test any high di-
+mensional real-world applications. Overall, we do not
+expect superior results that outperform state-of-the-
+art approaches in the first place, as viable circuit ar-
+chitectures for quantum machine learning are still an
+active and fast-moving field of research.
+Thus, both the real world applicability and the de-
+velopment of circuit architectures that indeed offer
+a benefit over classical ones should undergo further
+research attention. To empower real-world applica-
+tions, the use of hybrid quantum methods should also
+be kept in mind when pre-training large classification
+models like Resnet. Also, applying more advanced
+techniques to train the pre-processing or compression
+layer to take full advantage of the chosen quantum
+circuit should be examined. Therefore, auto-encoder
+architectures might be applicable to train a more gen-
+eralized mapping from the classical input-space to
+the quantum-space. Overall, we belief, that applying
+the proposed concepts and building upon SEQUENT,
+both valuable hybrid applications and beneficial quan-
+tum circuit architectures can be found.
+ACKNOWLEDGEMENTS
+This work is part of the Munich Quantum Valley,
+which is supported by the Bavarian state govern-
+ment with funds from the Hightech Agenda Bayern
+Plus and was partially funded by the German BMWK
+Project PlanQK (01MK20005I).
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+

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+An analytical approach to Bayesian evidence computation
+Juan Garc´ıa-Bellido
+Departamento de F´ısica Te´orica C-XI, Universidad Aut´onoma de Madrid,
+Cantoblanco, 28049 Madrid, Spain
+April 14th, 2005
+Abstract
+The Bayesian evidence is a key tool in model selection, allowing a comparison of models with differ-
+ent numbers of parameters. Its use in analysis of cosmological models has been limited by difficulties
+in calculating it, with current numerical algorithms requiring supercomputers. In this paper we give
+exact formulae for the Bayesian evidence in the case of Gaussian likelihoods with arbitrary correlations
+and top-hat priors, and approximate formulae for the case of likelihood distributions with leading non-
+Gaussianities (skewness and kurtosis). We apply these formulae to cosmological models with and without
+isocurvature components, and compare with results we previously obtained using numerical thermody-
+namic integration. We find that the results are of lower precision than the thermodynamic integration,
+while still being good enough to be useful.
+1
+Introduction
+Model selection refers to the statistical problem of deciding which model description of observational data
+is the best [1, 2]. It differs from parameter estimation, where the choice of a single model (i.e. choice of
+parameters to be varied) has already been made and the aim is to find their best-fitting values and ranges.
+While there have been widespread applications of parameter estimation techniques, usually likelihood fitting,
+to cosmological data, there has so far been quite limited application of model selection statistics [3, 4, 5].
+This is unfortunate, as model selection techniques are necessary to robustly distinguish between models
+with different numbers of parameters, and many of the most interesting issues in cosmology concern the
+desirability or otherwise of incorporating additional parameters to describe new physical effects.
+Within the context of Bayesian inference, model selection should be carried out using the Bayesian
+evidence [1, 2], which measures the probability of the model in light of the observational data (i.e. the
+average likelihood over the prior distribution). The Bayesian evidence associates a single number with each
+model, and the models can then be ranked in order of the evidence, with the ratios of those values interpretted
+as the relative probability of the models. This process sets up a desirable tension between model simplicity
+and ability to fit the data.
+Use of the Bayesian evidence has so far been limited by difficulties in calculating it.
+The standard
+technique is thermodynamic integration [6, 7], which varies the temperature in a Monte Carlo Markov Chain
+(MCMC) approach in order that the distribution is sampled in a way covering both posterior and prior
+distributions. However, in recent work [5] we showed that in order to obtain sufficiently-accurate results
+in a cosmological context, around 107 likelihood evaluations are required per model.
+Such analyses are
+CPU-limited by the time needed to generate the predicted spectra to compare with the data, and this
+requirement pushes the problem into the supercomputer class (for comparison, parameter estimation runs
+typically employ 105 to 106 likelihood evaluations).
+In this paper, we propose and exploit a new analytic method to compute the evidence based on an
+expansion of the likelihood distribution function.
+The method pre-supposes that the covariance of the
+posterior distribution has been obtained, for instance via an MCMC parameter estimation run, and in its
+1
+arXiv:2301.13783v1  [astro-ph.CO]  31 Jan 2023
+
+present form requires that the prior distributions of the parameters are uniform top-hat priors.1 While the
+method will not be applicable for general likelihood distributions, we include the leading non-gaussianities
+(skewness and kurtosis) in approximating the likelihood shape, with the expectation of obtaining good
+results whenever the likelihood distribution is sufficiently simple. Cosmological examples commonly exhibit
+likelihood distributions with only a single significant peak.
+We apply the method both to toy model examples and to genuine cosmological situations. In particular,
+we calculate the evidences for adiabatic and isocurvature models, which we previously computed using
+thermodynamic integration in Ref. [5]. We find that the discrepancies between the methods are typically no
+worse than 1 in ln(Evidence), meaning that the analytic method is somewhat less accurate than would be
+ideal, but is accurate enough to give a useful indication of model preference.
+2
+The Bayesian evidence
+The posterior probability distribution P(θ, M|D) for the parameters θ of the model M, given the data D,
+is related to the likelihood function L(D|θ, M) within a given set of prior distribution functions π(θ, M) for
+the parameters of the model, by Bayes’ theorem:
+P(θ, M|D) = L(D|θ, M) π(θ, M)
+E(D|M)
+,
+(1)
+where E is the Bayesian evidence, i.e. the average likelihood over the priors,
+E(D|M) =
+�
+dθ L(D|θ, M) π(θ, M) ,
+(2)
+where θ is a vector with n-components characterising the n independent parameters. The prior distribution
+function π contains all the information about the parameters before observing the data, i.e. our theoretical
+prejudices, our physical understanding of the model, and input from previous experiments.
+In the case of a large number of parameters (n ≫ 1), the evidence integral cannot be performed straight-
+forwardly and must be obtained either numerically or via an analytic approximation. Amongst numerical
+methods the most popular is thermodynamic integration [6, 7] but this can be computationally extremely
+intensive [5]. The simplest analytical approximation is the Laplace approximation, valid when the distribu-
+tion can be approximated by a multivariate Gaussian. This may hold when the quantity and quality of the
+data is optimal, but is likely to be valid only in limited cosmological circumstances.
+The Bayesian evidence is of interest because it allows a comparison of models amongst an exclusive and
+exhaustive set {Mi}i=1...N. We can compute the posterior probability for each hypothesis given the data D
+using Bayes theorem:
+P(Mi|D) ∝ E(D|Mi) π(Mi) ,
+(3)
+where E(D|Mi) is the evidence of the data under the model Mi, and π(Mi) is the prior probability of the
+ith model before we see the data. The ratio of the evidences for the two competing models is called the
+Bayes factor [8]
+Bij = E(D|Mi)
+E(D|Mj) ,
+(4)
+and this is also equal to the ratio of the posterior model probabilities if we assume that we do not favour
+any model a priori, so that π(M1) = π(M2) = ... = π(MN) = 1/N.
+The Bayes factor Eq. (4) provides a mathematical representation of Occam’s razor, because more complex
+models tend to be less predictive, lowering their average likelihood in comparison to simpler, more predictive
+models. More complex models can only be favoured if they are able to provide a significantly improved fit to
+the data. In simple cases where models give vastly different maximum likelihoods there is no need to employ
+model selection techniques, but they are essential for properly discussing cases where the improvement
+1An extension to gaussian priors should be feasible, but not one to arbitrary priors.
+2
+
+of fit is marginal. This latter situation is more or less inevitable whenever the possibility of requiring an
+additional parameter arises from new data, unless the new data is of vastly greater power than that preceding
+it; cosmological examples include the inclusion of spectral tilt, dark energy density variation, or the case
+explored later in this paper of trace isocurvature perturbations.
+In this paper we will obtain an analytical formula which approximates the Bayesian evidence by consid-
+ering the higher-order cumulants of the distribution in a systematic way. The advantage is that with these
+analytical formulae one can compute the evidence for a given model with an arbitrary number of parame-
+ters, given the hierarchy of cumulants of the distribution, assumed previously computed for the likelihood
+distribution function within the parameter estimation programme.
+The evidence needs to be calculated to sufficient precision for robust conclusions to be drawn.
+The
+standard interpretational scale, due to Jeffreys [1] and summarized in Ref. [5], strengthens its verdict roughly
+each time the difference in ln(Evidence) increases by one. The evidence therefore needs to be computed more
+accurately than this, with an uncertainty of 0.1 in ln(Evidence) easily sufficient, and a factor two worse than
+that acceptable. This accuracy requirement ensures that the relative model probabilities are little changed
+by the uncertainty.
+The first thing we need is to characterize the distribution function for the model with n parameters. Let
+f(x) be this function, and let us assume that it is properly normalized,
+� ∞
+−∞
+dnx f(x) = 1 .
+(5)
+Then, the p-point correlation function is given by
+⟨xi1 . . . xip⟩ =
+� ∞
+−∞
+dnx xi1 . . . xip f(x) .
+(6)
+From this distribution function one can always construct the generating functional, φ(u), as the Fourier
+transform
+φ(u) =
+� ∞
+−∞
+dnx ei u·x f(x) .
+(7)
+This function can be expanded as
+φ(u) = exp
+� ∞
+�
+p=1
+ip
+p! Ai1...ip ui1 . . . uip
+�
+,
+(8)
+where Ai1...ip are totally symmetric rank-p tensors. For instance, if we restrict ourselves to order 4, we can
+write
+φ(u) = exp
+�
+i µiui − 1
+2! Cij uiuj − i
+3! Bijk uiujuk + 1
+4! Dijkl uiujukul + · · · + in
+n! Ai1...in ui1 . . . uin
+�
+,
+(9)
+where µi is the mean value of variable xi; Cij is the covariance matrix; Bijk is the trilinear matrix associated
+with the third cumulant or skewness; Dijkl is the rank-4 tensor associated with the fourth cumulant or
+kurtosis, and Ai1...in is the rank-n tensor associated with the n-th cumulant. Their expressions in terms of
+n-point correlation functions can be obtained from Eq. (7), by realising that
+⟨xi1 . . . xin⟩ = (−i)n
+∂n��(u)
+∂ui1 . . . ∂uin
+����
+u=0
+.
+(10)
+For instance, the first-order term gives
+⟨xi⟩ = (−i) ∂φ(u)
+∂ui
+����
+u=0
+= µi .
+(11)
+3
+
+The second-order correlation function gives
+⟨xixj⟩ = (−i)2 ∂2φ(u)
+∂ui∂uj
+����
+u=0
+= Cij + µiµj ,
+(12)
+such that the covariance matrix is obtained, as usual, from
+Cij = ⟨xixj⟩ − ⟨xi⟩⟨xj⟩ .
+The third-order correlation function gives
+⟨xixjxk⟩ = (−i)3
+∂3φ(u)
+∂ui∂uj∂uk
+����
+u=0
+= Bijk + µiCjk + µjCki + µkCij + µiµjµk ,
+(13)
+such that the skewness matrix is obtained from
+Bijk = ⟨xixjxk⟩ − ⟨xi⟩⟨xjxk⟩ − ⟨xj⟩⟨xkxi⟩ − ⟨xk⟩⟨xixj⟩ + 2⟨xi⟩⟨xj⟩⟨xk⟩ .
+(14)
+The fourth-order correlation function gives
+⟨xixjxkxl⟩ = (−i)4
+∂4φ(u)
+∂ui∂uj∂uk∂ul
+����
+u=0
+=
+Dijkl + CijCkl + CikCjl + CilCjk
+(15)
++
+Bijkµl + Bijlµk + Bjklµi + Biklµj
++
+Cijµkµl + Cikµjµl + Cilµjµk
++
+Cjkµiµl + Cjlµiµk + Cklµiµj
++
+µiµjµkµl ,
+such that the kurtosis matrix is obtained from
+Dijkl
+=
+⟨xixjxkxl⟩ − ⟨xixj⟩⟨xkxl⟩ − ⟨xixk⟩⟨xjxl⟩ − ⟨xixl⟩⟨xjxk⟩
+(16)
+−
+⟨xixjxk⟩⟨xl⟩ − ⟨xixjxl⟩⟨xk⟩ − ⟨xixkxl⟩⟨xj⟩ − ⟨xjxkxl⟩⟨xi⟩
++
+2 ⟨xixj⟩⟨xk⟩⟨xl⟩ + 2 ⟨xixk⟩⟨xj⟩⟨xl⟩ + 2 ⟨xixl⟩⟨xj⟩⟨xk⟩ + 2 ⟨xjxk⟩⟨xi⟩⟨xl⟩
++
+2 ⟨xjxl⟩⟨xi⟩⟨xk⟩ + 2 ⟨xkxl⟩⟨xi⟩⟨xj⟩ − 6 ⟨xi⟩⟨xj⟩⟨xk⟩⟨xl⟩ ,
+and so on, for the higher order cumulants.
+3
+The Gaussian approximation
+Let us first evaluate the evidence for a multivariate Gaussian distribution, that is, one in which all the
+cumulants are zero except the covariance matrix Cij and the means µi. In this case, the generating functional
+and the distribution are given by
+φ(u) = exp
+�
+− iµiui − 1
+2 Cij uiuj
+�
+,
+(17)
+f(x) =
+1
+(2π)n
+� ∞
+−∞
+dnu e−i u·x φ(u)
+(18)
+=
+1
+(2π)n/2√
+det C
+exp
+�
+− 1
+2C−1
+ij (xi − µi)(xj − µj)
+�
+,
+(19)
+4
+
+which satisfies
+⟨xi⟩ = µi ,
+⟨xixj⟩ = Cij + µiµj ,
+⟨xixjxk⟩ = µ(iCjk) + µiµjµk ,
+. . .
+(20)
+where the subindices in parenthesis, (ijk), indicate a cyclic sum. Notice that all the n-point correlation
+functions can be written in terms of the first two moments of the distribution, and all the higher-order
+cumulants vanish.
+3.1
+Centred priors
+For initial calculations, we assume a top-hat prior and make the unrealistic assumption, to be lifted later,
+that it is centered at the mean value:
+π(x, a) ≡
+�
+(2a)−1
+−a < x − µ < a ,
+0
+otherwise .
+(21)
+Since the Fourier transform of a top-hat function is
+� ∞
+−∞
+dx eiux π(x, a) = sin au
+au
+exp[iµu] ,
+we can write the evidence either way
+E(a1, . . . , an)
+=
+� ∞
+−∞
+dnx f(x)
+n
+�
+i=1
+π(xi, ai) =
+n
+�
+i=1
+(2ai)−1
+� a1
+−a1
+dx1· · ·
+� an
+−an
+dxn f(˜x)
+(22)
+=
+1
+(2π)n
+� ∞
+−∞
+dnu φ(u)
+n
+�
+i=1
+sin aiui
+aiui
+.
+(23)
+In Eq. (22) we integrate over the displaced coordinate, ˜xi ≡ xi − µi, such that ⟨˜xi⟩ = 0 and ⟨˜xi˜xj⟩ = Cij.
+From now on, we ignore the tildes, and assume we have moved to those coordinates. Note that the choice
+of prior is not crucial. We could have chosen a Gaussian prior, and the result would not be very different,
+except that the window functions, sin z/z, would then be Gaussians. Let us now perform the integration
+Eq. (22) in the case of 1, 2 and then n variables.
+1 variable. Suppose the covariance is just C = σ2. The evidence is then
+E(a) =
+1
+2a σ
+√
+2π
+� a
+−a
+dx e− x2
+2σ2 = 1
+2π
+� ∞
+−∞
+du sin au
+au
+e− 1
+2 σ2u2 = 1
+2aErf
+�
+a
+σ
+√
+2
+�
+,
+(24)
+where Erf[x] is the error function, which asymptotes very quickly to one for x ≥ 2, or a ≥ 3σ. Therefore,
+the evidence of a model with centred top-hat prior of width 2a is well approximated by (2a)−1. The wider
+is the theoretical prior, the smaller is the evidence, as expected.
+2 variables. Suppose we have two correlated variables, x1 and x2, with covariance matrix
+C =
+�
+C11
+C12
+C12
+C22
+�
+=
+�
+σ2
+1
+ρσ1σ2
+ρσ1σ2
+σ2
+2
+�
+.
+(25)
+where the cross-correlation ρ is defined by
+ρ =
+⟨x1x2⟩
+�
+⟨x2
+1⟩⟨x2
+2⟩
+= ⟨x1x2⟩
+σ1σ2
+,
+5
+
+with σ1 and σ2 the corresponding quadratic dispersions. In this case, the normalized 2-dimensional distri-
+bution function is
+f(x) =
+1
+2πσ1σ2
+�
+1 − ρ2 exp
+�
+−1
+1 − ρ2
+� x2
+1
+2σ2
+1
+− ρx1x2
+σ1σ2
++ x2
+2
+2σ2
+2
+��
+,
+(26)
+which has the property that integrating (“marginalizing”) over one of the two variables, leaves a properly-
+normalized Gaussian distribution for the remaining variable,
+� ∞
+−∞
+dx2 f(x) =
+1
+σ1
+√
+2π e
+−
+x2
+1
+2σ2
+1 .
+(27)
+Let us now evaluate the evidence Eq. (22) by integrating first over the prior in x2,
+1
+2a2
+� a2
+−a2
+dx2 f(x) = e
+−
+x2
+1
+2σ2
+1
+σ1
+√
+2π ·
+1
+4a2
+�
+Erf
+� a2σ1 + ρσ2 x1
+σ1σ2
+�
+2(1 − ρ2)
+�
++ Erf
+� a2σ1 − ρσ2 x1
+σ1σ2
+�
+2(1 − ρ2)
+��
+.
+(28)
+The first term is the result we would have obtained if we had been marginalizing over x2; the second is a
+sum of error functions that still depend on x1, and modulates the marginalization. We can use the series
+expansion of the error function to second order,
+1
+2
+�
+Erf[a + x] + Erf[a − x]
+�
+= Erf[a] − 2a x2
+√π e−a2 + O(x4) ,
+to write Eq. (28) to order x2
+1 as
+1
+2a2
+� a2
+−a2
+dx2 f(x) = e
+−
+x2
+1
+2σ2
+1
+σ1
+√
+2π
+�
+�� 1
+2a2
+Erf
+�
+a2
+σ2
+�
+2(1 − ρ2)
+�
+−
+ρ2 x2
+1 e
+−
+a2
+2
+2σ2
+2(1−ρ2)
+2σ2
+1σ2(1 − ρ2)
+�
+2π(1 − ρ2)
+�
+�� .
+(29)
+Integrating now over the x1 prior, we finally obtain the evidence
+E(a1, a2)
+=
+1
+4a1a2
+� a1
+−a1
+dx1
+� a2
+−a2
+dx2 f(x)
+=
+1
+4a1a2
+Erf
+�
+a2
+σ2
+�
+2(1 − ρ2)
+�
+Erf
+�
+a1
+σ1
+√
+2
+�
+(30)
+−
+ρ2 e
+−
+a2
+2
+2σ2
+2(1−ρ2)
+2σ1σ2(1 − ρ2)
+�
+2π(1 − ρ2)
+Erf
+�
+a1
+σ1
+√
+2
+�
+2a1
++ ρ2 e
+−
+a2
+2
+2σ2
+2(1−ρ2) −
+a2
+1
+2σ2
+1
+4πσ2
+1σ2
+�
+1 − ρ2
+.
+Note that in the limit of no cross-correlations, ρ → 0, the integral factorizes and we can write an exact
+expression for the evidence,
+E(a1, a2)
+=
+1
+4a1a2
+1
+2πσ1σ2
+� a1
+−a1
+dx1
+� a2
+−a2
+dx2 e
+−
+x2
+1
+2σ2
+1
+−
+x2
+2
+2σ2
+2
+(31)
+=
+1
+4π2
+� ∞
+−∞
+du1
+� ∞
+−∞
+du2
+sin a1u1
+a1u1
+sin a2u2
+a2u2
+e− 1
+2 σ2
+1u2
+1− 1
+2 σ2
+2u2
+2
+(32)
+=
+1
+4a1a2
+Erf
+�
+a1
+σ1
+√
+2
+�
+Erf
+�
+a2
+σ2
+√
+2
+�
+.
+(33)
+6
+
+It happens, however, that even in the presence of cross-correlations, if the prior is wide (ai ≥ 2σi), then the
+terms proportional to exponentials are negligible and the evidence becomes, to very good approximation,
+E(a1, a2) =
+1
+4a1a2
+Erf
+�
+a2
+σ2
+�
+2(1 − ρ2)
+�
+Erf
+�
+a1
+σ1
+√
+2
+�
+.
+(34)
+Moreover, in that case, the error functions are very approximately given by 1.
+n variables. Suppose we have n correlated variables, x = (x1, . . . , xn), with covariance matrix
+Cn =
+�
+�
+�
+�
+�
+�
+�
+�
+C11
+C12
+. . .
+C1n
+C12
+C22
+. . .
+C2n
+...
+...
+...
+...
+C1n
+C2n
+. . .
+Cnn
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(35)
+In that case, the probability distribution function can be expressed as
+f(x) =
+1
+(2π)n/2√det Cn
+exp
+�
+− 1
+2xT C−1
+n x
+�
+,
+(36)
+which has the property that marginalizing over the last variable, xn, we obtain a correlated probability
+distribution function for the n − 1 variables, x = (x1, . . . , xn−1),
+f(x) =
+1
+(2π)(n−1)/2�
+det Cn−1
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+,
+(37)
+where the Cn−1 covariance matrix is given by Eq. (35) without the last column and the last row.
+We will now evaluate the evidence Eq. (22) for this multivariate Gaussian, starting with the integration
+over the last variable, xn,
+1
+2an
+� an
+−an
+dxn f(x)
+=
+1
+(2π)(n−1)/2�
+det Cn−1
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+×
+�
+1
+2an
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
++ O
+�
+e−
+a2
+n det Cn−1
+2 det Cn
+��
+.
+(38)
+Integrating now over the next variable, xn−1, we find
+1
+4anan−1
+� an
+−an
+dxn
+� an−1
+−an−1
+dxn−1 f(x) =
+1
+(2π)(n−2)/2�
+det Cn−2
+exp
+�
+− 1
+2 xT C−1
+n−2x
+�
+×
+�
+1
+4anan−1
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
+Erf
+�
+an
+√
+2
+�
+det Cn−2
+det Cn−1
+�
++ O
+�
+e−
+a2
+n det Cn−1
+2 det Cn
+��
+.
+(39)
+Continuing the integration over the priors, we end up with the evidence for the n-dimensional distribution,
+E(a1, . . . , an)
+=
+1
+�n
+p=1 2ap
+� a1
+−a1
+· · ·
+� an
+−an
+dnx f(x)
+=
+n
+�
+p=1
+1
+2ap
+Erf
+�
+ap
+√
+2
+�
+det Cp−1
+det Cp
+�
++ O
+�
+exp
+�
+−
+n
+�
+p=1
+a2
+p det Cp−1
+2 det Cp
+��
+,
+(40)
+7
+
+where the covariance matrices Cp are constructed as above, by eliminating the n−p last rows and columns, un-
+til we end up with C0 ≡ 1. Note that the approximation is very good whenever �n
+p=1(a2
+p det Cp−1)/(2 det Cp) ≫
+1, which is often the case. Note also that we recover the previous result Eq. (34) for the particular case
+n = 2.
+In the limit that the cross-correlation between the n variables vanishes, the evidence (40) reduces to the
+exact result
+E(a1, . . . , an) =
+n
+�
+p=1
+1
+2ap
+Erf
+�
+ap
+σp
+√
+2
+�
+.
+(41)
+Note that the evidence Eq. (40) reflects correctly the limit in which we eliminate the need for a new variable
+xn, by making its prior vanish,
+lim
+an→0 E(a1, . . . , an) = E(a1, . . . , an−1)
+1
+√
+2π
+�
+det Cn−1
+det Cn
+,
+(42)
+and thus we recover in that limit a properly-normalized distribution, f(x1, . . . , xn) → f(x1, . . . , xn−1), while
+the inspection of the likelihood function alone would not have been able to give a reasonable answer.
+On the other hand, in the case that our theoretical prejudice cannot assign a concrete prior to a given
+variable, we see that the evidence decreases as 1/2a as a increases. Therefore, the Bayesian evidence seems
+to be a very good discriminator between theoretical priors, and penalizes including too many parameters, a
+la Occam’s razor.
+3.2
+Uncentered priors
+It is unlikely that the priors will actually be centred on the mean of the distribution, as the priors are not
+supposed to know what the data will tell us. We therefore need to generalize the above for uncentred priors.
+We continue to assume that the priors are top hats.
+We also continue to assume for the moment that the probability distribution is well approximated by
+a Gaussian with mean value µ. We will then use displaced variables ˜xi = xi − µi, and write the Gaussian
+distribution function as in Eq. (36). The normalized top-hat prior is now uncentered with respect to the
+mean value,
+π(˜x; a, b) ≡
+�
+(a + b)−1
+−a < ˜x < b ,
+0
+otherwise .
+(43)
+For a single variable, the result is exact,
+E(a; b) =
+� ∞
+−∞
+dx f(x) π(x; a, b) =
+1
+2a + 2b
+�
+Erf
+�
+a
+σ
+√
+2
+�
++ Erf
+�
+b
+σ
+√
+2
+��
+.
+(44)
+where we are integrating over the displaced variable ˜x, from now on renamed as x. Note that we recover the
+result Eq. (24) for the centered prior case in the limit b → a.
+For two variables, with distribution function Eq. (26), the uncentered Bayesian evidence is
+E(a1, a2; b1, b2)
+=
+1
+(a1 + b1)(a2 + b2)
+� b1
+−a1
+dx1
+� b2
+−a2
+dx2 f(x1, x2)
+(45)
+=
+1
+(2a1 + 2b1)(2a2 + 2b2)
+��
+Erf
+�
+a1
+σ1
+√
+2
+�
++ Erf
+�
+b1
+σ1
+√
+2
+��
+(46)
+×
+�
+Erf
+�
+a2
+σ2
+�
+2(1 − ρ2)
+�
++ Erf
+�
+b2
+σ2
+�
+2(1 − ρ2)
+��
+−
+ρ
+2π
+�
+1 − ρ2
+�
+e
+−
+a2
+1
+2σ2
+1 − e
+−
+b2
+1
+2σ2
+1
+� �
+e
+−
+a2
+2
+2σ2
+2(1−ρ2) + e
+−
+b2
+2
+2σ2
+2(1−ρ2)
+��
+8
+
+The evidence for the multiple-variable case Eq. (36) is
+E(a, b) =
+� ∞
+−∞
+dnx f(x)
+n
+�
+i=1
+π(xi; ai, bi) =
+n
+�
+i=1
+(ai + bi)−1
+� b1
+−a1
+d˜x1· · ·
+� bn
+−an
+d˜xn f(˜x) .
+(47)
+Let us now evaluate it for the multivariate Gaussian Eq. (36), starting with the integration over the last
+variable, xn,
+1
+an + bn
+� bn
+−an
+dxn f(x) =
+1
+(2π)(n−1)/2�
+det Cn−1
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+1
+(2an + 2bn)
+×
+�
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
++ Erf
+�
+bn
+√
+2
+�
+det Cn−1
+det Cn
+�
++ O
+�
+e−
+a2
+n det Cn−1
+2 det Cn
++ e−
+b2
+n det Cn−1
+2 det Cn
+��
+(48)
+Integrating now over the next variable, xn−1, we find
+1
+(an + bn)(an−1 + bn−1)
+� bn
+−an
+dxn
+� bn−1
+−an−1
+dxn−1 f(x) =
+1
+(2π)(n−2)/2�
+det Cn−2
+exp
+�
+− 1
+2 xT C−1
+n−2x
+�
+1
+(2an + 2bn)(2an−1 + 2bn−1)
+(49)
+×
+��
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
++ Erf
+�
+bn
+√
+2
+�
+det Cn−1
+det Cn
+��
+(50)
+×
+�
+Erf
+�
+an−1
+√
+2
+�
+det Cn−2
+det Cn−1
+�
++ Erf
+�
+bn−1
+√
+2
+�
+det Cn−2
+det Cn−1
+��
+(51)
++ O
+�
+e−
+a2
+n det Cn−1
+2 det Cn
++ e−
+b2
+n det Cn−1
+2 det Cn
+�
+×
+�
+e
+−
+a2
+n−1 det Cn−2
+2 det Cn−1
++ e
+−
+b2
+n−1 det Cn−2
+2 det Cn−1
+��
+.
+Continuing the integration over the priors, we end up with the evidence for the n-dimensional distribution,
+E(a, b)
+=
+1
+�n
+p=1(ap + bp)
+� b1
+−a1
+· · ·
+� bn
+−an
+dnx f(x)
+=
+n
+�
+p=1
+1
+(2ap + 2bp)
+�
+Erf
+�
+ap
+√
+2
+�
+det Cp−1
+det Cp
+�
++ Erf
+�
+bp
+√
+2
+�
+det Cp−1
+det Cp
+��
+(52)
++ O
+� n
+�
+p=1
+�
+exp
+�
+− a2
+p det Cp−1
+2 det Cp
+�
++ exp
+�
+− b2
+p det Cp−1
+2 det Cp
+���
+,
+where the covariance matrices Cp are constructed as above, by eliminating the n−p last rows and columns, un-
+til C0 ≡ 1. Note that the approximation is very good whenever the exponents are large, �n
+p=1(a2
+p det Cp−1)/(2 det Cp) ≫
+1, which is often the case. Note also that we recover the expression of the evidence for the centered priors
+Eq. (40) in the limit b → a.
+Let us now evaluate the evidence for a distribution normalized to the maximum of the likelihood distri-
+bution,
+f(x) = Lmax exp
+�
+− 1
+2xT C−1
+n x
+�
+(53)
+9
+
+In this case, the evidence is given by Eq. (52), multiplied by a factor Lmax × (2π)n/2√det Cn from the nor-
+malization. We can then evaluate the logarithm of the evidence, ignoring the exponentially-small corrections,
+as
+ln E
+=
+ln Lmax + n
+2 ln(2π) + 1
+2 ln det Cn −
+n
+�
+p=1
+ln(2ap + 2bp)
++
+n
+�
+p=1
+ln
+�
+Erf
+�
+ap
+√
+2
+�
+det Cp−1
+det Cp
+�
++ Erf
+�
+bp
+√
+2
+�
+det Cp−1
+det Cp
+��
+.
+(54)
+Uncorrelated case. Suppose we have a multivariate Gaussian distribution without correlations between
+variables, i.e. Cij = σ2
+i δij is a diagonal matrix; then the evidence reads exactly,
+E(a, b) =
+1
+�n
+p=1(ap + bp)
+� b1
+−a1
+· · ·
+� bn
+−an
+dnx f(x) =
+n
+�
+p=1
+1
+2(ap + bp)
+�
+Erf
+�
+ap
+σp
+√
+2
+�
++ Erf
+�
+bp
+σp
+√
+2
+��
+,
+(55)
+where σp are the dispersions of each variable ˜xp, and thus the logarithm of the evidence becomes
+ln E = ln Lmax + n
+2 ln(2π) +
+n
+�
+p=1
+ln σp −
+n
+�
+p=1
+ln(2ap + 2bp) +
+n
+�
+p=1
+ln
+�
+Erf
+�
+ap
+σp
+√
+2
+�
++ Erf
+�
+bp
+σp
+√
+2
+��
+(56)
+Laplace approximation. The Laplacian approximation to the evidence assumes the distribution is a
+correlated Gaussian, and that the priors are large enough so that the whole distribution fits easily inside
+them, in which case the error functions are approximately unity and do not contribute to the evidence; from
+Eq. (54) we now have
+ln E = ln Lmax + n
+2 ln(2π) + 1
+2 ln det Cn −
+n
+�
+p=1
+ln ∆θp ,
+(57)
+where ∆θp = ap + bp is the parameter interval associated to the prior. In the next section we will compare
+the different approximations.
+4
+Non-Gaussian corrections
+The advantage of this method is that one can perform a systematic computation of the evidence of a given
+model with its own priors, given an arbitrary set of moments of the distribution. Here we will consider the
+first two beyond the covariance matrix, i.e. the skewness and the kurtosis terms, see Eq. (9).
+4.1
+Skewness
+Let us start with the first correction to the Gaussian approximation, the trilinear term Bijk. For this, we
+write the generating functional (9) as
+φ(u) = exp
+�
+i µiui − 1
+2! Cij uiuj − i
+3! Bijk uiujuk
+�
+.
+(58)
+10
+
+By performing a change of variable, ui = yi −i C−1
+ik (xk −µk), we can evaluate the Fourier transform integral
+and obtain the properly-normalized probability distribution function
+f(x)
+=
+1
+(2π)n/2√det Cn
+exp
+�
+− 1
+2xT C−1
+n x
+�
+×
+�
+1 − 1
+2Bijk C−1
+ij C−1
+kl xl + 1
+6Bijk C−1
+il C−1
+jmC−1
+kn xlxmxn
+�
+,
+(59)
+where xk are the displaced coordinates (xk − µk). This skewed distribution function satisfies
+⟨xi⟩ = 0 ,
+⟨xixj⟩ = Cij ,
+⟨xixjxk⟩ = Bijk ,
+⟨xixjxkxl⟩ = 0 ,
+. . .
+(60)
+as can be confirmed by direct evaluation. Let us now compute the evidence Eq. (22) for this skewed model.
+Since the extra terms in the parenthesis of Eq. (59) are both odd functions of x, when integrating over an
+even range like that of the centered top-hat prior Eq. (21), their contribution to the evidence vanish, and
+thus the final evidence for the skewed model does not differ from that of the Gaussian model Eq. (40). In
+case the prior is off-centered with respect to the mean, e.g. like in Eq. (43), then the contribution of the odd
+terms to the evidence would not vanish. Let us evaluate their contribution.
+For a single variable (n = 1), the correctly-normalized likelihood function can be written as
+f(x) = e−x2/2σ2
+σ
+√
+2π
+�
+1 − B x
+2σ4 + B x3
+6σ6
+�
+,
+satisfying ⟨x⟩ = 0, ⟨x2⟩ = σ2, ⟨x3⟩ = B, and the Bayesian integral can be computed exactly as
+E(a, b) =
+1
+2a + 2b
+�
+Erf
+�
+a
+σ
+√
+2
+�
++ Erf
+�
+b
+σ
+√
+2
+��
+− Bσ−3
+6
+√
+2π
+��
+1 − a2
+σ2
+�
+e− a2
+2σ2 −
+�
+1 − b2
+σ2
+�
+e− b2
+2σ2
+�
+1
+a + b .
+(61)
+Note that for even (centered) priors, with b = a, the evidence reduces to Eq. (24).
+For an arbitrary number of variables, the computation is more complicated. Let us start with the n-th
+variable and, in order to compute the integral, let us define the auxiliary function
+g(λ)
+=
+� bn
+−an
+dxn xn
+exp
+�
+− λ
+2 xT C−1
+n x
+�
+(2π)n/2√det Cn
+=
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+(2π)(n−1)/2�
+det Cn−1
+×
+×
+1
+λ
+√
+2π
+�
+exp
+�
+− λa2
+n
+2
+det Cn−1
+det Cn
+�
+− exp
+�
+− λb2
+n
+2
+det Cn−1
+det Cn
+��
+,
+(62)
+such that, using Erf′[x] =
+2
+√π e−x2,
+−2g′(λ = 1) =
+� bn
+−an
+dxn xn
+(xT C−1
+n x) exp
+�
+− 1
+2xT C−1
+n x
+�
+(2π)n/2√det Cn
+=
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+(2π)(n−1)/2�
+det Cn−1
+×
+×
+1
+√
+2π
+��
+2 + a2
+n
+det Cn−1
+det Cn
+�
+exp
+�
+− a2
+n
+2
+det Cn−1
+det Cn
+�
+−
+�
+2 + b2
+n
+det Cn−1
+det Cn
+�
+exp
+�
+− b2
+n
+2
+det Cn−1
+det Cn
+��
+.
+(63)
+Therefore, with the use of Eq. (63), the integral of the skewness-corrected distribution function Eq. (59) over
+the xn uncentered prior, becomes
+� bn
+−an
+dxn f(x) =
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+(2π)(n−1)/2�
+det Cn−1
+�
+1
+2
+�
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
++ Erf
+�
+bn
+√
+2
+�
+det Cn−1
+det Cn
+��
+− 1
+6Bijn C−1
+ij
+1
+√
+2π
+�
+det Cn−1
+det Cn
+��
+1 − a2
+n
+det Cn−1
+det Cn
+�
+e−
+a2
+n det Cn−1
+2 det Cn
+−
+�
+1 − b2
+n
+det Cn−1
+det Cn
+�
+e−
+b2
+n det Cn−1
+2 det Cn
+��
+.
+(64)
+11
+
+Let us define two new functions,
+Ei(ai, bi)
+=
+1
+2
+�
+Erf
+�
+ai
+√
+2
+�
+det Ci−1
+det Ci
+�
++ Erf
+�
+bi
+√
+2
+�
+det Ci−1
+det Ci
+��
+,
+(65)
+Fi(ai, bi)
+=
+1
+6
+√
+2π
+�
+det Ci−1
+det Ci
+��
+1 − a2
+i
+det Ci−1
+det Ci
+�
+e−
+a2
+i det Ci−1
+2 det Ci
+−
+�
+1 − b2
+i
+det Ci−1
+det Ci
+�
+e−
+b2
+i det Ci−1
+2 det Ci
+�
+.
+Integrating iteratively over xn−1, . . . , x1, we end up with the Bayesian evidence for the third-order-corrected
+probability distribution function f(x),
+E(a, b) =
+n
+�
+p=1
+Ep(ap, bp)
+(ap + bp)
+�
+1 −
+n
+�
+k=1
+Bijk C−1
+ij
+Fk(ak, bk)
+Ek(ak, bk)
+�
+.
+(66)
+Unless Bijk C−1
+ij
+is very large, the correction to the error function is exponentially suppressed, and we do
+not expect significant departures from the Gaussian case Eq. (40). Note also that if the prior is symmetric,
+it is easy to see that the skewness part of the integral vanishes, Fk(ak, bk) → 0, as can be checked explicitly
+by taking bk → ak.
+4.2
+Kurtosis
+The next correction beyond skewness is the fourth order moment or kurtosis, given by the Dijkl term in
+Eq. (9). Let us ignore for the moment the third order skewness and write
+φ(u) = exp
+�
+i µiui − 1
+2! Cij uiuj + 1
+4! Dijkl uiujukul
+�
+.
+(67)
+By performing the same change of variables, ui = yi − i C−1
+ik (xk − µk), we can now compute the Fourier
+transform and obtain the properly-normalized probability distribution function
+f(x)
+=
+1
+(2π)n/2√det Cn
+exp
+�
+− 1
+2xT C−1
+n x
+� �
+1 + 1
+8Dijkl C−1
+ij C−1
+kl
+−1
+4Dijkl C−1
+ij C−1
+kmC−1
+ln xmxn + 1
+24Dijkl C−1
+im C−1
+jn C−1
+kp C−1
+lq xmxnxpxq
+�
+.
+(68)
+Performing the integrals, it is easy to see that this distribution satisfies
+⟨xixj⟩ = Cij ,
+⟨xixjxkxl⟩ = Dijkl + CijCkl + CikCjl + CilCjk ,
+. . .
+(69)
+Note that in order for the new likelihood distribution (68) to be positive definite, it is required that
+DijklC−1
+ij C−1
+kl
+< 4, and if we impose that there is only one maximum at the center, then it must sat-
+isfy DijklC−1
+ij C−1
+kl < 2. These conditions impose bounds on the maximum possible deviation of the evidence
+from a that of a gaussian.
+Let us now compute the evidence Eq. (22) for this kurtosis model. The extra terms in the parenthesis of
+Eq. (68) are both even functions of x, and we cannot ignore them, even for centered priors.
+For a single variable (n = 1), the correctly-normalized likelihood function can be written as
+f(x) = e− x2
+2σ2
+σ
+√
+2π
+�
+1 + D
+8σ4 − D x2
+4σ6 + D x4
+24σ8
+�
+,
+satisfying ⟨x⟩ = 0, ⟨x2⟩ = σ2, ⟨x3⟩ = 0, ⟨x4⟩ = D + 3σ4, etc. The Bayesian integral can be computed exactly
+as
+E(a, b) =
+1
+2a + 2b
+�
+Erf
+�
+a
+σ
+√
+2
+�
++ Erf
+�
+b
+σ
+√
+2
+��
++ Dσ−4
+8
+√
+2π
+� a
+σ
+�
+1 − a2
+3σ2
+�
+e− a2
+2σ2 + b
+σ
+�
+1 − b2
+3σ2
+�
+e− b2
+2σ2
+�
+1
+a + b .
+(70)
+12
+
+For arbitrary number of variables, the computation is again much more complicated. Let us start with
+the n-th variable and, in order to compute the first integral, let us define a new auxiliary function
+h(λ)
+=
+� bn
+−an
+dxn
+exp
+�
+− λ
+2 xT C−1
+n x
+�
+(2π)n/2√det Cn
+=
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+(2π)(n−1)/2�
+det Cn−1
+×
+×
+1
+2
+√
+λ
+�
+Erf
+�
+an
+√
+λ
+√
+2
+�
+det Cn−1
+det Cn
+�
++ Erf
+�
+bn
+√
+λ
+√
+2
+�
+det Cn−1
+det Cn
+��
+,
+(71)
+such that,
+−2h′(λ = 1)
+=
+� bn
+−an
+dxn
+(xT C−1
+n x) exp
+�
+− 1
+2xT C−1
+n x
+�
+(2π)n/2√det Cn
+=
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+(2π)(n−1)/2�
+det Cn−1
+×
+×
+�
+1
+2
+�
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
++ Erf
+�
+bn
+√
+2
+�
+det Cn−1
+det Cn
+��
+(72)
+−
+1
+√
+2π
+�
+det Cn−1
+det Cn
+�
+an exp
+�
+− a2
+n
+2
+det Cn−1
+det Cn
+�
++ bn exp
+�
+− b2
+n
+2
+det Cn−1
+det Cn
+���
+.
+4h′′(λ = 1)
+=
+� bn
+−an
+dxn
+(xT C−1
+n x)2 exp
+�
+− 1
+2xT C−1
+n x
+�
+(2π)n√det Cn
+=
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+(2π)(n−1)/2�
+det Cn−1
+×
+×
+�
+3
+2
+�
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
++ Erf
+�
+bn
+√
+2
+�
+det Cn−1
+det Cn
+��
+(73)
+−
+3
+√
+2π
+�
+det Cn−1
+det Cn
+�
+an exp
+�
+− a2
+n
+2
+det Cn−1
+det Cn
+�
++ bn exp
+�
+− b2
+n
+2
+det Cn−1
+det Cn
+��
+−
+a2
+n
+√
+2π
+�det Cn−1
+det Cn
+�3/2 �
+an exp
+�
+− a2
+n
+2
+det Cn−1
+det Cn
+�
++ bn exp
+�
+− b2
+n
+2
+det Cn−1
+det Cn
+���
+.
+Therefore, with the use of Eqs. (72) and (73), the integral of the kurtosis-corrected distribution function (68)
+over the xn prior, becomes
+� bn
+−an
+dxn f(x) =
+exp
+�
+− 1
+2xT C−1
+n−1x
+�
+(2π)(n−1)/2�
+det Cn−1
+�
+1
+2
+�
+Erf
+�
+an
+√
+2
+�
+det Cn−1
+det Cn
+�
++ Erf
+�
+bn
+√
+2
+�
+det Cn−1
+det Cn
+��
++
+(74)
++ 1
+8Dijkl C−1
+ij C−1
+kl
+1
+√
+2π
+�
+det Cn−1
+det Cn
+�
+an
+�
+1 − a2
+n
+3
+det Cn−1
+det Cn
+�
+e−
+a2
+n det Cn−1
+2 det Cn
++ bn
+�
+1 − b2
+n
+3
+det Cn−1
+det Cn
+�
+e−
+b2
+n det Cn−1
+2 det Cn
+��
+.
+We can now define a new function
+Gi(ai, bi) =
+1
+8
+√
+2π
+�
+det Ci−1
+det Ci
+�
+ai
+�
+1 − a2
+i
+3
+det Ci−1
+det Ci
+�
+e−
+a2
+i det Ci−1
+2 det Ci
+− bi
+�
+1 − b2
+i
+3
+det Ci−1
+det Ci
+�
+e−
+b2
+i det Ci−1
+2 det Ci
+�
+.
+(75)
+Integrating iteratively over xn−1, . . . , x1, we end up with the Bayesian evidence for the fourth-order-corrected
+probability distribution function f(x),
+E(a, b) =
+n
+�
+p=1
+Ep(ap, bp)
+(ap + bp)
+�
+1 + Dijkl C−1
+ij C−1
+kl
+n
+�
+m=1
+Gm(am, bm)
+Em(am, bm)
+�
+.
+(76)
+13
+
+so, unless Dijkl C−1
+ij C−1
+kl
+is very large, the correction to the error function is exponentially suppressed, and
+we do not expect significant departures from the Gaussian case, Eq. (40).
+In order to compare models it is customary to compute the logarithm of the evidence. Let us assume that
+we are given a likelihood distribution function normalized by the maximum likelihood, and with corrections
+up to fourth order,
+f(x) = Lmax exp
+�
+− 1
+2xT C−1
+n x
+� �
+1 + 1
+8Dijkl C−1
+ij C−1
+kl
+�−1�
+1 − 1
+2Bijk C−1
+ij C−1
+kl xl + 1
+6Bijk C−1
+il C−1
+jmC−1
+kn xlxmxn
++ 1
+8Dijkl C−1
+ij C−1
+kl − 1
+4Dijkl C−1
+ij C−1
+kmC−1
+ln xmxn + 1
+24Dijkl C−1
+im C−1
+jn C−1
+kp C−1
+lq xmxnxpxq
+�
+.
+(77)
+Note that it is normalized so that the maximum corresponds to the mean-centered distribution, i.e. x = 0.
+In this case, the evidence of the normalized distribution is given by
+E(a, b) = Lmax (2π)n/2�
+det Cn
+�
+1 + 1
+8Dijkl C−1
+ij C−1
+kl
+�−1
+×
+(78)
+n
+�
+p=1
+Ep(ap, bp)
+(ap + bp)
+�
+1 −
+n
+�
+k=1
+Bijk C−1
+ij
+Fk(ak, bk)
+Ek(ak, bk) + Dijkl C−1
+ij C−1
+kl
+n
+�
+m=1
+Gm(am, bm)
+Em(am, bm)
+�
+.
+We can then evaluate the logarithm of the evidence by
+ln E
+=
+ln Lmax + n
+2 ln(2π) + 1
+2 ln det Cn − ln
+�
+1 + 1
+8Dijkl C−1
+ij C−1
+kl
+�
+−
+n
+�
+p=1
+ln(2ap + 2bp)
++
+n
+�
+p=1
+ln
+�
+Erf
+�
+ap
+√
+2
+�
+det Cp−1
+det Cp
+�
++ Erf
+�
+bp
+√
+2
+�
+det Cp−1
+det Cp
+��
+(79)
++ ln
+�
+1 −
+n
+�
+k=1
+Bijk C−1
+ij
+Fk(ak, bk)
+Ek(ak, bk) + Dijkl C−1
+ij C−1
+kl
+n
+�
+m=1
+Gm(am, bm)
+Em(am, bm)
+�
+.
+Note that the condition DijklC−1
+ij C−1
+kl
+< 2 constrains the maximum amount that the kurtosis corrections
+can contribute to the evidence.
+Uncorrelated case. In the case where the likelihood distribution had no correlations among the different
+variables, the exact expression for the Bayesian evidence is
+ln E = ln Lmax + n
+2 ln(2π) +
+n
+�
+p=1
+ln σp −
+n
+�
+p=1
+ln(2ap + 2bp) +
+n
+�
+p=1
+ln
+�
+Erf
+�
+ap
+σp
+√
+2
+�
++ Erf
+�
+bp
+σp
+√
+2
+��
+(80)
+− ln
+�
+1 + 1
+8Diijj σ−2
+i
+σ−2
+j
+�
++ ln
+�
+1 −
+n
+�
+k=1
+Biik σ−2
+k
+Fk(ak, bk)
+Ek(ak, bk) + Diijj σ−2
+i
+σ−2
+j
+n
+�
+m=1
+Gm(am, bm)
+Em(am, bm)
+�
+,
+where σp are the corresponding dispersions of variables xp, and the functions Ei, Fi and Gi are the corre-
+sponding limiting functions of Eqs. (65) and (75) for uncorrelated matrices.
+5
+Model comparison
+Finally we turn to specific applications of the formalism discussed above. Initially we will carry out some
+toy model tests of its performance, and then examine real cosmological applications for which we previously
+obtained results by thermodynamic integration [5].
+14
+
+Figure 1: This figure shows the calculated evidence as a function of the number of likelihood evaluations.
+Note that the horizontal axis is logarithmic. The solid line corresponds to the thermodynamic integration.
+The dotted line and dot-dashed lines are the analytical methods with and without non-Gaussian corrections
+applied. The horizontal dashed line is the number obtained by the direct integration. The upper two panels
+correspond to Lg, while the lower two to Lng. The left-hand side panels correspond to wide flat priors of
+(−7, 10) on both parameters, while the right-hand side to the narrow priors of (−2, 3) on both parameters.
+See text for discussion.
+5.1
+A baby-toy model comparison
+We begin with a very simple two-dimensional toy model. The purpose of this section is to illustrate the
+ineffectiveness of the thermodynamic integration and to give an indication of the performance of the method
+we propose here. In addition, the two-dimensional model is simple enough to allow a brute-force direct
+numerical integration of evidence allowing us to check the accuracy at the same time. We use the following
+two forms of likelihood:
+Lg(x, y)
+=
+exp
+�
+−2x2 − 2(y − 1)2 − xy
+2
+�
+(81)
+Lng(x, y)
+=
+exp
+�
+−2x2 − 2(y − 1)2 − xy
+2
+�
++ exp
+�
+−2x2 − 2y2 − 3xy
+2
+�
+(82)
+The subscripts g and ng indicate the Gaussian and non-Gaussian cases respectively.
+Firstly, we calculate the evidence by the analytical method using Eqs. (56) and (80) and covariance
+15
+
+matrices inferred from sampling the likelihood using the vanilla Metropolis–Hastings algorithm with fixed
+proposal widths. Chains ranging from few to several million samples were used. We also calculate evidence
+using thermodynamic algorithm explained in Ref. [5]. Again, we vary algorithm parameters to get evidence
+values of varying accuracy. The resulting evidence as a function of number of likelihood evaluations is plotted
+in the Figure 1, together with the correct value inferred by direct numerical integration. The number of
+likelihood evaluations is crucial as this is the time-limiting step in the cosmological parameter estimation
+and model comparison exercises. The results are what could have been anticipated. We note that the size
+of the prior does not seem to be of crucial importance. This is comforting, given that the analytical method
+requires the knowledge of the true covariance information, while we can only supply a covariance matrix
+estimated from the prior-truncated likelihood. We also note that the thermodynamic integration converges
+to the correct value in all cases. However, it does so after very many likelihood evaluations; typically about
+a million or so even for a two-dimensional problem. The analytical method becomes limited by systematics
+already by the ten-thousand samples.
+For Gaussian case, there is no systematic by construction, while
+the non-gaussian case suffers a systematic of about 0.1 in ln E. The non-Gaussian correction reduces the
+error by about a half and thus correctly estimates the uncertainty associated with the purely Gaussian
+approximation. In the case of wide priors, the only non-Gaussian correction of an appreciable size is the
+ln(1 + DijklC−1
+ij C−1
+kl /8).
+5.2
+A toy model comparison
+We now proceed by calculating the Bayesian evidence for simple toy models with 5 and 6 parameters, shown
+in Table I. The purpose is to compare results with those obtained from thermodynamic integration again,
+but this time using a model that bears more resemblance to a typical problem one encounters in cosmology.
+Parameter
+Mean
+Prior Range
+Model
+x1
+0.022
+[0.0001, 0.044]
+toy5,toy6
+x2
+0.12
+[0.001, 0.3]
+toy5,toy6
+x3
+1.04
+[0.8, 1.4]
+toy5,toy6
+x4
+0.1
+[0.01, 0.3]
+toy5,toy6
+x5
+3.1
+[2.6, 3.6]
+toy5,toy6
+x6
+0.98
+[0.5, 1.5]
+toy6
+Table 1:
+The parameters used in the analytical evaluation of the toy model evidences, with 5 and 6
+parameters respectively. The maximum likelihod of the toy models is taken (arbitrarily) to be Lmax = 1.
+Beginning with the five-parameter model, we assume first that it has an uncorrelated multivariate Gaus-
+sian likelihood distribution. In this case the aim is to test the thermodynamic integration method, which
+gives ln Enum
+toy5 = −8.65 ± 0.03, while the exact expression gives ln Eana
+toy5 = −8.66. Therefore, we conclude
+that the thermodynamic integration method is rather good in obtaining the correct evidence of the model.
+The Laplace approximation Eq. (57) also fares well for uncorrelated distributions, ln ELap
+toy5 = −8.67.
+We now consider a likelihood function with a correlated covariance matrix Cij, with the same mean
+values and dispersions as the previous case, but with significant correlations. The analytic formula needed,
+Eq. (54), is no longer exact,2 and gives ln Eana
+toy5c = −7.32. For comparison thermodynamic integration gives
+ln Enum
+toy5c = −7.28 ± 0.06, again in perfect agreement within errors. In this case the Laplace approximation
+fails significantly, ln ELap
+toy5c = −6.89, the reason being that the correlations chosen bring the posterior into
+significant contact with the edges of the priors.
+Let us now return to the uncorrelated case and include a new parameter, x6, as in Table I, and evaluate the
+different evidences that appear because of this new parameter, in order to see the sensitivity to systematic
+errors in the evaluation of the Bayesian evidence and their effects on model comparison. The numerical
+2One could rotate the parameter basis to remove the correlations, but then the priors wouldn’t be top-hats.
+16
+
+result is ln Enum
+toy6 = −10.75 ± 0.03, while the exact analytical expression gives ln Eana
+toy6 = −10.74, in perfect
+agreement, within errors. The Laplace approximation Eq. (57) again fares well for uncorrelated distributions,
+ln ELap
+toy6 = −10.74.
+When the likelihood function has large correlations, and the priors are not too large, the naive Laplace
+approximation, Eq. (57), fares less well than the analytical approximation, Eq. (54).
+5.3
+A real model comparison
+In this subsection we will make use of the results obtained in Ref. [5], where we evaluated the evidence for
+5- and 6-parameter adiabatic models, and for three 10-parameter mixed adiabatic plus isocurvature models.
+The prior ranges used are given in Table II. The latter models give a marginally better fit to the data but
+require more parameters, which is exactly the situation where model selection techniques are needed to draw
+robust conclusions. In Ref. [5] we used thermodynamic integration to compute the evidence and showed that
+the isocurvature models ware less favoured than the adiabatic ones, but only at a mild significance level.3
+Beginning with the simplest adiabtic model, which uses the Harrison–Zel’dovich spectrum, we have
+used the analytical formulae above, Eq. (54), together with the covariance matrix provided by the cosmoMC
+programme [10], and obtained ln Eana
+ad
+= −854.07, while the thermodynamical integration gave ln Enum
+ad
+=
+−854.1±0.1 [5]. The agreement is excellent; this is because the distribution function for the adiabatic model
+is rather well approximated by a Gaussian, and the priors are rather large, so the formula Eq. (54) is very
+close to that obtained in the Laplace approximation, ln ELap
+ad
+= −854.08.
+Parameter
+Mean
+Prior Range
+Model
+ωb
+0.022
+[0.018, 0.032]
+AD-HZ,AD-ns,ISO
+ωdm
+0.12
+[0.04, 0.16]
+AD-HZ,AD-ns,ISO
+θ
+1.04
+[0.98, 1.10]
+AD-HZ,AD-ns,ISO
+τ
+0.17
+[0, 0.5]
+AD-HZ,AD-ns,ISO
+ln[1010Rrad]
+3.1
+[2.6, 4.2]
+AD-HZ,AD-ns,ISO
+ns
+1.0
+[0.8, 1.2]
+AD-ns,ISO
+niso
+1.5
+[0, 3]
+ISO
+δcor
+1.5
+[−0.14, 0.4]
+ISO
+√α
+0
+[−1, 1]
+ISO
+β
+0
+[−1, 1]
+ISO
+Table 2:
+The parameters used in the models; see Ref. [5] for nomenclature and other details. For the
+AD-HZ model ns was fixed to 1 and niso, δcor, α and β were fixed to 0. In the AD-ns model, ns also varies.
+Every isocurvature model holds the same priors for the whole set of parameters.
+However the analytic method fares less well for the adiabatic model with varying ns, with both the
+analytic and Laplace methods giving ln EAD−ns = −853.4, while the numerical method gives the smaller
+value -854.1, a discrepency of nearly unity.
+Turning now to the iscurvature cases, we found an extremely good result for the CDI model, gaining from
+Eq. (54) the value ln Eana
+cdi = −855.08, while the thermodynamical integration gives ln Enum
+cdi
+= −855.1 ± 0.1.
+This is surprising, given the relatively large non-gaussianities for at least three variables: niso, β and δcor,
+whose priors are not centered with respect to the mean.
+However the NID case shows much less good
+agreement, with a discrepency of 0.6. That suggests that the closeness of the CDI comparison is to some
+extent a statistical fluke, with the underlying method less accurate.
+A summary of the different models can be found in Table 3.
+3Recently Trotta [9] used a different technique to analyze a restricted class of isocurvature model featuring just one extra
+parameter, and found it highly disfavoured. The different conclusion is primarily due to the very different prior he chose on
+the isocurvature amplitude, such that almost all the models under the prior are domintaed by isocurvature modes and in poor
+agreement with the data.
+17
+
+Model
+ln Lmax
+ln Enum
+ln Eana
+ln ELap
+toy5
+0
+−8.65 ± 0.03
+−8.66
+−8.67
+toy5c
+0
+−7.28 ± 0.06
+−7.32
+−6.89
+toy6
+0
+−10.75 ± 0.03
+−10.74
+−10.74
+toy6c
+0
+−9.73 ± 0.06
+−9.71
+−9.63
+AD
+−840.78
+−854.1 ± 0.1
+−854.1
+−854.1
+AD-ns
+−838.50
+−854.1 ± 0.1
+−853.4
+−853.4
+CDI
+−838.05
+−855.1 ± 0.2
+−855.1
+−854.5
+NID
+−836.60
+−855.1 ± 0.2
+−854.5
+−854.5
+NIV
+−842.53
+−855.1 ± 0.3
+−854.9
+−854.9
+Table 3:
+The different models, both toy and real, with their maximum likelihoods and evidences.
+5.4
+Savage–Dickey method
+Another numerical method for evidence calculation is the Savage–Dickey method, first described in Ref. [11]
+and recently used in Ref. [9]. This technique allows one to calculate the evidence ratio of two models from a
+simple and quick analysis of the Markov chains used for parameter estimation, provided that the models are
+nested; i.e., that one of them is included in the parameter space of the other. For instance, the AD model
+is nested within the AD-ns model, and the AD and AD-ns models are both nested within the CDI, NID
+and NIV ones. In the context of Markov chains, the Savage–Dickey method is essentially a measure of how
+much time the sampler spends in the nested model, weighted by the respective volumes of the two models.
+When the outer model has extra parameters, this method relies on approximating the nested model as a
+model with negligibly narrow priors in directions of extra parameters. We note, however, that when many
+extra parameters are present, this method must fail for reasons similar to those why grid-based parameter
+estimation approaches fail with models with many parameters. The MCMC parameter estimation simply
+does not have high enough dynamic range to probe the two models given the large prior volume ratio.
+The AD and AD-ns models differ by one parameter. Using the same AD+ns samples as for the analytic
+method (i.e., the samples from which we extracted the covariance matrix), we obtained ln(EAD/EAD+ns) =
+0.03. The result from the precise thermodynamical integration, ln(EAD/EAD−ns) = 0 ± 0.1 is in excellent
+agreement. The AD-ns and CDI (or NID, NIV) models differ by four parameters. With most simple choices
+of parametrization (including in particular the isocurvature and cross-correlation tilts), the AD-ns is not a
+point, but a hypersurface within the parameter space of the isocurvature models (i.e. α = 0 and other three
+parameters act as dummy, unconstrained, parameters which do not affect the evidence). In these cases, the
+evidence ratios given by the Savage–Dickey method do not converge as the priors of the extra parameters
+are tightened up around the nested model, although they match thermodynamically-determined values to
+within a unit of ln E.
+6
+Discussion and Conclusions
+We have developed an analytical formalism for computing the Bayesian evidence in the case of an arbitrary
+likelihood distribution with a hierarchy of non-Gaussian corrections, and with arbitrary top-hat priors,
+centered or uncentered. This analysis can be of great help for the problem of model comparison in the
+present context of cosmology, where observational data is still unable to rule out most extensions of the
+standard model based on the ΛCDM inflationary paradigm.
+As an application of the exact and approximate formulae obtained for the Bayesian evidence of a model
+with approximately Gaussian likelihood distributions, we have compared the value predicted analytically
+with that computed with a time-consuming algorithm based on the thermodynamical integration approach.
+The values obtained analytically agree surprisingly well with those obtained numerically. While one can
+estimate the magnitude of the higher order corrections for the analytical formulae, it is very difficult to
+18
+
+estimate the systematic effects of the numerical approach. Thus, with this analytical method we can test
+for systematics in the thermodynamical integration approach. So far, the values obtained agree, so it seems
+that the numerical approach is a good tool for estimating the evidence. However, it takes considerable effort
+and machine time to do the correct evaluation, and therefore, we propose the use of the analytical estimate,
+whose corrections are well under control, in the sense that one can compute the next order corrections and
+show that they are small.
+Note added: Many years after my work was finished, a book appeared [12] which thoroughly discussed
+Bayesian Methods in Cosmology.
+References
+[1] H. Jeffreys, Theory of Probability, 3rd ed, Oxford University Press (1961).
+[2] D. J. C. MacKay, Information theory, inference and learning algorithms, Cambridge University Press
+(2003).
+[3] A. Jaffe, Astrophys. J. 471, 24 (1996); P. S. Drell, T. J. Loredo, and I. Wasserman I, Astrophys. J. 530,
+593 (2000); M. V. John and J. V. Narlikar, Phys. Rev. D 65, 043506 (2002); M. P. Hobson, S. L. Bridle,
+and O. Lahav, Mon. Not. Roy. Astr. Soc. 335, 377 (2002); A. Slosar et al., Mon. Not. Roy. Astr. Soc.
+341, L29 (2003); T. D. Saini, J. Weller, and S. L. Bridle, Mon. Not. Roy. Astr. Soc. 348, 603 (2004);
+A. Niarchou, A. H. Jaffe, and L. Pogosian, Phys. Rev. D 69, 063515 (2004); P. Marshall, N. Rajguru,
+and A. Slosar, Phys. Rev. D 73, 067302 (2006).
+[4] A. R. Liddle, Mon. Not. Roy. Astr. Soc. 351, L49 (2004).
+[5] M. Beltran, J. Garc´ıa-Bellido, J. Lesgourgues, A. R. Liddle and A. Slosar, Phys. Rev. D 71, 063532
+(2005).
+[6] J. J. K. ´O’Ruanaidh and W. J. Fitzgerald, Numerical Bayesian Methods Applied to Signal Processing,
+Springer–Verlag, New York (1996).
+[7] M. P. Hobson and C. McLachlan, Mon. Not. Roy. Astr. Soc. 338, 765 (2003).
+[8] R. E. Kass and A. E. Raftery, Journ. Amer. Stat. Assoc. 90, 773 (1995).
+[9] R. Trotta, Mon. Not. Roy. Astr. Soc. 378, 72 (2007).
+[10] A. Lewis and S. Bridle, Phys. Rev. D66, 103511 (2002).
+[11] J. M. Dickey, Ann. Math. Stat 42, 204 (1971).
+[12] M. P. Hobson, A. H. Jaffe, A. R. Liddle, P. Mukherjee & D. Parkinson, Bayesian Methods in Cosmology,
+Cambridge University Press (2010).
+19
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+Engineering sub-Poisson light in a simple mirror and beam
+splitter system
+Sun-Hyun Youn∗
+Department of Physics, Chonnam National University, Gwangju 500-757, Korea
+Abstract
+Vacuum fluctuation, which is the intrinsic nature of an electric field can be measured via homo-
+dyne detection. Moreover, electric field intensity fluctuation are also related to vacuum fluctuations.
+Squeezed vacuum and sub-Poisson light can be obtained by controlling the vacuum fluctuation us-
+ing noble nonlinear interaction. Based on the squeezed vacuum by inserting a mirror on the unused
+part of the beam splitter was proposed in 1994, we present the mode matching method for the
+vacuum and light fields. Light intensity fluctuations also can be reduced by inserting a mirror on
+the unused part of the beam splitter. To obtain sub-Poisson light as a function of the distance
+between the mirror and detector, a detector with a thinner active layer than the wavelength is
+required.
+PACS numbers: 03.67.-a,03.70.+k, 03.65.Yz
+Keywords: Quantum optics, Squeezed State, Vacuum fluctuation, Sub-Poisson, Beam splitter and Mirror
+∗ E-mail: [email protected], fax: +82-62-530-3369
+1
+arXiv:2301.01455v1  [quant-ph]  4 Jan 2023
+
+I.
+INTRODUCTION
+When a single photon is in a particular mode, according to the particle nature of light,
+photons will be sequentially found in that mode. The probability of finding a photon is
+proportional to the absolute square of the wave function related to the electromagnetic
+wave. Vacuum fluctuations are related to the spatial characteristics of the electromagnetic
+wave. The spontaneous decay caused by the vacuum can be suppressed in cavities [1]. The-
+oretical and experimental studies have beem conducted on methods to change the vacuum
+fluctuations near mirrors[2–4].
+In this study, in contrast to previous studies on the vacuum noise characteristics of
+light using a homodyne detector, we calculate the intensity fluctuations when photons are
+directly measured using photon counter. The obtained results are similar to those obtained
+in previous studies, but herein we predict the results considering mode matching in the
+experiment.
+In section II, the fluctuation of light that can be measured using a detector is calculated
+with a mirror placed on one side of the beam splitter. In section III, an experimental device
+is proposed for perfect mode matching, and in the last section, the practical limits of the
+vacuum fluctuation near the mirror are discussed.
+II.
+VACUUM FLUCTUATION NEAR A MIRROR.
+An electric field can be written as
+ˆEL = ˆEcl + ˆEQ,
+(1)
+where
+ˆEcl = i
+�
+ℏω
+2ϵ0V (αei(ωt−k0z) − α∗ei(ωt−k0z))⃗x,
+ˆEQ = i
+�
+k
+�
+ℏωk
+2ϵ0V (ˆbke−i(ωkt−kz) − ˆb†
+kei(ωkt−kz))⃗x.
+(2)
+Here, k0 and ω are the wave number and angular frequency of the laser, respectively, ℏ and
+ϵ0 have usual meanings, and V is the normalization volume[5]. Considering the laser mode
+2
+
+FIG. 1: Vacuum mode relations in the beam splitter with a mirror. BS: Beam splitter, M: mirror
+in Fig. 1, the modes aout
+1
+and aout
+2
+can be written as
+aout
+1
+=
+√
+Tb +
+√
+Rc,
+aout
+2
+= −
+√
+Rb +
+√
+Tc,
+(3)
+where the modes c and cout can be written as
+c =
+�
+Tmd −
+�
+Rmcout,
+cout =
+√
+Ra1 +
+√
+Ta2.
+Then the electric field in fluctuating vacuum modes at a1 is
+ˆE(+)
+vac,1 =
+�
+k
+i
+�
+ℏωk
+4ϵ0V {
+√
+Tˆb†
+kei(ωkt−kZ1) + µˆa†
+1,kei(ωkt+kz1)
+−R
+�
+Rmˆa†
+1,kei(ωkt−kz1) −
+√
+RT
+�
+Rmˆa†
+2,kei(ωkt−kz1) +
+�
+RTm ˆd†
+kei(ωkt−kZM)} (4)
+where Rm(Tm) is the reflectance(transmittance) of the mirror and R(T) is the reflectance
+(transmittance) of the beam splitter, z1(Z1) is the distance from the mirror (laser) to the
+detector. ZM is related to the vacuum source behind the mirror and it can be any number.
+We add the factor
+1
+√
+2 for the normalization of the vacuum fluctuation. The vacuum mode
+(ˆa†
+1ei(ωt−kz1)) at the detector is the reflected vacuum mode (ˆa†
+1ei(ωt+kz1)) at the mirror. If two
+modes are perfectly matched the µ in Eq. 4 is 1 and the two counterpropagating modes
+yield the standing wave mode[2, 3]. If µ = 0, the fluctuation value from Eq. 7 becomes |α|2T
+2 ,
+3
+
+b
+BS
+α2
+C
+d
+ino
+↑
+M
+ino
+a1
+ait is the square of the constant dc current T|α|2
+2 . In other words, if we directly measure the
+fluctuation of the laser intensity, the fluctuation is dependent on the distance (z1) between
+the mirror and the detector.
+Even in photo counting experiments, the photon number
+fluctuation is related to the vacuum fluctuation, therefor, the photon number fluctuation is
+also depend on the distance z1.
+If we used the photodetetion theory [6] with instantaneous response of the photodetector
+[7],
+ˆI1 = {
+√
+T ˆE(+)
+cl
++ ˆE(+)
+vac,1} × {
+√
+T ˆE(−)
+cl
++ ˆE(−)
+vac,1},
+(5)
+where we normalize the photocurrent. If the electric field of the local oscillator is considerably
+greater than the vacuum field, the terms containig α have physical significance. When the
+constant dc current T|α|2
+2
+is neglected, Eq. 5 yields
+ˆIo
+1(z1, Z1) = |α|
+√
+2[
+√
+Teiφ{(µe−ik(Z1+z1) − e−ik(Z1−z1)R
+�
+Rm)ˆa1 − e−ik(Z1−z1)ˆa2}
++
+√
+Te−iφ{(µeik(Z1+z1) − eik(Z1−z1)R
+�
+Rm)ˆa†
+1 − e−k(Z1−z1)ˆa†
+2}
++ eiφTˆb + e−iφTˆb† + eiφeik(ZM−Z1)�
+TRTm ˆd + e−iφe−ik(ZM−z1)�
+TRTm ˆd†],
+(6)
+We then evaluate the square of the photocurrent to determine the fluctuation. After
+squaring Eq. 6, we find the photocurrent fluctuation as follows:
+⟨(ˆIo
+1)2⟩ = |α|2T
+2
+{1 + µ2 − 2µR
+�
+Rm cos(2kz1)}
+(7)
+If µ = 0, the fluctuation value from Eq. 7 becomes |α|2T
+2 , which is the square of the con-
+stant dc current
+√
+T|α|
+√
+2 . In other words, if we directly measure the laser intensity fluctuation,
+the fluctuation is dependent on the distance (z1) between the mirror and detector. Even
+in the photo counting experiment, the photon number fluctuation is related to the vacuum
+fluctuation; therefore, the photon number fluctuation is also dependent on the distance z1.
+If we consider practical limits such as finite linewidth and finite absorption length, Eq. 7
+will change as follows[2, 8].
+⟨(ˆIo
+1)2⟩P = |α|2T
+2
+{1 + µ2 − 2µR
+�
+Rme−z2
+1∆k2
+× κ[cos(2k0z1 + φ0) − e−κD cos(2k0(z1 + D) + φ0)]
+�
+4k2
+0 + κ2
+},
+(8)
+4
+
+where ∆k is the line width of the local oscillator beam with Gaussian line width distribution
+functions. κ is the absorption coefficient, D is the detector active length, and φ0 = arctan 2k
+κ .
+We assumed that the probability that a photon is converted into an electron hole pair at
+distance η from the surface of the detector’s active region is κe−κη[9].
+The two coefficients √Rm and µ depend on the mode matching condition. Even when
+we used the total mirror, if the mode from the mirror is not perfectly matched with the
+mode from the laser, the effective reflectance √Rm can not be 1. Furthermore, the mode
+a1 to the mirror is reflected by the mirror and then meets at the detector. At the detector,
+if two counter-propagating modes are not exactly matched, the coefficient µ cannot be 1.
+To evaluate this mode matching condition, we assume that the amplitude envelope of the
+electromagnetic wave in the transverse plane is given by a Gaussian function.
+Considering the Gaussian modes [10]
+E(ρ, z) = E0
+w0
+w(z) exp[−
+ρ2
+w(z)2] exp[−ikz − ik
+ρ2
+2R(z) + iζ(z)]
+(9)
+, where w0 is the radius of the beam waist and
+w(z) = w0
+�
+1 + ( z
+z0
+)2
+R(z) = z(1 + (z0
+z )2)
+ζ(z) = tan−1 z
+z0
+(10)
+and z0 is defined as follows:
+z0 = π
+λw2
+0.
+(11)
+First, we assume that the laser and vacuu modes have the same beam waist w0 at the
+detector. Then the laser and vacuum modes are perfectly matched; thus, √Rm = 1. On
+the other hand, the vacuum Ev(0) starting from the detector propagates to the mirror and
+reflects at the mirror. The returned vacuum Ev(2z1) is not the same Ev(0). The coefficient
+µ can be calculated as follow:
+µ =
+| < Ev(0)Ev(2z1)∗ > |
+�
+< Ev(0)2 >< Ev(2z1)2 >
+=
+(1 + 4z2
+1
+z2
+0 )
+1
+4
+(1 + 5z2
+1
+z2
+0 + 4 z4
+1
+z4
+0 )
+1
+4
+(12)
+5
+
+FIG. 2:
+Mode matching value µ as a function of w0 and z1.
+In Fig. 2, µ is plotted as a function of z1 and w0, where z1 is the distance between the
+mirror and detector We assume that the detector and mirror are large enough that all the
+waves are detected and reflected. If the distance between the mirror and detector and the
+size of the beam waist are small enough, the coefficient µ remains near 1.
+If we consider the case where the vacuum field has waist at the mirror, the coefficient µ
+automatically becomes 1 due to the symmetry, but the vacuum field Ev(z1) at the detector
+does not matche the laser field EL(0). We assumed that the laser field has beam waist w0 at
+the detector, and the vacuum field has a beam waist wm at the mirror. Then the effective
+reflectance √Rm becomes
+�
+Rm =
+| < Ev(z1)EL(0)∗ > |
+�
+< Ev(z1)2 >< EL(0)2 >
+=
+√
+2
+�
+wm
+w0 (1 + z2
+1
+z2m)
+1
+4
+({(1 + w2m
+w2
+0 )2 + z2
+1
+z2
+0 }{1 + z2
+1
+z2m})
+1
+4
+,
+(13)
+where zm = π
+λw2
+m.
+In Fig. 3, √Rm is plotted as a function of z1 and wm, where z1 is the distance between
+the mirror and detector. We set w0 to 100λ. Additionally, we also assume that the detector
+and mirror are large enough that all the waves are detected and reflected. The coefficient
+√Rm can be 1 only when the distance between the mirror and detector is small and the size
+of the beam waist is sufficiently small.
+6
+
+μ
+1
+0.5
+3
+4
+5
+log(
+0m
+3
+入
+log()
+7
+2FIG. 3: Mode matching value √Rm as a function of w0 and z1, with w0 equal to 100λ
+The mode matching condition is crucial for detecting the modulation effect of the vacuum
+fluctuation near the mirror, as denoted by Eq. 8. With the usual setup, we can not satisfy
+the conditions µ = 1 and √Rm = 1. In the next section, we suggest a noble experimental
+setup that satisfies two mode-matching conditions.
+III.
+SET UP FOR MODE MATCHING
+For a laser that has a Gaussian transverse mode, we have to establish a vacuum mode that
+also has a Gaussian transverse mode. Fig. 4 displays the setup for perfect mode matching
+between the laser light mode and a vacuum mode.
+The laser used in the experiment passes through lens L1 and is divided into two by
+the beam splitter (BS1). The laser is a Gaussian beam and it proceeds according to the
+Gaussian approximation. The light passing through BS1 and traveling to mirror M2 reaches
+the partial mirror B and yields a beam waist on the L3 side surface of B. Similarly, the
+light reflecting from the mirror M1 passes through the partial reflector A and yields a beam
+waist on the L2 side surface of A.
+The light passing through A and B passes through the L2 and L3 of the same focal length,
+respectively, and yields another beam waist on the detector surface. The transmittance of
+light passing through A from M1 is almost 0, and the reflectance of light stemming from the
+L2 side is almost 1. In this way, if the mode is perfectly matched using the light passing
+through B and A, an experimental setup can be established wherein one side of the beam
+splitter BS2 is a mirror (A).
+7
+
+/Rm
+1
+0.5
+3
+4
+5
+3
+1og
+wm
+log()
+入
+7
+2Using this method, the degree of mode matching can be increased compared to that
+when the experiment is performed by simply placing a plane mirror on one side of the beam
+splitter. Additionally the experimental constraints caused by the mode matching can be
+overcome. The experimental setup in Fig. 4 enables the measurement of how the vacuum
+fluctuations of the light passing through the beam splitter change when a mirror is placed
+on one side of the beam splitter.
+FIG. 4: Mode matching setup
+IV.
+CONCLUSION AND DISCUSSION.
+The quantum nature of photons is highly dependent on their vacuum fluctuations. Vac-
+uum fluctuations can be directly measured via homodyne detection. The fluctuation of one
+quadrature of the vacuum can be less than that of the usual vacuum, e.g., squeezed vacuum.
+Light intensity fluctuations are also dependent on vacuum fluctuations. Sub-Poisson light
+can be generated by controlling the vacuum fluctuations based on the nonlinear interaction
+of light and matter. In this study, we proposed the modulation of vacuum fluctuations by
+inserting a mirror on the unused part of the beam splitter in a homodyne measuring system.
+Furthermore, we calculated the effect of the line width of the laser and the thickness of the
+detector layer. The line width can be practically reduced to modulate vacuum fluctuations,
+but the decrease of the thickness of the detector to modulate vacuum fluctuations is chal-
+lenging. We calculated the effect of mode matching between the vacuum and light fields and
+8
+
+BS1
+M1
+A
+L1
+L2
+DD1
+M2
+BS2
+B
+L3
+D2showed that the degree of mode matching obtained by adding a simple mirror in the unused
+beam splitter may not be sufficient to modulate the vacuum fluctuations. We present the
+perfect mode matching method for the vacuum and light fields. Then, the light intensity
+fluctuations can be reduced by inserting a beam splitter and a mirror. We still require a
+detector with an active layer thinner than the wavelength to obtain a sub-Poisson light as a
+function of the distance between the mirror and detector. We expect that our simple method
+of reducing vacuum fluctuations will play a great role in quantum information science.
+[1] W. Jhe, A Anderson, E. A. Hinds, D. Meschede, L. Moi, and S. Haroche, Phys. Rev. Lett.
+58, 666 (1987)
+[2] S. H. Youn, J. H. Lee, J. S. Chang, Opt. and Quant. Elec. 27, 355 (1995)
+[3] S. H. Youn, J. H. Lee, J. S. Chang, International Workshop on Squeezed States and Uncer-
+tainty Relations, N95-13921 (1994)
+[4] S. A. Wadood, J. T, Schultz, A. N. Vamivakas, and C.R. Stroud Jr, J. of Mod. Opt. 66, 1116
+(2019)
+[5] A. Yariv, Quantum Electronics 3rd ed., John Wiley & Sons. Inc, (1989)
+[6] P. D. Drummond, Phys. Rev. A 35, 4253 (1987).
+[7] B. Yurke, Phys. Rev. A 32, 311 (1985)
+[8] A. E. Siegman, Laser (Oxford University Press, Oxford, 1986 )
+[9] S. M. Sze,
+Semiconductor Devices Physics and Technology (AT&T Bell Lab. Murray Hill,
+New Jersey, 1985)
+[10] B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics ( Wiley, Nw York, 1991)
+9
+

4dAzT4oBgHgl3EQfffz3/content/tmp_files/load_file.txt

@@ -0,0 +1,179 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf,len=178
+page_content='Engineering sub-Poisson light in a simple mirror and beam  splitter system  Sun-Hyun Youn∗  Department of Physics, Chonnam National University, Gwangju 500-757, Korea  Abstract  Vacuum fluctuation, which is the intrinsic nature of an electric field can be measured via homo-  dyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Moreover, electric field intensity fluctuation are also related to vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  Squeezed vacuum and sub-Poisson light can be obtained by controlling the vacuum fluctuation us-  ing noble nonlinear interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Based on the squeezed vacuum by inserting a mirror on the unused  part of the beam splitter was proposed in 1994, we present the mode matching method for the  vacuum and light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Light intensity fluctuations also can be reduced by inserting a mirror on  the unused part of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' To obtain sub-Poisson light as a function of the distance  between the mirror and detector, a detector with a thinner active layer than the wavelength is  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  PACS numbers: 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='-a,03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='+k, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='Yz  Keywords: Quantum optics, Squeezed State, Vacuum fluctuation, Sub-Poisson, Beam splitter and Mirror  ∗ E-mail: sunyoun@jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='kr, fax: +82-62-530-3369  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='01455v1  [quant-ph]  4 Jan 2023  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  INTRODUCTION  When a single photon is in a particular mode, according to the particle nature of light,  photons will be sequentially found in that mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The probability of finding a photon is  proportional to the absolute square of the wave function related to the electromagnetic  wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Vacuum fluctuations are related to the spatial characteristics of the electromagnetic  wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The spontaneous decay caused by the vacuum can be suppressed in cavities [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The-  oretical and experimental studies have beem conducted on methods to change the vacuum  fluctuations near mirrors[2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  In this study, in contrast to previous studies on the vacuum noise characteristics of  light using a homodyne detector, we calculate the intensity fluctuations when photons are  directly measured using photon counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The obtained results are similar to those obtained  in previous studies, but herein we predict the results considering mode matching in the  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  In section II, the fluctuation of light that can be measured using a detector is calculated  with a mirror placed on one side of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' In section III, an experimental device  is proposed for perfect mode matching, and in the last section, the practical limits of the  vacuum fluctuation near the mirror are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  VACUUM FLUCTUATION NEAR A MIRROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  An electric field can be written as  ˆEL = ˆEcl + ˆEQ,  (1)  where  ˆEcl = i  �  ℏω  2ϵ0V (αei(ωt−k0z) − α∗ei(ωt−k0z))⃗x,  ˆEQ = i  �  k  �  ℏωk  2ϵ0V (ˆbke−i(ωkt−kz) − ˆb†  kei(ωkt−kz))⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  (2)  Here, k0 and ω are the wave number and angular frequency of the laser, respectively, ℏ and  ϵ0 have usual meanings, and V is the normalization volume[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Considering the laser mode  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 1: Vacuum mode relations in the beam splitter with a mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' BS: Beam splitter, M: mirror  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 1, the modes aout  1  and aout  2  can be written as  aout  1  =  √  Tb +  √  Rc,  aout  2  = −  √  Rb +  √  Tc,  (3)  where the modes c and cout can be written as  c =  �  Tmd −  �  Rmcout,  cout =  √  Ra1 +  √  Ta2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  Then the electric field in fluctuating vacuum modes at a1 is  ˆE(+)  vac,1 =  �  k  i  �  ℏωk  4ϵ0V {  √  Tˆb†  kei(ωkt−kZ1) + µˆa†  1,kei(ωkt+kz1)  −R  �  Rmˆa†  1,kei(ωkt−kz1) −  √  RT  �  Rmˆa†  2,kei(ωkt−kz1) +  �  RTm ˆd†  kei(ωkt−kZM)} (4)  where Rm(Tm) is the reflectance(transmittance) of the mirror and R(T) is the reflectance  (transmittance) of the beam splitter, z1(Z1) is the distance from the mirror (laser) to the  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' ZM is related to the vacuum source behind the mirror and it can be any number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  We add the factor  1  √  2 for the normalization of the vacuum fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The vacuum mode  (ˆa†  1ei(ωt−kz1)) at the detector is the reflected vacuum mode (ˆa†  1ei(ωt+kz1)) at the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' If two  modes are perfectly matched the µ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 4 is 1 and the two counterpropagating modes  yield the standing wave mode[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' If µ = 0, the fluctuation value from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 7 becomes |α|2T  2 ,  3  b  BS  α2  C  d  ino  ↑  M  ino  a1  ait is the square of the constant dc current T|α|2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' In other words, if we directly measure the  fluctuation of the laser intensity, the fluctuation is dependent on the distance (z1) between  the mirror and the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  Even in photo counting experiments, the photon number  fluctuation is related to the vacuum fluctuation, therefor, the photon number fluctuation is  also depend on the distance z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  If we used the photodetetion theory [6] with instantaneous response of the photodetector  [7],  ˆI1 = {  √  T ˆE(+)  cl  + ˆE(+)  vac,1} × {  √  T ˆE(−)  cl  + ˆE(−)  vac,1},  (5)  where we normalize the photocurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' If the electric field of the local oscillator is considerably  greater than the vacuum field, the terms containig α have physical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' When the  constant dc current T|α|2  2  is neglected, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 5 yields  ˆIo  1(z1, Z1) = |α|  √  2[  √  Teiφ{(µe−ik(Z1+z1) − e−ik(Z1−z1)R  �  Rm)ˆa1 − e−ik(Z1−z1)ˆa2}  +  √  Te−iφ{(µeik(Z1+z1) − eik(Z1−z1)R  �  Rm)ˆa†  1 − e−k(Z1−z1)ˆa†  2}  + eiφTˆb + e−iφTˆb† + eiφeik(ZM−Z1)�  TRTm ˆd + e−iφe−ik(ZM−z1)�  TRTm ˆd†],  (6)  We then evaluate the square of the photocurrent to determine the fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' After  squaring Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 6, we find the photocurrent fluctuation as follows:  ⟨(ˆIo  1)2⟩ = |α|2T  2  {1 + µ2 − 2µR  �  Rm cos(2kz1)}  (7)  If µ = 0, the fluctuation value from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 7 becomes |α|2T  2 , which is the square of the con-  stant dc current  √  T|α|  √  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' In other words, if we directly measure the laser intensity fluctuation,  the fluctuation is dependent on the distance (z1) between the mirror and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Even  in the photo counting experiment, the photon number fluctuation is related to the vacuum  fluctuation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' therefore, the photon number fluctuation is also dependent on the distance z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  If we consider practical limits such as finite linewidth and finite absorption length, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 7  will change as follows[2, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  ⟨(ˆIo  1)2⟩P = |α|2T  2  {1 + µ2 − 2µR  �  Rme−z2  1∆k2  × κ[cos(2k0z1 + φ0) − e−κD cos(2k0(z1 + D) + φ0)]  �  4k2  0 + κ2  },  (8)  4  where ∆k is the line width of the local oscillator beam with Gaussian line width distribution  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' κ is the absorption coefficient, D is the detector active length, and φ0 = arctan 2k  κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  We assumed that the probability that a photon is converted into an electron hole pair at  distance η from the surface of the detector’s active region is κe−κη[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  The two coefficients √Rm and µ depend on the mode matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Even when  we used the total mirror, if the mode from the mirror is not perfectly matched with the  mode from the laser, the effective reflectance √Rm can not be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Furthermore, the mode  a1 to the mirror is reflected by the mirror and then meets at the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' At the detector,  if two counter-propagating modes are not exactly matched, the coefficient µ cannot be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  To evaluate this mode matching condition, we assume that the amplitude envelope of the  electromagnetic wave in the transverse plane is given by a Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  Considering the Gaussian modes [10]  E(ρ, z) = E0  w0  w(z) exp[−  ρ2  w(z)2] exp[−ikz − ik  ρ2  2R(z) + iζ(z)]  (9)  , where w0 is the radius of the beam waist and  w(z) = w0  �  1 + ( z  z0  )2  R(z) = z(1 + (z0  z )2)  ζ(z) = tan−1 z  z0  (10)  and z0 is defined as follows:  z0 = π  λw2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  (11)  First, we assume that the laser and vacuu modes have the same beam waist w0 at the  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Then the laser and vacuum modes are perfectly matched;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' thus, √Rm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' On  the other hand, the vacuum Ev(0) starting from the detector propagates to the mirror and  reflects at the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The returned vacuum Ev(2z1) is not the same Ev(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The coefficient  µ can be calculated as follow:  µ =  | < Ev(0)Ev(2z1)∗ > |  �  < Ev(0)2 >< Ev(2z1)2 >  =  (1 + 4z2  1  z2  0 )  1  4  (1 + 5z2  1  z2  0 + 4 z4  1  z4  0 )  1  4  (12)  5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 2:  Mode matching value µ as a function of w0 and z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 2, µ is plotted as a function of z1 and w0, where z1 is the distance between the  mirror and detector We assume that the detector and mirror are large enough that all the  waves are detected and reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' If the distance between the mirror and detector and the  size of the beam waist are small enough, the coefficient µ remains near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  If we consider the case where the vacuum field has waist at the mirror, the coefficient µ  automatically becomes 1 due to the symmetry, but the vacuum field Ev(z1) at the detector  does not matche the laser field EL(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' We assumed that the laser field has beam waist w0 at  the detector, and the vacuum field has a beam waist wm at the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Then the effective  reflectance √Rm becomes  �  Rm =  | < Ev(z1)EL(0)∗ > |  �  < Ev(z1)2 >< EL(0)2 >  =  √  2  �  wm  w0 (1 + z2  1  z2m)  1  4  ({(1 + w2m  w2  0 )2 + z2  1  z2  0 }{1 + z2  1  z2m})  1  4  ,  (13)  where zm = π  λw2  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 3, √Rm is plotted as a function of z1 and wm, where z1 is the distance between  the mirror and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' We set w0 to 100λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Additionally, we also assume that the detector  and mirror are large enough that all the waves are detected and reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The coefficient  √Rm can be 1 only when the distance between the mirror and detector is small and the size  of the beam waist is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  6  μ  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='5  3  4  5  log(  0m  3  入  log()  7  2FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 3: Mode matching value √Rm as a function of w0 and z1, with w0 equal to 100λ  The mode matching condition is crucial for detecting the modulation effect of the vacuum  fluctuation near the mirror, as denoted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' With the usual setup, we can not satisfy  the conditions µ = 1 and √Rm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' In the next section, we suggest a noble experimental  setup that satisfies two mode-matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  SET UP FOR MODE MATCHING  For a laser that has a Gaussian transverse mode, we have to establish a vacuum mode that  also has a Gaussian transverse mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 4 displays the setup for perfect mode matching  between the laser light mode and a vacuum mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  The laser used in the experiment passes through lens L1 and is divided into two by  the beam splitter (BS1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The laser is a Gaussian beam and it proceeds according to the  Gaussian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The light passing through BS1 and traveling to mirror M2 reaches  the partial mirror B and yields a beam waist on the L3 side surface of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Similarly, the  light reflecting from the mirror M1 passes through the partial reflector A and yields a beam  waist on the L2 side surface of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  The light passing through A and B passes through the L2 and L3 of the same focal length,  respectively, and yields another beam waist on the detector surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The transmittance of  light passing through A from M1 is almost 0, and the reflectance of light stemming from the  L2 side is almost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' In this way, if the mode is perfectly matched using the light passing  through B and A, an experimental setup can be established wherein one side of the beam  splitter BS2 is a mirror (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  7  /Rm  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='5  3  4  5  3  1og  wm  log()  入  7  2Using this method, the degree of mode matching can be increased compared to that  when the experiment is performed by simply placing a plane mirror on one side of the beam  splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Additionally the experimental constraints caused by the mode matching can be  overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The experimental setup in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 4 enables the measurement of how the vacuum  fluctuations of the light passing through the beam splitter change when a mirror is placed  on one side of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 4: Mode matching setup  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  CONCLUSION AND DISCUSSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  The quantum nature of photons is highly dependent on their vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Vac-  uum fluctuations can be directly measured via homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The fluctuation of one  quadrature of the vacuum can be less than that of the usual vacuum, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=', squeezed vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  Light intensity fluctuations are also dependent on vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Sub-Poisson light  can be generated by controlling the vacuum fluctuations based on the nonlinear interaction  of light and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' In this study, we proposed the modulation of vacuum fluctuations by  inserting a mirror on the unused part of the beam splitter in a homodyne measuring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  Furthermore, we calculated the effect of the line width of the laser and the thickness of the  detector layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' The line width can be practically reduced to modulate vacuum fluctuations,  but the decrease of the thickness of the detector to modulate vacuum fluctuations is chal-  lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' We calculated the effect of mode matching between the vacuum and light fields and  8  BS1  M1  A  L1  L2  DD1  M2  BS2  B  L3  D2showed that the degree of mode matching obtained by adding a simple mirror in the unused  beam splitter may not be sufficient to modulate the vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' We present the  perfect mode matching method for the vacuum and light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Then, the light intensity  fluctuations can be reduced by inserting a beam splitter and a mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' We still require a  detector with an active layer thinner than the wavelength to obtain a sub-Poisson light as a  function of the distance between the mirror and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' We expect that our simple method  of reducing vacuum fluctuations will play a great role in quantum information science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Jhe, A Anderson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
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+page_content=' Meschede, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Moi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Haroche, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
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+page_content=' T, Schultz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Vamivakas, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Stroud Jr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' of Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' 66, 1116  (2019)  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Yariv, Quantum Electronics 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=', John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Inc, (1989)  [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Drummond, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' A 35, 4253 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content='  [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Yurke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' A 32, 311 (1985)  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Siegman, Laser (Oxford University Press, Oxford, 1986 )  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Sze,  Semiconductor Devices Physics and Technology (AT&T Bell Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Murray Hill,  New Jersey, 1985)  [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Saleh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}
+page_content=' Teich, Fundamentals of Photonics ( Wiley, Nw York, 1991)  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'}

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+1
+Plug-in Channel Estimation with Dithered
+Quantized Signals in Spatially Non-Stationary
+Massive MIMO Systems
+Tianyu Yang1, Johannes Maly2,3, Sjoerd Dirksen4,
+and Giuseppe Caire1
+Abstract
+As the array dimension of massive MIMO systems increases to unprecedented levels, two problems
+occur. First, the spatial stationarity assumption along the antenna elements is no longer valid. Second,
+the large array size results in an unacceptably high power consumption if high-resolution analog-to-
+digital converters are used. To address these two challenges, we consider a Bussgang linear minimum
+mean square error (BLMMSE)-based channel estimator for large scale massive MIMO systems with
+one-bit quantizers and a spatially non-stationary channel. Whereas other works usually assume that the
+channel covariance is known at the base station, we consider a plug-in BLMMSE estimator that uses an
+estimate of the channel covariance and rigorously analyze the distortion produced by using an estimated,
+rather than the true, covariance. To cope with the spatial non-stationarity, we introduce dithering into the
+quantized signals and provide a theoretical error analysis. In addition, we propose an angular domain
+fitting procedure which is based on solving an instance of non-negative least squares. For the multi-user
+data transmission phase, we further propose a BLMMSE-based receiver to handle one-bit quantized data
+signals. Our numerical results show that the performance of the proposed BLMMSE channel estimator
+is very close to the oracle-aided scheme with ideal knowledge of the channel covariance matrix. The
+BLMMSE receiver outperforms the conventional maximum-ratio-combining and zero-forcing receivers
+in terms of the resulting ergodic sum rate.
+1Communications and Information Theory Group (CommIT), Technische Universit¨at Berlin, 10587 Berlin, Germany (e-mail:
+{tianyu.yang, caire}@tu-berlin.de).
+2Ludwig-Maximilians-Universit¨at Munich, 80333 Munich, Germany (e-mail: [email protected]).
+3Munich Center for Machine Learning (MCML).
+4Utrecht University, 3584 CD Utrecht, Netherlands (e-mail: [email protected]).
+This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which
+this version may no longer be accessible.
+arXiv:2301.04641v1  [cs.IT]  11 Jan 2023
+
+2
+Index Terms
+Extra-Large Scale Massive MIMO, Spatially Non-Stationary, One-Bit Quantization, Dithering, Buss-
+gang Linear MMSE (BLMMSE).
+I. INTRODUCTION
+Massive multiple-input-multiple-output (MIMO) has been vastly researched and considered as
+an essential technology in 5G wireless communication systems within sub-6 GHz bands [1–3].
+Benefiting from the large number (tens to hundreds) of antennas at the base station (BS) array,
+dozens of users can be served in the same time-frequency slots. This results in higher spectrum
+and energy efficiency due to the spatial multiplexing and high array gain [1, 4]. Theory shows that
+by increasing the array dimension, i.e., the number of antenna elements, it is possible to achieve
+higher data rates and to mitigate the impacts of inter-cell interference and thermal noise [5].
+Nevertheless, as the array dimension increases, two new challenges occur. First, some inherent
+properties of the channel environment change compared to small-scale MIMO, so that the basic
+assumptions of massive MIMO design are no longer valid for large arrays. Specifically, most
+existing massive MIMO works are based on the assumption of a spatially stationary channel,
+where all antenna elements observe the same far-field propagation from the channel scatters [6–9].
+However, in the very large antenna array regime, spatial non-stationarity has been experimentally
+observed [10]. Two main reasons for this non-stationarity are further discussed in [11–13]. First,
+with large arrays, the distance between the BS array and some scattering clusters may be smaller
+than the Rayleigh distance. As a consequence, user signals impinging onto the BS array cannot
+be assumed as far-field propagation and have a spherical wavefront instead of a plane wavefront.
+Second, due to the physical large size of the array, some of the scattering clusters may only be
+visible to a part of the array. Furthermore, a new deployment of large arrays that are usually
+integrated into large structures, e.g., along the walls of buildings [14], was considered as an
+extension of massive MIMO and referred to as extra-large scale massive MIMO (XL-MIMO) in
+[15]. It is also pointed out in [15] that due to the large dimension (tens of meters) of XL-MIMO,
+spatially non-wide sense stationary (non-WSS) characteristics appear along the array.
+Aside from the spatially non-WSS property of large antenna arrays, a second major concern is
+the hardware cost and power consumption of high-resolution analog-to-digital converters (ADCs).
+Commercial high-resolution ADCs (12 to 16 bits) are expensive and their power consumption
+
+3
+grows exponentially in terms of the number of quantization bits [16]. This problem is even more
+severe for wideband systems, where the power consumption of high-resolution ADCs increases
+linearly with the signal bandwidth due to required higher sampling rates [17]. To alleviate the
+issue of high power consumption, low-resolution ADCs (e.g., 1-3 bits) are utilized for massive
+MIMO systems [18–20] and it is shown in [18, 19] that the capacity loss due to the coarse
+quantization is approximately equal to only π/2 at low signal-to-noise ratios (SNRs). In massive
+MIMO systems the SNR per antenna element may be relatively low, while still achieving an
+overall large spectral efficiency over data stream due to the large number of antennas per user
+(data stream), such that both spatial multiplexing gain and array gain are achieved.
+A. Contributions
+Accurate estimation of channel state information (CSI) at the BS is a key factor to achieve
+the potential benefits of massive MIMO systems. Taking the spatially non-WSS property into
+account, an adaptive grouping sparse Bayesian learning scheme was proposed for uplink channel
+estimation in [21]. A model-driven deep learning-based channel reconstruction scheme was
+proposed in [22]. On the other hand, many recent works have investigated channel estimators
+with one-bit quantized signals in massive MIMO systems, see e.g., [23, 24] and references
+therein. Very recently, in [25] a covariance recovery scheme for one-bit sampled non-stationary
+signals with time-varying sampling thresholds was proposed, where a modified arcsine law was
+further generalized to fit the non-stationary case. However, the study in [25] is not within the
+massive MIMO regime. To the best of our knowledge, no work has addressed the problem of
+channel estimation with both low-resolution quantization and spatially non-WSS channels in
+massive MIMO systems. To fill this gap, in this paper we adopt the Bussgang linear minimum
+mean square estimation (BLMMSE) method that was initially proposed in [23], and propose
+a BLMMSE-based “plug-in” channel estimator for one-bit Massive MIMO systems with the
+spatially non-WSS property.1 Our main contributions are summarized as follows.
+• We adopt a BLMMSE channel estimator to deal with the one-bit quantized signal. However,
+in contrast to [23] that assumes exact knowledge of the channel covariance at the BS, we
+1By “plug-in” we mean that the BLMMSE requires the knowledge of the channel covariance, which is typically assumed to
+be known. However, in our setting, also the channel covariance needs to be estimated from low-resolution quantized samples.
+The plug-in estimator consists of using the estimated covariance “as if” it were the true one, in the BLMMSE.
+
+4
+propose a “plug-in” version that instead uses an estimate of the channel covariance. Our first
+contribution is a theoretical analysis of the distortion caused by the use of this estimate: we
+estimate the mean squared distance between the BLMMSE estimate based on an estimate of
+the channel covariance versus the BLMMSE estimate based on the true channel covariance
+(see Lemma 1).
+• Our second contribution is a method to estimate the channel covariance matrix based on
+one-bit samples. We introduce dithering into the one-bit ADCs to cope with the non-Toeplitz
+structure of the channel covariance matrix (resulting from the spatially non-WSS channel).
+We propose a covariance estimator based on dithered quantized samples and derive bounds
+on the estimation error in terms of the maximum norm and the Frobenius norm (Theorem 1).
+By combining this result with the aforementioned bound on the MSE achieved by the
+BLMMSE with given estimated covariance, we derive a bound on the expected MSE of the
+channel estimator in terms of the number of samples used to estimate the channel covariance
+(Theorem 2).
+• We empirically further enhance the proposed channel covariance estimator by exploiting
+the angle domain of the spatially non-WSS channel. Using dictionary functions in the angle
+domain, we formulate the channel covariance estimation as a non-negative least-squares
+problem (NNLS), which can be efficiently solved by a standard numerical NNLS solver,
+e.g., [26], even for very large problem dimensions.
+• We design a linear receiver for the uplink (UL) data transmission phase, based on an estimate
+of the channel matrix obtained in the training phase, to achieve better rate detection and thus
+improve the ergodic sum rate of multi-user severing. Contrary to the conventional maximum-
+ratio-combining (MRC) and zero-forcing (ZF) receivers that do not take the quantization into
+account, the proposed receiver considers the Bussgang decomposition of one-bit quantized
+data signals and uses BLMMSE-based estimation with knowledge of only the estimated
+channel covariance matrix.
+• Our numerical results show that the proposed BLMMSE channel estimator, which uses the
+proposed channel covariance estimator based on dithered quantized samples, is superior
+to benchmark methods and achieves a performance very close to the performance of an
+oracle-aided scheme using the true channel covariance matrix. The proposed BLMMSE-
+based receiver also significantly outperforms MRC and ZF receivers as expected due to its
+
+5
+specific consideration of quantized signals.
+B. Organization
+The rest of this paper is organized as follows. In Section II, we introduce the channel model
+with the spatially non-stationary property. Section III is devoted to the analysis of the BLMMSE
+channel estimator and our results on channel covariance estimation from one-bit quantized
+samples. In Section IV, we propose the BLMMSE receiver for the data transmission phase
+to obtain a higher sum rate. The numerical results are then provided in Section V. Finally, in
+Section VI we conclude our work and provide a discussion of possible future research directions.
+C. Notation
+For any N ∈ N we write [N] = {1, 2, . . . , N}. We use lower-case, bold lower-case, and
+bold upper-case letters to denote scalars, column vectors, and matrices, respectively. The trace,
+transpose and Hermitian transpose are respectively denoted by tr(·), (·)T and (·)H. E[·] returns the
+mathematical expectation. diag(A) gives a diagonal matrix with diagonal of A, while diag(a)
+denotes the diagonal matrix with diagonal equal to a. We denote the M × M identity matrix by
+IM. The i-th element of a vector a is denoted by [a]i, while the i-th row and column of a matrix
+A are respectively denoted by [A]i,· and [A]·,i. An all-zero matrix is denoted by 0. ∥a∥2 denotes
+the Euclidean norm of a vector a. ∥A∥F, ∥A∥, and ∥A∥∞ denote the Frobenius, operator, and
+maximum norms of a matrix A. We use ⟨A, B⟩F := tr(AHB) to denote the Frobenius inner
+product. We furthermore use a ≲ b to abbreviate a ≤ Cb, for some absolute constant C > 0.
+II. SYSTEM MODEL WITH SPATIALLY NON-STATIONARY CHANNEL
+Consider a BS equipped with M antennas in a uniform linear array (ULA). We assume that
+the channel scattering clusters consist of common and local clusters, where common clusters
+are visible to all antennas while local clusters are only visible to a sub-array. An illustration of
+the considered scattering geometry is shown in Fig. 1. The channel vector resulting from the
+contribution of the common clusters at the n-th time slot is given by
+hc
+n =
+Lc
+�
+i=1
+ρc
+i(n)a(θc
+i),
+(1)
+where Lc denotes the total number of multipaths in the common clusters, ρc
+i(n) ∼ CN(0, γc
+i )
+is the i-th complex channel gain of the common clusters with its power γc
+i , θc
+i is the i-th
+
+6
+User
+Local Cluster
+BS 
+ULA
+Local Cluster
+Common 
+Cluster
+Fig. 1: Illustration of the studied large-scale Massive MIMO system in ULA with spatially non-
+WSS channel, where local clusters are only visible to a part of antenna elements while the
+common clusters are visible to the whole array.
+angle of arrival (AoA), and where a(θ) ∈ CM×1 is the steering vector, whose m-th entry is
+[a(θ)]m
+= ejπ(m−1) sin(θ) by assuming that the antenna spacing is equal to half of the carrier
+wavelength, for all m ∈ M, where M = [M] is the antenna index set of all M antennas.
+Assume that there are L local clusters and that each local cluster is visible to a consecutive sub-
+array. The i-th local cluster is thus visible to M l
+i antennas with index set Ml
+i, where M l
+i = |Ml
+i|.
+The channel vector resulting from the contribution of the paths in the i-th local cluster at the
+n-th time slot is given by
+hl
+n,i =
+Ll
+j
+�
+j=1
+ρl
+ij(n)Sia(θl
+ij),
+(2)
+where Ll
+i denotes the total number of multipaths in the i-th local cluster, ρl
+ij(n) ∼ CN(0, γl
+ij) is
+the j-th complex channel gain of the i-th local cluster with its power γl
+ij, θl
+ij is the AoA, and
+where Si ∈ CM×M is the diagonal selection matrix indicating the visible sub-array of the i-th
+local cluster, whose diagonal is defined as
+[Si]m,m =
+�
+�
+�
+�
+�
+1,
+m ∈ Ml
+i
+0,
+m ∈ M \ Ml
+i.
+(3)
+We further assume that the channel gains of different paths in all common and local clusters at
+
+7
+each time slot n, {ρc
+i(n)}Lc
+i=1 and
+�
+ρl
+ij(n)
+�Ll
+i
+j=1, ∀i ∈ [L], are uncorrelated2. Note that we implicitly
+assume that the channel geometry and visibility of all clusters do not change over the channel
+geometry coherent time Tc, which is a much longer time period than the channel coherent time
+(see [30] and references therein). Concretely, the Angular Power Spectrum (APS) {γc
+i }Lc
+i=1 and
+�
+γl
+ij
+�Ll
+i
+j=1, ∀i ∈ [L] along with the AoAs {θc
+i}Lc
+i=1 and
+�
+θl
+ij
+�Ll
+j
+j=1 , ∀i ∈ [L] as well as the selection
+matrices {Si}L
+i=1 are constant over Tc. Under these assumptions, the total channel vector at n-th
+time slot hn and the corresponding total channel covariance matrix Ch are given by
+hn = hc
+n +
+L
+�
+i=1
+hl
+n,i,
+(4)
+Ch = E[hnhH
+n] = Chc +
+L
+�
+i=1
+Chl
+i
+(5)
+=
+Lc
+�
+i=1
+γc
+i a(θc
+i)a(θc
+i)H +
+L
+�
+i=1
+Ll
+j
+�
+j=1
+γl
+ijSia(θl
+ij)a(θl
+ij)HSH
+i
+(6)
+= Acdiag(γc)(Ac)H +
+L
+�
+i=1
+SiAl
+idiag(γl
+i)(Al
+i)HSH
+i ,
+(7)
+where Ac := [a(θc
+1), . . . , a(θc
+Lc)], γc := [γc
+1, . . . , γc
+Lc]T and Al
+i := [a(θl
+i,1), . . . , a(θl
+i,Ll
+i)], γl
+i :=
+[γl
+i,1, . . . , γl
+i,Ll
+i]T, ∀i ∈ [L].
+The total channel power gain of all common clusters and all local clusters are given by
+P c =
+Lc
+�
+i=1
+γc
+i
+and
+P l =
+L
+�
+i=1
+P l
+i =
+L
+�
+i=1
+Ll
+j
+�
+j=1
+γl
+ij,
+(8)
+where we assume that P c and P l are normalized such that max(diag(Ch)) = 1. To help
+readability, Table I summarizes the model notation.
+III. CHANNEL ESTIMATION WITH ONE-BIT SAMPLES
+For a generic user under a normalized pilot, the BS receives at the n-th time slot the signal
+yn = hn + nn,
+(9)
+where hn ∼ CN(0, Ch) is the M × 1 channel vector and nn ∼ CN(0, N0IM) is additive white
+Gaussian noise (AWGN) with noise power N0. The SNR is thus defined by 1/N0 due to the
+2Note that this is a standard assumption specified in, e.g., the channel models of 3GPP standard TR 38.901 [27] and TR
+25.996 [28]. This assumption is also implicitly included in the documentation of the well-known channel simulator QuaDRiGa
+[29].
+
+8
+L
+Number of local clusters
+Lc, Ll
+i
+Number of multipaths of common and local clusters
+ρc
+i, ρl
+ij
+Complex channel gain of common and local clusters
+γc
+i, γl
+ij
+APS of common and local clusters
+Si
+Diagonal selection matrices of local clusters
+θc
+i, θl
+ij
+AoAs of common and local clusters
+Ac, Al
+i
+Matrices of steering vectors of common and local clusters
+TABLE I: Summary of the used notations
+assumption that max(diag(Ch)) = 1. After one-bit ADC the quantized signal becomes
+rn = Q(yn),
+(10)
+where Q(·) is a suitable one-bit quantizer that is applied separately to the real and imaginary
+part. One popular instance for such a quantizer is the complex-sign operator [31]
+rnd
+n = csign(yn) = 1
+√
+2
+�
+sign(Re(yn)) + j sign(Im(yn))
+�
+,
+(11)
+which quantizes the entries of Re(yn) and Im(yn) independently, i.e., the sign-function
+sign: R → {−1, 1}
+sign(x) =
+�
+�
+�
+�
+�
+1
+x ≥ 0
+−1
+x < 0
+(12)
+acts componentwise (memoryless scalar quantization). We use the superscript “nd” (“non-dithered”)
+in (11) since the sign-function is applied directly to the samples without dithering. Note that
+this type of one-bit quantization looses any scaling information.
+A. Bussgang LMMSE channel estimator
+We consider channel estimation for a generic time slot. Thus, we ignore the time slot index
+n for simplicity. In order to estimate the channel vector h from a quantized sample r, we first
+transfer the nonlinear quantizer operation to a statistically equivalent linear formulation via the
+well-known Bussgang decomposition [32], which yields
+r = Q(y) = Ay + q
+(13)
+= Ah + An + q
+(14)
+= Ah + �n,
+(15)
+where the linear operator A is called the Bussgang gain, q is a mean-zero random vector that
+is uncorrelated with y, and �n := An+q is the total noise. To enforce q to be uncorrelated with
+
+9
+y, the Bussgang gain A is chosen to minimize the power of the equivalent quantization noise
+[33] such that
+A = E
+�
+ryH�
+E
+�
+yyH�−1 = CryC−1
+y ,
+(16)
+where Cry = E
+�
+ryH�
+denotes the covariance between the quantized signal r and the received
+signal y. The so-called BLMMSE estimator [23] of the channel vector h given the quantized
+signal r is then expressed as
+�hBLM = ChrC−1
+r r.
+(17)
+Note that this is not the optimal MMSE estimator since q is not Gaussian noise. We however
+know that the vector q is uncorrelated with the vector y and one can prove that q is also
+uncorrelated with the channel vector h, see [23, App. A] for the proof of E[hqH] = 0. Thus, h
+is uncorrelated with the total noise �n and consequently we obtain from (15) that
+Chr = ChAH.
+(18)
+Similarly as in [23], A and Cr can be easily computed as follows. For the one-bit quantizer in
+(11) and Gaussian inputs, Cry is given as [34], [35, Ch.12]
+Cry =
+�
+2
+πdiag(Cy)− 1
+2Cy
+(19)
+and combining (19) and (16) we obtain
+A =
+�
+2
+πdiag(Cy)− 1
+2.
+(20)
+Furthermore, Cr can be obtained using the map Parcsine(·) by the arcsine law [34, 36] as
+Cr = Parcsine(Cy)
+= 2
+π
+�
+arcsin
+�
+diag(Cy)− 1
+2Re(Cy)diag(Cy)− 1
+2
+�
++ j arcsin
+�
+diag(Cy)− 1
+2Im(Cy)diag(Cy)− 1
+2
+� �
+.
+(21)
+With the BLMMSE estimator in hand, (17) has a closed form that only depends on Cy =
+Ch + N0IM. Whereas the noise power N0 is normally assumed to be known at the BS3, the
+channel covariance matrix Ch still needs to be estimated from samples to finally apply the
+BLMMSE estimator (17). Consider an “plug-in estimator” �Cy of Cy, we define the estimated
+3This can be achieved via, e.g., low rate control channel.
+
+10
+channel vector as
+�h = �Chr �C−1
+r r,
+(22)
+where �Chr and �Cr are the estimators of Chr and Cr obtained by replacing Cy by its estimator
+�Cy, i.e.,
+�Chr = �Ch �AH,
+�Ch = �Cy−N0IM,
+�A =
+�
+2
+πdiag(�Cy)− 1
+2,
+�Cr = Parcsine(�Cy). (23)
+The following lemma controls the estimation error in (17) if an estimator �Cy of Cy is used.
+Lemma 1: There are absolute constants c1, c2, C > 0 such that the following holds. Let
+θ ∈ (0, 1) be fixed. Assume that
+����
+�
+diag(Cy)− 1
+2Cydiag(Cy)− 1
+2
+�
+i,j
+���� ≤ 1 − θ,
+for all i ̸= j
+(24)
+and
+min
+i∈[M] |[Cy]i,i| ≥ θ,
+λmin(Cr) ≥ θ,
+(25)
+where λmin(·) gives the minimal eigenvalue of the matrix. Consider εF > 0, ε∞ > 0 such that
+∥�Cy − Cy∥F < εF,
+∥�Cy − Cy∥∞ < ε∞
+and assume that
+ε∞ ≤ c1 min
+�
+εF
+∥Cy∥F
+,
+θ3
+∥Cy∥∞
+, θ, 1
+�
+and εF ≤ c2 min
+�
+θ4,
+θ6∥Ch∥F
+max{1, ∥Ch∥} ∥Cy∥
+�
+.
+(26)
+Then,
+E
+�����h − �hBLM���
+2
+2
+�
+≤ Cθ−6 max{1, ∥Ch∥} ∥Ch∥FεF,
+(27)
+where the expectation is taken with respect to r.
+Proof: See Appendix A.
+Remark 1: Let us briefly comment on the assumptions in Lemma 1. As the construction
+of the BLMMSE involves the inverses of diag(Cy) and Cr, it is to be expected that in the
+situation that these matrices are near-singular, a small error in the estimation of the covariance
+can lead to a large difference between �h and �hBLM. This expected behaviour is quantified in
+Lemma 1 using the parameter θ. The lower bound on λmin(Cr) is an implicit condition on
+Cy. To give a more explicit condition, let us write offdiag(Cr) for the off-diagonal part. Using
+that ∥ arcsin(B)∥ ≤ π
+2∥B∥ if ∥B∥∞ ≤ 1 (see [37, Supplementary material]), we can make the
+potentially crude estimate
+λmin(Cr) ≥ λmin(diag(Cr)) − ∥ offdiag(Cr)∥ ≥ 1 − ∥ offdiag(Cy)∥,
+(28)
+
+11
+so that it is sufficient if
+∥ offdiag(Cy)∥ ≤ 1 − θ.
+(29)
+Note that the latter condition also implies (24). Finally, let us comment on the condition linking
+ε∞ and εF in (26). In the application that follows, we will see that the ℓ∞-error achieved by the
+estimator �Cy is a factor M smaller than the achieved Frobenius norm error. As a consequence,
+the relation between ε∞ and εF will be satisfied.
+♦
+B. Channel covariance estimation from quantized samples
+In this part, we present an approach to estimate the covariance matrix Cy from a finite number
+of samples so that we can use the estimate �Cy to apply the plug-in BLMMSE channel estimator in
+(22). Assume that the BS collects N unquantized i.i.d. samples {yn}N
+n=1 for covariance estimation
+and applies coarse quantization in the ADCs. In the case of a spatially WSS channel, the diagonal
+of Ch is constant and the (non-dithered) one-bit samples rnd
+n defined in (11) can be used. Defining
+the sample covariance of the quantized samples
+�Cnd
+r = 1
+N
+N
+�
+n=1
+rnd
+n
+�
+rnd
+n
+�H ,
+(30)
+the true covariance matrix Cy can then be estimated via the arcsin-law [31, 36, 37]
+�Cnd
+y = sin
+�π
+2 Re
+�
+�Cnd
+r
+��
++ j sin
+�π
+2 Im
+�
+�Cnd
+r
+��
+.
+(31)
+Due to the spatially non-WSS property in our model, however, it is seen from the formulation in
+(6) that the channel covariance may have a non-constant diagonal and non-Toeplitz structure. In
+such a scenario, the estimator in (31) will perform poorly since it enforces a constant diagonal.
+To overcome this limitation of the quantizer csign, we will introduce random dithering [38–
+40]. The beneficial effect of dithering in memoryless one-bit quantization was recently rigorously
+analyzed in the context of one-bit compressed sensing, see, e.g., [41–46]. We will adapt a
+covariance estimator from [37] that uses two-bit dithered quantized samples. Specifically, we
+assume that the real and imaginary parts of each entry are quantized independently with two
+independent dithers, so that we are given the (dithered) four-bit samples
+�
+Re(rd
+n), Im(rd
+n), Re(�rd
+n), Im(�rd
+n)
+�
+:=
+(32)
+�
+sign
+�
+Re(yn) + τ Re
+n
+�
+, sign
+�
+Im(yn) + τ Im
+n
+�
+, sign
+�
+Re(yn) + �τ Re
+n
+�
+, sign
+�
+Im(yn) + �τ Im
+n
+��
+,
+
+12
+RF
+S/H
+1-bit ADC
+DSP
+S/H
+Switch
+S/H
+1-bit ADC
+S/H
+Switch
+DG 
+Fig. 2: Illustration of the implementation of dithered one-bit quantizer in the m-th antenna chain.
+where the real dithering vectors τ Re
+n , τ Im
+n , �τ Re
+n , �τ Im
+n
+∈ RM, for n ∈ [N], are independent and
+uniformly distributed in [−λ, λ]M and λ > 0 is a tuning parameter. An example of an im-
+plementation of such a dithered quantization design is illustrated in Fig. 2. The real part and
+imaginary part of the received signal after the radio frequency (RF) circuits are sampled and
+stored separately in two sample-and-hold (S/H) circuits. Then, a switch is used to extract in turn
+the signals from two S/H circuits and forward them to the one-bit ADC. Meanwhile, a dithering
+signal generated by the dithering generator (DG) is added into the one-bit ADC and the analog
+signal is dithered quantized. For instance, if the switches connect the a points, the DG generates
+random dithering signals τ Re and τ Im. Oppositely, if the b points are connected, ��τ Re and �τ Im
+are generated from DG. The quantized signals of all antenna chains will be processed in the
+digital signal processor (DSP). Note that we use S/H circuits to avoid using two one-bit ADCs
+for each real or imaginary part signal. Also, this circuit can be directly used for non-dithered
+one-bit quantization by fixing the connection of switches and turn off the DG.
+Given N dithered quantized samples from (32), we can estimate Cy via
+�Cd
+y = 1
+2
+�Cd + 1
+2
+�
+�Cd�H
+,
+(33)
+where
+�Cd = λ2
+N
+N
+�
+n=1
+rd
+n
+��rd
+n
+�H
+(34)
+is an asymmetric version of the sample covariance matrix scaled with λ2. We can now quantify
+the approximation of Cy by �Cd
+y for all random vectors y ∈ CM with S-subgaussian coordinates.
+Definition 1: We say that a random vector y ∈ CM with covariance matrix Cy has S-
+
+13
+subgaussian coordinates if, for all p ≥ 2 and j ∈ [M],
+max
+��
+E
+�
+|[Re(y)]j|p�� 1
+p,
+�
+E
+�
+|[Im(y)]j|p�� 1
+p�
+≤ S√p∥Cy∥
+1
+2∞.
+(35)
+Note that if y ∈ CM is complex Gaussian with mean zero, then both Re(y) and Im(y) are
+mean-zero real Gaussian vectors with covariance matrix 1
+2Re(Cy). Hence, y has S-subgausian
+coordinates for some absolute constant S. The following estimates, which complement operator
+norm error bounds derived in [37], are tailored to be used in Lemma 1.
+Theorem 1: Let y ∈ CM be a mean-zero random vector vector with covariance matrix
+E
+�
+yyH�
+= Cy and S-subgaussian coordinates. Let y1, ..., yN
+i.i.d.
+∼ y. Then there exists a constant
+c > 0 which only depends on S such that if λ2 ≳ log(N)∥Cy∥∞, the covariance estimator �Cd
+y
+fulfills, for any t ≥ 0, with probability at least 1 − 8e−cNt
+����Cd
+y − Cy
+���
+∞ ≲ λ2
+�
+log(M) + t
+N
+.
+(36)
+and
+����Cd
+y − Cy
+���
+F ≲ λ2
+�
+M 2(log(M) + t)
+N
+.
+(37)
+Proof: See Appendix B.
+By combining Theorem 1 with Lemma 1, we can derive a bound on the expected estimation
+error of the channel vector in terms of the number of samples N used to estimate Cy.
+Theorem 2: There exist constants c1, . . . , c4 > 0 depending only on S such that the following
+holds. Let y ∈ CM be a zero-mean random vector with covariance matrix Cy and S-subgaussian
+coordinates. Let y1, ..., yN
+i.i.d.
+∼ y. Suppose that Cy, Cr, and θ ∈ (0, 1) satisfy (24) and (25).
+Further suppose that λ2 ≥ c1 log(N)∥Cy∥∞ and
+N ≥ c2λ4M 2�
+θ−6 +θ−12∥Ch∥−2
+F
+max{1, ∥Ch∥2}∥Cy∥2�
+max{1, ∥Cy∥2
+∞}
+�
+log(M)+t
+�
+. (38)
+Then, for any t ≥ 0, with probability at least 1 − 8e−c3Nt
+E
+�����h − �hBLM���
+2
+2
+�
+≤ c4λ2θ−6M max{1, ∥Cy∥∞} max{1, ∥Ch∥} ∥Ch∥F
+�
+log(M) + t
+N
+.
+(39)
+Proof: See Appendix C.
+Remark 2: Theorem 2 implies that the parameter λ of the uniform distribution of the dithering
+vectors must be carefully tuned. Developing a corresponding method is thus desirable. We defer
+this open point to future work.
+♦
+
+14
+C. APS-based channel covariance estimation
+Let us now revisit the problem of estimating the channel covariance based on an estimator
+�Cy of Cy. Previously we used the basic estimator for the channel covariance given by
+�Ch =
+�
+�Cy − N0IM
+�
+,
+(40)
+which may not necessarily be a positive semi-definite matrix. To heuristically improve the
+performance of this basic estimator, we further exploit the angle domain and apply a commonly
+considered APS-based covariance fitting to estimate the APS and subsequently enhance the
+estimate of the channel covariance, see e.g., [9, 47–49]. Specifically, assuming that the visible
+antennas of all scattering clusters are known at the BS, i.e., the BS has the exact knowledge of
+the selection matrices {Si}L
+i=1. Using G Dirac delta functions that are equally spaced in the angle
+domain with AoAs {θi}G
+i=1 as dictionary, the channel covariance matrix can be approximated as
+�Ch(�γ) = �Adiag(�γL+1)�AH +
+L
+�
+i=1
+Si �Adiag(�γi)�AHSH
+i ,
+(41)
+where �A := [a(θ1), . . . , a(θG)] and we define �γ ∈ R �G
++ := [�γT
+1 , . . . , �γT
+L+1]T as the non-negative
+coefficients to be estimated, where �G = (L + 1)G. Then, we estimate the coefficients by fitting
+the parametric channel covariance �Ch(�γ) to the basic estimator �Ch in terms of the Frobenius
+norm. We denote the estimated coefficients by
+�γ⋆ = arg min
+�γ∈R �
+G
++
+����Ch(�γ) − �Ch
+���
+2
+F .
+(42)
+By defining b := vec(�Ci
+h) and B :=
+�
+Bl
+1, . . . , Bl
+L, Bc�
+, where
+Bc :=
+�
+vec(a(θ1)a(θ1)H), . . . , vec(a(θG)a(θG)H)
+�
+,
+(43)
+Bl
+i :=
+�
+vec(Sia(θ1)a(θ1)HSH
+i ), . . . , vec(Sia(θG)a(θG)HSH
+i )
+�
+,
+∀i ∈ [L],
+(44)
+we see that �γ⋆ can be obtained by solving the nonnegative least squares (NNLS) problem
+min
+�γ∈R �
+G
++
+∥B�γ − b∥2
+2 .
+(45)
+This problem can be efficiently solved using a variety of convex optimization techniques (see,
+e.g., [50, 51]). In our simulations, we use the novel MATLAB NNLS solver [26] which is much
+faster and stabler than the built-in MATLAB function lsqnonneg, especially under large problem
+dimensions as considered here. With the estimated ASF �γ⋆ in hand, we obtain �C⋆
+h := �Ch(�γ⋆)
+as the final estimator of the channel covariance.
+
+15
+IV. DATA TRANSMISSION RATE
+In the UL data transmission phase, we assume K users simultaneously transmit their data.
+The received signal at the BS before quantization is given by
+yD = Hs + nD,
+(46)
+where H ∈ CM×K is the channel matrix of K users, s ∼ CN(0, IK) is the vector of the data
+signals of K users and nD ∼ CN(0, N0IM) is additive white Gaussian noise. Note that we use
+the subscript ‘D’ to indicate the signal during data transmission. After the one-bit non-dithered
+quantization, the quantized signal is given by
+rD = Q(yD) = ADHs + ADnD + qD,
+(47)
+where AD =
+�
+2
+π(diag(HHH+N0IM))− 1
+2 is the Bussgang gain calculated similarly as in (16) and
+(20). By treating interference as noise, we apply a linear receiver WH to separate the quantized
+signal into K streams as
+�s = WHrD = WHADHs + WH(ADnD + qD).
+(48)
+The data signal of the k-th user is then decoded by the k-th element of �s as
+�sk = wH
+k ADhksk + wH
+k
+K
+�
+i̸=k
+ADhisi + wH
+k (ADnD + qD),
+(49)
+where wk and hk are the k-th columns of the W and H, respectively. The covariance matrix of
+the statistically equivalent quantizer noise qD is given by
+CqD = CrD − ADCyDAH
+D,
+(50)
+where CyD = HHH+N0IM and the covariance matrix CrD of rD can be obtained via the arcsine
+law as CrD = Parcsine(CyD). Note that the quantizer noise qD is non-Gaussian. Considering the
+worst case by treating qD as Gaussian distributed with the same covariance matrix CqD, we can
+obtain a lower bound of the optimistic ergodic sum rate4 of K users [53], given as
+Rsum =
+K
+�
+k=1
+E
+�
+log2
+�
+1 +
+|wH
+k ADhk|2
+�K
+i̸=k |wH
+k ADhi|2 + N0∥wH
+k AD∥2
+2 + wH
+k CqDwk
+��
+.
+(51)
+Now, we consider the design of the linear receiver W. Using the proposed plug-in channel
+estimator, the channel matrix H is estimated as �H. Then, the conventional MRC and ZF receivers
+4Note that the “optimistic ergodic sum rate” is an upper bound assuming Gaussian signaling since it assumes that the useful
+signal coefficient and the interference variance are perfectly known [52].
+
+16
+are given as
+WH
+MRC = �HH,
+(52)
+WH
+ZF =
+�
+�HH �H
+�−1 �HH.
+(53)
+Note that the conventional MRC and ZF receivers might not perform so well due to the quantized
+signal. Therefore, we propose a BLMMSE receiver that takes directly into account the quantized
+signal by considering its Bussgang decomposition, which is expected to yield better performance.
+Specifically, the BLMMSE receiver is given as
+WH
+BLM = CsrDC−1
+rD ,
+(54)
+where
+CsrD = E
+�
+srH
+D
+�
+= HHAH
+D,
+CrD = E
+�
+rDrH
+D
+�
+= Parcsine(CyD).
+(55)
+In practice, using the estimated channel matrix �H, the BLMMSE receiver WH
+BLM is given as
+WH
+BLM = �HH �AH
+D
+�
+Parcsine
+�
+�CyD
+��−1
+,
+(56)
+where �AD =
+�
+2
+π(diag( �H �HH + N0IM))− 1
+2 and �CyD = �H �HH + N0IM.
+V. SIMULATION RESULTS
+In our simulation, we take M = 256 antennas at the BS in ULA. The channel consists of Lc =
+3 multipaths in common clusters and L = 2 local clusters, where each local cluster is composed
+of three multipaths, i.e., Ll
+i = 3, i = 1, 2. The first local cluster is visible to the first quarter
+of antennas and the second local cluster is visible to the last quarter of antennas, i.e., Ml
+1 =
+{1, 2, . . . , M
+4 }, Ml
+2 = { 3M
+4 +1, 3M
+4 +2, . . . , M}. The AoAs of the common clusters and the first
+and second local clusters are uniformly and randomly generated from [−60, 60], [−60, 0] and
+[0, 60] degrees, respectively, i.e., θc
+i ∼ U(−60, 60), ∀i ∈ [Lc], θl
+1,j ∼ U(−60, 0), ∀j ∈ [Ll
+1], θl
+2,j ∼
+U(0, 60), ∀j ∈ [Ll
+2]. The APS of all multipaths are randomly generated with the constraints P c =
+0.3, P l
+1 = 0.7, and P l
+2 = 0.5. Note that this setting satisfies the assumption of max(diag(Ch)) =
+1. The SNR is set to 10dB (equivalently, N0 = 0.1). We consider three different estimators �Cy
+for Cy: the estimator (31), based on non-dithered one-bit quantized samples, the estimator (33),
+based on dithered one-bit quantized samples, and finally, as a reference, the sample covariance
+matrix of the unquantized samples. In the first two cases with quantized samples, we use the
+estimator to produce the APS-based estimator �C⋆
+h of the channel covariance Ch, as is detailed
+in Section III-C. All results presented below are averaged over 10 random channel geometry
+realizations, each with 20 groups of N i.i.d. random channel realizations.
+
+17
+A. Channel covariance estimation
+Given an estimator �C⋆
+h of the channel covariance matrix, we first evaluate it in terms of the
+normalized Frobenius norm error, which is given by
+ENF = E
+�
+��
+���Ch − �C⋆
+h
+���
+2
+F
+∥Ch∥2
+F
+�
+�� .
+(57)
+The numerical results under different values of λ are shown in Fig. 3a. It is seen that the choice
+of λ significantly influences the results of dithered quantization. A larger number of samples N
+can result in a more robust covariance estimation in terms of tuning λ. Moreover, the results
+under various numbers of samples N are shown in Fig. 3b, where 3 different choices of λ for
+dithered quantization are included. It is observed that the Frobenius norm errors of the dithered
+estimators are much smaller than the ones of the non-dithered estimators over a large range of
+number of samples N. It is also seen that by finding a proper λ, the estimation performance
+can be much improved. Furthermore, in the regime of a small number of samples (e.g., when
+N < 100 in our case), the results for the estimator based on dithered quantized samples are even
+better than the sample covariance of the unquantized samples. This shows that our algorithm
+has a benefit in practical cases with limited number of samples.
+B. Channel vector estimation via BLMMSE
+Next, we numerically evaluate the BLMMSE-based channel vector estimator in terms of the
+normalized MSE, which is given by
+ENMSE =
+E
+����h − �h
+���
+2
+2
+�
+tr(Ch)
+.
+(58)
+Given an estimated channel covariance, we calculate the NMSE with 100 i.i.d. random channel
+realizations. The averaged results under various λ and various N are depicted in Fig. 4a and
+Fig. 4b, respectively, where the lower bound is given by the BLMMSE estimator obtained using
+the true channel covariance. It is observed again that the choice of λ significantly influences the
+estimation performance of the dithered case. By applying a proper λ (e.g., λ = 1 in Fig. 4b)
+the channel estimate can be considerably improved compared to the non-dithered case for a
+large range of number of samples. Moreover, it is seen from Figs. 3a and 4a that the trend of
+tuning λ in BLMMSE based channel estimation is different from the trend in channel covariance
+estimation. In the range of 1 ≤ λ ≤ 2, the results of BLMMSE-based channel estimation are no
+
+18
+longer as robust as the results in covariance estimation even under large N. The optimal choices
+of λ for the two estimation problems are also different, e.g., λ⋆ ≈ 1.5 in Fig. 3a and λ⋆ ≈ 1.2
+in Fig. 4a under N = 500.
+C. Ergodic sum rate evaluation
+Finally, we evaluate the proposed scheme in terms of the ergodic sum rate given in (51). We
+test with K = 4 users and assume that the channel geometry of all users follows the setting
+described at the beginning of this section. For each channel estimation we test with MRC, ZF, and
+BLMMSE receivers given in (52), (53), and (56), respectively. Similarly as in the previous part,
+beside the non-dithered and dithered schemes we also provide results based on the true channel
+covariance matrix. However, unlike for the channel MSE criterion used in the previous part, the
+use of the true covariance is not guaranteed to yield a better sum rate, as a channel estimator
+with smaller MSE does not necessarily yield a higher sum rate. Furthermore, we provide results
+based on the true channel vectors as upper bounds.
+We first present the resulting sum rates under various λ with N = 50 samples for covariance
+estimation and BLMMSE channel estimation in Fig. 5a. It is firstly observed that the BLMMSE
+receiver performs much better than the ZF and MRC receivers. This is expected since the
+BLMMSE receiver takes the quantization into account whereas the conventional ZF and MRC
+receivers do not. Next, we observe that the results based on the true covariance do not always
+provide the largest sum rate as previously explained. Specifically, among the results with ZF
+receivers (the dashed lines) the non-dithered one is the best. Since the ZF and MRC receivers are
+designed to deal with non-quantized received signals and perform much worse than the BLMMSE
+receiver, we now focus on the results of the BLMMSE receiver, which is also individually
+depicted in Fig. 5b with one more case of N = 500 samples. It is noticed again that a larger
+number of samples N makes the results with dithering in terms of the sum rate not only better
+but also more robust against λ. From Fig. 5b it is seen that the highest sum rate obtained by
+the proposed scheme with dithering (under N = 500 and λ ≈ 0.6) is very close to the result
+based on the true covariance matrix. This shows the advantage of the proposed scheme for both
+channel estimation and multi-user receivers under one-bit quantization.
+Finally, we focus on the influence of the number of samples N. The sum rates under various
+N of MRC, ZF, and BLMMSE are depicted in Fig. 6a and of only BLMMSE with 3 different
+
+19
+0.5
+1
+1.5
+2
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+ENF
+Unquantized, N = 50
+Unquantized, N = 500
+Non-Dithered, N = 50
+Non-Dithered, N = 500
+Dithered, N = 50
+Dithered, N = 500
+(a) ENF v.s. λ
+101
+102
+103
+N
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+ENF
+Unquantized
+Non-Dithered
+Dithered,  = 0.66667
+Dithered,  = 1
+Dithered,  = 1.5
+(b) ENF v.s. N
+Fig. 3: Normalized Frobenius-norm error of channel covariance under various λ in (a) and i.i.d.
+samples N in (b).
+choices of λ for the dithered case are depicted in Fig. 6b. In Fig. 6a we see a similar behavior as
+before: the BLMMSE receiver produces better sum rates than the ZF and MRC receivers over a
+large range of N. It is seen from Fig. 6b that under the best λ⋆ ≈ 0.6 the proposed scheme with
+dithering produces sum rates comparable to the results based on the true channel covariance when
+N ≥ 100. It is additionally observed from Fig. 6b that as the number of samples N increases
+the difference between the results obtained with different λ is decreasing. This indicates again
+that under larger N the results with dithering are more robust to variations in λ.
+VI. CONCLUSION AND DISCUSSION
+In this work, we proposed a plug-in channel estimator for massive MIMO systems with
+spatially non-stationary channels and one-bit quantizers. We analyzed the quantized signal via
+the Bussgang decomposition and analyzed the distortion produced by using an estimated, rather
+than the true, channel covariance in the construction of the BLMMSE estimator of the channel.
+To obtain an estimate of the covariance of the spatially non-stationary channel, we introduced
+a channel covariance estimator based on dithered quantized samples and theoretically analyzed
+its performance. We further enhanced this estimator using an APS-based NNLS solution. Our
+numerical results showed large performance gains of the proposed scheme with dithering in
+terms of both channel vector and covariance estimation. Finally, we proposed a BLMMSE-
+based receiver tailored to one-bit quantized data signals for the multi-user data transmission
+
+20
+0.5
+1
+1.5
+2
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+ENMSE
+Non-Dithered, N = 50
+Non-Dithered, N = 500
+Dithered, N = 50
+Dithered, N = 500
+True Covariance
+(a) EMMSE v.s. λ
+101
+102
+103
+N
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+ENMSE
+Non-Dithered
+Dithered,  = 0.66667
+Dithered,  = 1
+Dithered,  = 1.5
+True Covariance
+(b) EMMSE v.s. N
+Fig. 4: Normalized MSE of channel vectors via BLMMSE under various λ in (a) and i.i.d.
+samples N in (b).
+0.5
+1
+1.5
+2
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Rsum
+Non-Dithered, N = 50, MRC
+Non-Dithered, N = 50, ZF
+Non-Dithered, N = 50, BLMMSE
+Dithered, N = 50, MRC
+Dithered, N = 50, ZF
+Dithered, N = 50, BLMMSE
+True Covariance, MRC
+True Covariance, ZF
+True Covariance, BLMMSE
+True Channel, MRC
+True Channel, ZF
+True Channel, BLMMSE
+(a) Rsum v.s. λ via MRC, ZF, BLMMSE receivers
+0.5
+1
+1.5
+2
+12.6
+12.7
+12.8
+12.9
+13
+13.1
+13.2
+13.3
+13.4
+13.5
+Rsum
+Non-Dithered, N = 50, BLMMSE
+Non-Dithered, N = 500, BLMMSE
+Dithered, N = 50, BLMMSE
+Dithered, N = 500, BLMMSE
+True Covariance, BLMMSE
+(b) Rsum v.s. λ via BLMMSE receiver
+Fig. 5: Ergodic sum rate of K = 4 users under various λ via MRC, ZF and BLMMSE receivers
+in (a) and enlarged view of results via BLMMSE receiver in (b).
+phase and showed in numerical experiments that it outperforms the conventional MRC and ZF
+receivers in terms of the resulting ergodic sum rate.
+There are two important aspects of our work that can be improved. First, we observed in
+the numerical experiments that the hyperparameter λ of the dithering generation influences the
+channel estimation significantly. Even though this influence was observed to diminish as the
+sample size increases, it is still of significant interest to develop a data-driven method to optimally
+
+21
+101
+102
+103
+N
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Rsum
+Non-Dithered, MRC
+Non-Dithered, ZF
+Non-Dithered, BLMMSE
+Dithered,  = 0.66667, MRC
+Dithered,  = 0.66667, ZF
+Dithered,  = 0.66667, BLMMSE
+True Covariance, MRC
+True Covariance, ZF
+True Covariance, BLMMSE
+True Channel, MRC
+True Channel, ZF
+True Channel, BLMMSE
+(a) Rsum v.s. N via MRC, ZF, BLMMSE receivers
+101
+102
+103
+N
+12.4
+12.6
+12.8
+13
+13.2
+13.4
+13.6
+Rsum
+Non-Dithered, BLMMSE
+Dithered,  = 0.66667, BLMMSE
+Dithered,  = 1, BLMMSE
+Dithered,  = 1.5, BLMMSE
+True Covariance, BLMMSE
+(b) Rsum v.s. N via BLMMSE receiver
+Fig. 6: Ergodic sum rate of K = 4 users under various number of i.i.d. samples N via MRC,
+ZF and BLMMSE receivers in (a) and enlarged view of results via BLMMSE receiver in (b).
+tune λ. Second, in the proposed APS-based channel covariance estimation scheme, the visibility
+of local clusters is assumed to be known at the BS. In practice, however, the visibility of local
+clusters is usually not easy to estimate. It is therefore desirable to develop a scheme without
+the assumption of known visibility of local clusters. We will investigate these two questions in
+future work.
+APPENDIX A
+PROOF OF LEMMA 1
+Recall that
+�hBLM = ChrC−1
+r r = ChAHC−1
+r r = (Cy − N0I)AHC−1
+r r
+(59)
+and
+�h =
+�
+�Cy − N0I
+�
+�AH �C−1
+r r,
+(60)
+where �A and �Cr are defined like A and Cr with Cy being replaced by �Cy. Let us abbreviate
+�hBLM = Mr
+and
+�h = �
+Mr.
+(61)
+Consider α, β, γ > 0 such that mini∈[M] |[Cy]i,i| ≥ α, λmin(Cr) ≥ γ, and
+����
+�
+diag(Cy)− 1
+2Cydiag(Cy)− 1
+2
+�
+i,j
+���� ≤ 1 − β,
+for all i ̸= j.
+(62)
+In particular, θ ≤ min{α, β, γ}. We start by writing
+E
+�����h − �hBLM���
+2
+2
+�
+= E
+�
+tr
+�
+�hBLM �
+�hBLM�H
+− �hBLM�hH − �h
+�
+�hBLM�H
++ �h�hH
+��
+(63)
+
+22
+= tr
+�
+MCr
+�
+M − �
+M
+�H
++ �
+MCr
+�
+�
+M − M
+�H�
+(64)
+=
+�
+MCr + �
+MCr,
+�
+M − �
+M
+��
+F
+(65)
+= 2
+�
+MCr,
+�
+M − �
+M
+��
+F −
+��
+�
+M − M
+�
+Cr,
+�
+M − �
+M
+��
+F
+(66)
+≤ 2∥MCr∥F ∥M − �
+M∥F + ∥Cr∥ ∥M − �
+M∥2
+F
+(67)
+Observe that ∥A∥ ≤
+1
+√α by assumption such that
+∥MCr∥F = ∥ChAH∥F ≤ ∥Ch∥F ∥AH∥ ≤
+1
+√α∥Ch∥F.
+(68)
+Moreover, using that ∥ arcsin(B)∥ ≤ π
+2∥B∥ if ∥B∥∞ ≤ 1 (see [37, Supplementary Material, Eq.
+(4)]), we find
+∥Cr∥ ≤
+���diag(Cy)− 1
+2Re(Cy)diag(Cy)− 1
+2
+��� +
+���diag(Cy)− 1
+2Im(Cy)diag(Cy)− 1
+2
+���
+(69)
+≤ 2
+���diag(Cy)− 1
+2
+��� ∥Cy∥
+���diag(Cy)− 1
+2
+��� ≤ 2
+α ∥Cy∥ .
+(70)
+We conclude that
+E
+�����h − �hBLM���
+2
+2
+�
+≲ α− 1
+2∥Ch∥F∥M − �
+M∥F + α−1 ∥Cy∥ ∥M − �
+M∥2
+F.
+(71)
+We will now show that
+���M − �
+M
+���
+F ≤ κ ≤ α
+1
+2 ∥Ch∥F
+∥Cy∥ ,
+(72)
+so that we obtain
+κ
+√α∥Ch∥F as a final estimate.
+We start by estimating
+���M − �
+M
+���
+F =
+���(Cy − N0I)AHC−1
+r
+−
+�
+�Cy − N0I
+�
+�AH �C−1
+r
+���
+F
+(73)
+≤
+���Cy − �Cy
+���
+F ∥A∥
+��C−1
+r
+�� +
+����Cy − N0I
+���
+F
+���A − �A
+���
+��C−1
+r
+��
+(74)
++ ∥�Cy − N0I∥∥�A∥∥C−1
+r
+− �C−1
+r ∥F.
+(75)
+The first term is clearly bounded by γ−1α− 1
+2εF. To estimate the second term, we note that
+����Cy − N0I
+���
+F ≤ ∥Ch∥F +
+����Cy − Cy
+���
+F ≤ ∥Ch∥F + εF and
+����Cy − N0I
+��� ≤ ∥Ch∥ + εF.
+(76)
+Furthermore, we use that
+Z−1
+1
+− Z−1
+2
+= Z−1
+1 (Z2 − Z1)Z−1
+2
+(77)
+for any invertible Z1, Z2 of the same dimensions. This yields
+����A − A
+��� = 2
+π
+���diag(�Cy)− 1
+2
+�
+diag(Cy)
+1
+2 − diag(�Cy)
+1
+2
+�
+diag(Cy)− 1
+2
+���
+(78)
+
+23
+≤ 2
+π
+���diag(�Cy)− 1
+2
+���
+���diag(Cy)
+1
+2 − diag(�Cy)
+1
+2
+���
+���diag(Cy)− 1
+2
+���
+(79)
+By assumption, we have
+���diag(Cy)− 1
+2
+��� =
+�
+1
+mini[Cy]i,i
+≤
+�
+1
+α.
+(80)
+Moreover, since
+����Cy − Cy
+���
+∞ ≤ α
+2 by (26), we find
+min
+i [�Cy]i,i ≥ min
+i [Cy]i,i −
+����Cy − Cy
+���
+∞ ≥ α
+2
+(81)
+and so
+���diag(�Cy)− 1
+2
+��� ≤
+�
+2
+α.
+(82)
+Note that this also implies that ∥�A∥ ≲
+1
+√α. Using that |√x − √y| ≤ |x−y|
+√c
+if x ≥ c > 0, y ≥ 0,
+we find
+���diag(Cy)
+1
+2 − diag(�Cy)
+1
+2
+��� ≤
+�
+1
+α
+���Cy − �Cy
+���
+∞ =
+�
+1
+αε∞,
+(83)
+and hence
+���A − �A
+��� ≤ 4
+πα− 3
+2ε∞.
+(84)
+Let us finally estimate the last term on the right hand side of (73). Write cij = [Cy]i,j, ˆcij =
+[�Cy]i,j and observe that
+�����
+ˆcij
+�
+ˆciiˆcjj
+−
+cij
+√ciicjj
+����� ≤
+�����
+ˆcij − cij
+�
+ˆciiˆcjj
+����� + |cij| 1
+√ˆcii
+�����
+1
+�
+ˆcjj
+−
+1
+√cjj
+����� + |cij|
+1
+�
+ˆcjj
+����
+1
+√ˆcii
+−
+1
+√cii
+���� (85)
+≲ 1
+α
+����Cy − Cy
+���
+∞ + ∥Cy∥∞
+1
+α2
+����Cy − Cy
+���
+∞
+(86)
+≲ ∥Cy∥∞
+1
+α2
+����Cy − Cy
+���
+∞ ≤ β
+2
+(87)
+as
+ε∞ ≲ β
+α2
+∥Cy∥∞
+.
+(88)
+By (62), this implies that
+����
+�
+diag(�Cy)− 1
+2 �Cydiag(�Cy)− 1
+2
+�
+i,j
+���� ≤ 1 − β
+2 ,
+for all i ̸= j.
+(89)
+Clearly, for any
+| arcsin(x) − arcsin(y)| ≤ Lβ|x − y|,
+(90)
+for all x, y ∈ (−1 + β
+2, 1 − β
+2) where
+Lβ =
+sup
+0≤z<1− β
+2
+�
+1
+1 − z2 =
+�
+1
+1 − (1 − β
+2)2 ≤
+� 2
+β .
+(91)
+
+24
+Together with (62) and (89) this yields
+∥Cr − �Cr∥F ≲ β−1/2 ���diag(�Cy)− 1
+2Re(�Cy)diag(�Cy)− 1
+2 − diag(Cy)− 1
+2Re(Cy)diag(Cy)− 1
+2
+���
+F
++ β−1/2 ���diag(�Cy)− 1
+2Im(�Cy)diag(�Cy)− 1
+2 − diag(Cy)− 1
+2Im(Cy)diag(Cy)− 1
+2
+���
+F .
+(92)
+Now observe that
+���diag(�Cy)− 1
+2Re(�Cy)diag(�Cy)− 1
+2 − diag(Cy)− 1
+2Re(Cy)diag(Cy)− 1
+2
+���
+F
+≤
+���diag(�Cy)− 1
+2 − diag(Cy)− 1
+2
+��� ∥�Cy∥F
+���diag(�Cy)− 1
+2
+���
++
+���diag(Cy)− 1
+2
+��� ∥�Cy − Cy∥F
+���diag(�Cy)− 1
+2
+���
++
+���diag(Cy)− 1
+2
+��� ∥Cy∥F
+���diag(�Cy)− 1
+2 − diag(Cy)− 1
+2
+���
+(93)
+≲ α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF
+(94)
+and analogously,
+���diag(�Cy)− 1
+2Im(�Cy)diag(�Cy)− 1
+2 − diag(Cy)− 1
+2Im(Cy)diag(Cy)− 1
+2
+���
+F
+≲ α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF.
+(95)
+Hence,
+���Cr − �Cr
+���
+F ≲ β− 1
+2α−2∥Cy∥Fε∞ + β− 1
+2(α−2ε∞ + α−1)εF.
+(96)
+By our assumptions on ε∞ and εF, the right hand side is bounded by γ/2 and hence the
+assumption ∥C−1
+r ∥ ≤ γ−1 implies that ∥�C−1
+r ∥ ≤ 2γ−1. Using now again (77) we finally arrive
+at
+���C−1
+r
+− �C−1
+r
+���
+F ≲ β− 1
+2γ−2 �
+α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF
+�
+.
+(97)
+Combining all our estimates in (73), we find
+���M − �
+M
+���
+F ≲ γ−1α− 1
+2εF + γ−1(∥Ch∥F + εF)α− 3
+2ε∞
++ (∥Ch∥ + εF)α− 1
+2β− 1
+2γ−2�
+α−2∥Cy∥Fε∞ +
+�
+α−2ε∞ + α−1�
+εF
+�
+.
+(98)
+Since
+ε∞ ≤ min
+�
+εF
+∥Cy∥F
+, 1
+�
+,
+(99)
+we can estimate the right hand side by
+κ := c α− 5
+2β− 1
+2γ−2 max{1, ∥Ch∥}εF,
+(100)
+
+25
+for an absolute constant c > 0. Clearly,
+κ ≤ α− 1
+2 ∥Ch∥F
+∥Cy∥
+(101)
+by our assumption on εF, which completes the proof.
+APPENDIX B
+PROOF OF THEOREM 1
+In the proof of Theorem 1 we will use the following lemmas. The first one bounds the bias
+of (34) in terms of λ.
+Lemma 2: Let S > 0. There exist constants c1, c2 > 0 depending only on S such that the fol-
+lowing holds. Let y ∈ CM be a mean-zero random vector with covariance matrix E
+�
+yyH�
+= Cy
+and S-subgaussian coordinates. Let λ > 0 and let rRe = sign(Re(y)+τ Re), rIm = sign(Im(y)+
+τ Im), �rRe = sign(Re(y)+�τ Re), and �rIm = sign(Im(y)+�τ Im), where τ Re, τ Im, �τ Re, �τ Im are inde-
+pendent and uniformly distributed in [−λ, λ]M and independent of y. Abbreviate r = rRe +jrIm
+and �r = �rRe + j�rIm. Then,
+��λ2E
+�
+r�rH�
+− Cy
+��
+∞ ≤ c1(λ2 + ∥Cy∥∞)e
+−c2λ2
+∥Cy∥∞ .
+(102)
+Proof: The proof of this lemma is a straightforward extension of [37, Lemma 17] to the
+complex domain. We include it for the convenience of the reader. First note that
+��λ2E
+�
+r�rH�
+− Cy
+��
+∞
+=
+��λ2E
+�
+(rRe + jrIm)(�rRe + j�rIm)H�
+− E
+�
+(Re(y) + jIm(y))(Re(y) + jIm(y))H���
+∞
+≤
+��λ2E
+�
+rRe(�rRe)T�
+− E
+�
+Re(y)Re(y)T���
+∞ +
+��λ2E
+�
+rRe(�rIm)T�
+− E
+�
+Re(y)Im(y)T���
+∞
++
+��λ2E
+�
+rIm(�rRe)T�
+− E
+�
+Im(y)Re(y)T���
+∞ +
+��λ2E
+�
+rIm(�rIm)T�
+− E
+�
+Im(y)Im(y)T���
+∞
+(103)
+Since y has S-subgaussian coordinates, we get from (35) that ∥[Re(y)]i∥ψ2, ∥[Im(y)]i∥ψ2 ≤
+S∥Cy∥
+1
+2∞, for any i ∈ [M], where ∥ · ∥ψ2 denotes the subgaussian norm. Applying [37, Lemma
+17] for U = [Re(y)]i and V = [Re(y)]j yields
+���λ2E
+�
+sign
+�
+[Re(y)]i + [τ Re]i
+�
+· sign
+�
+[Re(y)]j + [�τ Re]j
+��
+− E
+�
+[Re(y)]i[Re(y)]j
+����
+≲ (λ2 + S2∥Cy∥∞)e
+−c
+λ2
+S2∥Cy∥∞ .
+(104)
+Since this holds for any choice of i, j ∈ [M], the first term on the right-hand side of (103)
+satisfies the claimed bound. The three other terms can be treated in the same way such that our
+claim follows.
+
+26
+The second lemma is a simple concentration inequality that applies to dithered samples of real
+distributions.
+Lemma 3: There exist absolute constants c1, c2 > 0 such that the following holds. Let y, �y ∈
+RM be random vectors. Let y1, ..., yN
+i.i.d.
+∼ y, let �y1, ..., �yN
+i.i.d.
+∼ �y, and let τ 1, . . . , τ N, �τ 1, . . . , �τ N
+be independent and uniformly distributed in [−λ, λ], for λ > 0. Define rk = sign(yk + τ k) and
+�rk = sign(�yk + �τ k). If N ≥ c1 log(M), then
+Pr
+������
+λ2
+N
+N
+�
+k=1
+rk�rT
+k − E
+�
+rk�rT
+k
+�
+�����
+∞
+≥
+�
+λ4
+�
+c1
+log(M)
+N
++ t
+��
+≤ 2e−c2Nt.
+(105)
+In particular, the claim holds if y = �y and yi = �yi, for all i ∈ [N].
+Proof: Write Rk
+i,j = [rk]i[�rk]j for i, j ∈ [M]. Since |Rk
+i,j − E[Rk
+i,j]| ≤ 2 for all i, j, k, the
+bound is trivial for t ≥ 4. Moreover, by Bernstein’s inequality for bounded random variables
+(see, e.g., [54, Theorem 2.8.4]), we find for any u ≤ 8λ2
+Pr
+�
+1
+N
+�����
+N
+�
+k=1
+λ2 �
+Rk
+i,j − E[Rk
+i,j]
+�
+����� ≥ u
+�
+≤ 2e
+−c min
+�
+N2u2
+σ2
+i,j
+, Nu
+2λ2
+�
+(106)
+≤ 2e
+−cN min
+�
+u2
+λ4 , u
+λ2
+�
+≤ 2e−c2N u2
+λ4 ,
+(107)
+as
+σ2
+i,j :=
+N
+�
+k=1
+λ4E
+��
+Rk
+i,j − E[Rk
+i,j]
+�2�
+=
+N
+�
+k=1
+λ4 �
+E
+�
+(Rk
+i,j)2�
+−
+�
+E[Rk
+i,j]
+�2�
+≤ λ4N.
+(108)
+Hence, for any given t < 4 we can set u =
+�
+λ4
+�
+c1
+log(M)
+N
++ t
+�
+and note that u ≤ 8λ2 as
+N ≥ c1 log(M). By applying the union bound over all M 2 entries we obtain the result.
+Proof of Theorem 1: By the triangle inequality,
+����Cd
+y − Cy
+���
+∞ ≤
+����Cd
+y − E
+�
+�Cd
+y
+����
+∞ +
+���E
+�
+�Cd
+y
+�
+− Cy
+���
+∞
+(109)
+Write r = rRe + jrIm and �r = �rRe + j�rIm, where rRe = sign(Re(y) + τ Re), rIm = sign(Im(y) +
+τ Im), �rRe = sign(Re(y) + �τ Re), and �rIm = sign(Im(y) + �τ Im). By Lemma 2,
+���E
+�
+�Cd
+y
+�
+− Cy
+���
+∞ =
+��λ2E
+�
+r�rH�
+− Cy
+��
+∞ ≲
+�
+λ2 + ∥Cy∥∞
+�2 e
+−c2λ2
+∥Cy∥∞ ≲ λ2
+√
+N
+,
+(110)
+where we have used that λ2 ≳ log(N)∥Cy∥∞. To estimate the first term in (109), observe that
+����Cd
+y − E
+�
+�Cd
+y
+����
+∞ =
+����Cd − E
+�
+�Cd����
+∞
+(111)
+=
+�����
+�
+λ2
+N
+N
+�
+k=1
+(rRe
+k + jrIm
+k )(�rRe
+k + j�rIm
+k )H
+�
+− E
+�
+λ2
+N
+N
+�
+k=1
+(rRe
+k + jrIm
+k )(�rRe
+k + j�rIm
+k )H
+� �����
+∞
+
+27
+≤
+�����
+�
+λ2
+N
+N
+�
+k=1
+rRe
+k (�rRe
+k )T
+�
+− E
+�
+λ2
+N
+N
+�
+k=1
+rRe
+k (�rRe
+k )T
+������
+∞
++
+�����
+�
+λ2
+N
+N
+�
+k=1
+rRe
+k (�rIm
+k )T
+�
+− E
+�
+λ2
+N
+N
+�
+k=1
+rRe
+k (�rIm
+k )T
+������
+∞
++
+�����
+�
+λ2
+N
+N
+�
+k=1
+rIm
+k (�rRe
+k )T
+�
+− E
+�
+λ2
+N
+N
+�
+k=1
+rIm
+k (�rRe
+k )T
+������
+∞
++
+�����
+�
+λ2
+N
+N
+�
+k=1
+rIm
+k (�rIm
+k )T
+�
+− E
+�
+λ2
+N
+N
+�
+k=1
+rIm
+k (�rIm
+k )T
+������
+∞
+(112)
+Using Lemma 3 for each of the four terms and applying a union bound, we get
+Pr
+�����Cd
+y − E
+�
+�Cd
+y
+����
+∞ ≳ λ2
+�
+log(M) + t
+N
+�
+≤ 8e−cNt,
+(113)
+and thus the first statement of Theorem 1. The second statement follows trivially using
+����Cd
+y − Cy
+���
+F ≤ M
+����Cd
+y − Cy
+���
+∞ .
+(114)
+APPENDIX C
+PROOF OF THEOREM 2
+By Theorem 1, we can apply Lemma 1 with
+ε∞ ∼ λ2
+�
+log(M) + t
+N
+,
+εF ∼ M max{1, ∥Cy∥∞}λ2
+�
+log(M) + t
+N
+Note that we do not pick the “minimal setting” for εF suggested by Theorem 1: the additional
+factor max{1, ∥Cy∥∞} ensures that ε∞ ≲ εF/∥Cy∥F holds (as required in (26)). It remains to
+note that all other conditions on ε∞ and εF in Lemma 1 under the stated assumption on N. This
+completes the proof.
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+

6tAyT4oBgHgl3EQfcvc9/content/tmp_files/2301.00288v1.pdf.txt

@@ -0,0 +1,3087 @@
+arXiv:2301.00288v1  [math.AP]  31 Dec 2022
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR
+NON-MONOTONIC SHEAR FLOWS
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+Abstract. We give a proof of linear inviscid damping and vorticity depletion for non-monotonic
+shear flows with one critical point in a bounded periodic channel. In particular, we obtain
+quantitative depletion rates for the vorticity function without any symmetry assumptions.
+Dedicated to Carlos Kenig, on the occasion of his 70th birthday.
+Key Words: Inviscid damping, vorticity depletion, non-monotonic shear flows.
+Mathematics Subject Classification: 35B40, 35Q31, 35P25
+Contents
+1.
+Introduction
+1
+2.
+Spectral property and representation formula
+5
+3.
+Bounds on the Green’s function and modified Green’s function
+7
+4.
+The limiting absorption principle
+13
+5.
+Bounds on ψι
+k,ǫ: the non-degenerate case
+18
+6.
+Bounds on ψι
+k,ǫ: the degenerate case
+20
+7.
+Proof of Theorem 1.2
+27
+References
+30
+1. Introduction
+The study of stability problems in mathematical analysis of fluid dynamics has a long and
+distinguished history, dating back to the work of Kelvin [18], Orr [25] and Rayleigh [26] among
+many others, and continuing to the present day. Hydrodynamical stability problems can be
+considered in both two and three dimensions. In this paper we work with two dimensional
+inviscid flows.
+For the Euler equations, there are significant recent progresses on the asymptotic stability
+of monotonic shear flows and vortices, assuming spectral stability, see for example [9, 30, 34,
+35, 14, 16, 3, 17, 22, 28] for linear results.
+The main mechanism of stabilization is the so
+called “inviscid damping”, which refers to the transfer of energy of vorticity to higher and
+higher frequencies leading to decay of the stream and velocity functions, as t → ∞. Extending
+the linearized stability analysis for inviscid fluid equations to the full nonlinear setting is a
+challenging problem, and the only available results are on spectrally stable monotonic shear
+The first author was supported in part by NSF grant DMS-2007008. The second author is partially supported
+by a UC Davis startup grant. The third author was supported in part by NSF grant DMS-1945179.
+1
+
+2
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+flows [2, 23, 10, 12], and on point vortices [11]. We refer also to the recent review article [13]
+for a more in-depth discussion of recent developments of both linear and nonlinear inviscid
+damping.
+Many physically important shear flows are not monotonic, such as Poiseuille flow and Kol-
+mogorov flows. For such flows on the linear inviscid level, there is an additional significant
+physical phenomenon called “vorticity depletion” which refers to the asymptotic vanishing of
+vorticity as t → ∞ near the critical point where the derivative of the shear flow is zero, first
+predicted in Bouchet and Morita [5], and proved rigorously in Wei-Zhang-Zhao [31]. A similar
+phenomenon was proved in Bedrossian-Coti Zelati-Vicol [3] for the case of vortices. See also
+[17] by the first and third author for a refined description of the dynamics in Gevrey spaces as
+a step towards proving nonlinear vortex symmetrization.
+In [31] by Wei-Zhang-Zhao, sharp linear inviscid damping estimates and quantitative deple-
+tion estimates were obtained for an important class of “symmetric shear flows” in a periodic
+channel (see also [32] by Wei-Zhang-Zhao for a similar result for Kolmogorov flow). When
+no symmetry is assumed, only qualitative bounds are available.
+Heuristically the general
+case should be similar to the symmetric one, since the main vorticity depletion mechanism
+is completely local and asymptotically all shear flows approach symmetric ones at the (non-
+degenerate) critical points. However there are significant difficulties in using the approach of
+[31] to extend the quantitative depletion bounds of [31] to the general case, as the argument
+in [31] relies heavily on decomposition of functions into odd and even parts, which are specific
+to symmetric shear flows.
+In this paper we prove linear inviscid damping estimates and quantitative vorticity depletion
+estimates for a class of stable non-monotonic shear flows with one non-degenerate critical
+point. The main new features of our results are that we do not need symmetry condition on
+the background shear flow, and that our formulation on quantitative depletion for vorticity
+function seem to be new even for general symmetric shear flows (see however Wei-Zhang-Zhao
+[32] which contains a sharp depletion rate at the critical points for Kolmogorov flow), see
+Theorem 1.2 below for the precise statements.
+We begin with the description of our main
+equations and theorem.
+1.1. Main equations. Consider the two dimensional Euler equation linearized around a shear
+flow (b(y), 0), in the periodic channel (x, y, t) ∈ T × [0, 1] × [0, ∞):
+∂tω + b(y)∂xω − b′′(y)uy = 0,
+div u = 0
+and
+ω = −∂yux + ∂xuy,
+(1.1)
+with the natural non-penetration boundary condition uy|y=0,1 = 0.
+For the linearized flow,
+�
+T×[0, 1]
+ux(x, y, t) dxdy and
+�
+T×[0, 1]
+ω(x, y, t) dxdy are conserved quan-
+tities. In this paper, we will assume that
+�
+T×[0,1]
+ux
+0(x, y) dxdy =
+�
+T×[0,1]
+ω0 dxdy = 0.
+These assumptions can be dropped by adjusting b(y) with a linear shear flow C0y + C1. Then
+one can see from the divergence free condition on u that there exists a stream function ψ(t, x, y)
+with ψ(t, x, 0) = ψ(t, x, 1) ≡ 0, such that
+ux = −∂yψ, uy = ∂xψ.
+(1.2)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS
+3
+The stream function ψ can be solved through
+∆ψ = ω,
+ψ|y=0,1 = 0.
+(1.3)
+We summarize our equations as follows
+
+
+
+∂tω + b(y)∂xω − b′′(y)∂xψ = 0,
+∆ψ(t, x, y) = ω(t, x, y),
+ψ(t, x, 0) = ψ(t, x, 1) = 0,
+(ux, uy) = (−∂yψ, ∂xψ),
+(1.4)
+for t ≥ 0, (x, y) ∈ T × [0, 1].
+Our goal is to understand the long time behavior of ω(t) as t → ∞, with Sobolev regular
+initial vorticity ω0.
+1.2. The main results. We describe more precisely the main assumptions and our main
+conclusion. The main conditions we shall assume on the shear flow b(y) ∈ C4([0, 1]) are as
+follows.
+Assumption 1.1. We assume that the background flow b(y) ∈ C4([0, 1]) satisfies the following
+conditions.
+(1)
+S := {y ∈ [0, 1] : b′(y) = 0} = {y∗} ⊂ (0, 1).
+(1.5)
+In addition, b′′(y∗) ̸= 0.
+(2) For k ∈ Z\{0}, the linearized operator Lk : L2(0, 1) → L2(0, 1) defined as
+Lkg(y) := b(y)g(y) + b′′(y)
+� 1
+0
+Gk(y, z)g(z) dz
+(1.6)
+has no discrete eigenvalues nor generalized embedded eigenvalues. In the above Gk is
+the Green’s function for k2 −
+d2
+dy2 on the interval (0, 1) with zero Dirichlet boundary
+condition.
+We refer to section 2 below for the definition and more discussion about generalized embed-
+ded eigenvalues.
+Our main result is the following theorem.
+Theorem 1.2. Assume that ω(t, ·) ∈ C([0, ∞), H4(T×[0, 1])) with the associated stream func-
+tion ψ(t, ·) is the unique solution to (1.4), with initial data ω0 ∈ H4(T × [0, 1]) satisfying for
+all y ∈ [0, 1],
+�
+T
+ω0(x, y) dx = 0.
+(1.7)
+Then we have the following bounds.
+(i) Inviscid damping estimates:
+∥ψ(t, ·)∥L2(T×[0,1]) ≲
+1
+⟨t⟩2 ∥ω0∥H4(T×[0,1]),
+(1.8)
+∥ux(t, ·)∥L2(T×[0,1]) ≲ 1
+⟨t⟩∥ω0∥H4(T×[0,1]),
+∥uy(t, ·)∥L2(T×[0,1]) ≲
+1
+⟨t⟩2 ∥ω0∥H4(T×[0,1]).
+(1.9)
+(ii) Vorticity depletion estimates: there exists a decomposition
+ω(t, x, y) := ωloc(t, x, y) + ωnloc(t, x, y),
+(1.10)
+
+4
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+where for (x, y, t) ∈ T × [0, 1] × [0, ∞),
+|ωloc(t, x, y)| ≲ |y − y∗|7/4∥ω0∥H4(T×[0,1]),
+|ωnloc(t, x, y)| ≲
+1
+⟨t⟩7/8 ∥ω0∥H4(T×[0,1]).
+(1.11)
+1.3. Remarks and main ideas of proof. We have the following remarks on Theorem 1.2.
+Firstly, in the above theorem we have not tracked the minimal regularity required for the
+bounds (1.8), (1.9) and (1.11) to hold, and a more careful argument can probably significantly
+reduce the number of derivatives needed on the initial data ω0. Secondly, we note also that
+the argument here can be applied to non-monotonic shear flows with multiple non-degenerate
+points, although the presentation will be more complicated.
+Thirdly, a more sophisticated
+analysis may yield a sharper rate of vorticity depletion with rate
+|ωloc(t, x, y)| ≲ |y − y∗|2−,
+|ωnloc(t, x, y)| ≲ ⟨t⟩−1+.
+It is not clear to us though if one can reach the optimal rates of |y − y∗|2 and ⟨t⟩−1.
+We briefly explain the main ideas of the proof.
+By a standard spectral representation formula, see (2.7), it suffices to study the spectral
+density functions and the associated Rayleigh equation (2.8). There are two main cases to
+consider. When the spectral parameter λ is not close to the critical value b(y∗), the situation
+is similar to monotonic shear flows and can be treated as in [14]. The main new case is when
+the spectral parameter λ is close to the critical value b(y∗). In this case, the Rayleigh equation
+(2.8) is very singular, and the potential term
+b′′(y)
+b(y)−λ+iǫ has a quadratic singularity roughly of
+the form
+2
+(y−y∗)2+(λ−b(y∗))+iǫ for y close to y∗.
+The key observation here, as in [17], is that the potential term
+b′′(y)
+b(y)−λ+iǫ is critically singular
+and has real part with a favorable sign for 1 ≫ |y − y∗| ≫ |λ − b(y∗)|1/2, which needs to be
+incorporated as part of the main term. We therefore define a modified Green’s function for
+the main term, see (3.12)-(3.13), which has strong vanishing conditions near y = y∗, leading
+ultimately to vorticity depletion. After extracting the main terms in the Rayleigh equation
+(2.8), the rest of the terms can be treated as compact perturbations, and can be bounded using
+a limiting absorption principle, see Lemma 4.4, thanks to the spectral assumption 1.1.
+The limiting absorption principle provides preliminary bounds on the spectral density func-
+tions ψι
+k,ǫ(y, λ) with ι ∈ {±}. To obtain the desired quantitative decay rates, we take up to
+two derivatives in λ of the spectral density functions, and again use the limiting absorption
+principle to estimate the resulting derivatives, after extracting the main singular terms. The
+procedure is more or less straightforward but the calculations are quite lengthy. We refer to
+[14] also for similar calculations in a simpler setting. Lastly, we note that there are important
+cancellations between ψ+
+k,ǫ(y, λ) and ψ−
+k,ǫ(y, λ) in the limit ǫ → 0+, which is the reason why we
+need two versions of the limiting absorption principle, see Lemma 4.4, with different weighted
+spaces.
+1.4. Notations. We summarize here some notations that are specific for this paper for the
+reader’s conveniences.
+For positive numbers α, β, we set α ∧ β := min{α, β}.
+We denote
+for d > 0, Σd := {b(y) :
+y ∈ [y∗ − d, y∗ + d]}, Sd := [y∗ − d, y∗ + d].
+We also denote
+Σ := {b(y) : y ∈ [0, 1]} and I := [0, 1]. For k ∈ Z\{0}, we define for f ∈ H1(I) the norm
+∥f∥H1
+k(I) := ∥f∥L2(I) + |k|−1∥f ′∥L2(I).
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS
+5
+2. Spectral property and representation formula
+Taking Fourier transform in x in the equation (1.4) for ω, we obtain that
+∂tωk + ikb(y)ωk − ikb′′(y)ψk = 0,
+(2.1)
+for k ∈ Z, t ≥ 0, y ∈ [0, 1]. In the above, ωk and ψk are the k-th Fourier coefficients of ω, ψ in
+x respectively. For each k ∈ Z\{0}, recall from (1.6) that for any g ∈ L2(0, 1),
+Lkg(y) = b(y)g(y) + b′′(y)
+� 1
+0
+Gk(y, z)g(z)dz,
+(2.2)
+where Gk is the Green’s function for the operator k2− d2
+dy2 on (0, 1) with zero Dirichlet boundary
+condition. Then (2.1) can be reformulated abstractly as
+∂tωk + ikLkωk = 0.
+(2.3)
+In contrast to the spectral property of the linearized operator around monotonic shear flows,
+the spectral property of Lk is less understood, especially on the generation of discrete eigen-
+values and embedded eigenvalues. From general spectral theory, we know that the spectrum
+of Lk consists of the continuous spectrum
+Σ :=
+�
+b(y) : y ∈ [0, 1]
+�
+,
+(2.4)
+together with some discrete eigenvalues with nonzero imaginary part which can only accumulate
+at the set of continuous spectrum Σ. Unlike the case of monotonic shear flows where the discrete
+eigenvalues can accumulate only at inflection points of the background shear flow, there appears
+no simple characterization of the possible accumulation points for non-monotonic shear flows.
+Recall that λ ∈ Σ is called an embedded eigenvalue if there exists a nontrivial g ∈ L2(0, 1),
+such that
+Lkg = λg.
+(2.5)
+For non-monotonic shear flows, this definition is too restrictive, as accumulation points of
+discrete eigenvalues may no longer be embedded eigenvalues. To capture the discrete eigen-
+values, we recall the following definition of “generalized embedded eigenvalues”, which can be
+found already in [31], adapted to our setting.
+Definition 2.1. We call λ ∈ Σ a generalized embedded eigenvalue, if one of the following
+conditions is satisfied.
+• λ is an embedded eigenvalue.
+• λ ̸= b(y∗) and there exists a nontrivial ψ ∈ H1
+0(0, 1) : (0, 1) → C such that in the sense
+of distributions on (0, 1),
+(k2 − ∂2
+y)ψ(y) + P.V.b′′(y)ψ(y)
+b(y) − λ + iπ
+�
+z∈[0,1], b(z)=λ
+b′′(z)ψ(z)
+|b′(z)|
+δ(y − z) = 0.
+(2.6)
+We remark that our assumption that the critical point y∗ of b(y) being non-degenerate
+implies that the sum in (2.6) is finite, and that the spectral assumption 1.1 is satisfied if b′′ > 0
+on [0, 1].
+
+6
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+Proposition 2.2. Suppose that k ∈ Z\{0} and ωk
+0 ∈ L2([0, 1]). Then the stream function
+ψk(t, y) for k ∈ Z\{0}, y ∈ [0, 1], t ≥ 0 has the representation
+ψk(t, y) = − 1
+2πi lim
+ǫ→0+
+�
+Σ
+e−ikλt �
+ψ−
+k,ǫ(y, λ) − ψ+
+k,ǫ(y, λ)
+�
+dλ,
+(2.7)
+where ψι
+k,ǫ(y, λ) for ι ∈ {+, −}, y ∈ [0, 1], λ ∈ Σ, k ∈ Z\{0}, and sufficiently small ǫ ∈
+[−1/4, 1/4]\{0}, are the solutions to
+−k2ψι
+k,ǫ(y, λ) + d2
+dy2 ψι
+k,ǫ(y, λ) −
+b′′(y)
+b(y) − λ + iιǫψι
+k,ǫ(y, λ) =
+−ωk
+0(y)
+b(y) − λ + iιǫ,
+(2.8)
+with zero Dirichlet boundary condition.
+Proof. By standard theory of spectral projection, from (2.3), we obtain that for y ∈ [0, 1],
+ωk(t, y) =
+1
+2πi lim
+ǫ→0+
+�
+Σ
+eiλt ��
+(λ + kLk − iǫ)−1 − (λ + kLk + iǫ)−1�
+ωk
+0
+�
+(y) dλ.
+(2.9)
+We then obtain for y ∈ [0, 1],
+ψk(t, y) = − 1
+2πi lim
+ǫ→0+
+�
+Σ
+e−ikλt
+� 1
+0
+Gk(y, z)
+×
+��
+(−λ + Lk − iǫ)−1 − (−λ + Lk + iǫ)−1�
+ωk
+0
+�
+(z) dzdλ
+= − 1
+2πi lim
+ǫ→0+
+�
+Σ
+e−ikλt �
+ψ−
+k,ǫ(y, λ) − ψ+
+k,ǫ(y, λ)
+�
+dλ.
+(2.10)
+In the above, for y ∈ [0, 1] and λ ∈ Σ,
+ψ+
+k,ǫ(y, λ) :=
+� 1
+0
+Gk(y, z)
+�
+(−λ + Lk + iǫ)−1ωk
+0
+�
+(z) dz,
+ψ−
+k,ǫ(y, λ) :=
+� 1
+0
+Gk(y, z)
+�
+(−λ + Lk − iǫ)−1ωk
+0
+�
+(z) dz.
+(2.11)
+Therefore for ι ∈ {+, −}, y ∈ [0, 1], λ ∈ Σ,
+�
+k2 − d2
+dy2
+�
+ψι
+k,ǫ(y, y0) = (−λ + Lk + iιǫ)−1ωk
+0(y),
+(2.12)
+which implies
+ωk
+0(y) =(−λ + Lk + iιǫ)
+�
+k2 − d2
+dy2
+�
+ψι
+k,ǫ(y, λ)
+=(b(y) − λ + iιǫ)
+�
+k2 − d2
+dy2
+�
+ψι
+k,ǫ(y, λ) + b′′(y)ψι
+k,ǫ(y, λ).
+(2.13)
+It follows from (2.13) that ψ+
+k,ǫ(y, λ), ψ−
+k,ǫ(y, λ) satisfy (2.8). The proposition is now proved.
+□
+Remark 2.3. The existence of ψι
+k,ǫ for sufficiently small ǫ ̸= 0 follows from our spectral
+assumptions, which imply the solvability of (2.8) for sufficiently small ǫ ̸= 0, see also (4.9).
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS
+7
+3. Bounds on the Green’s function and modified Green’s function
+3.1. Elementary properties of the standard Green’s function. For integers k ∈ Z\{0},
+recall that the Green’s function Gk(y, z) solves
+− d2
+dy2 Gk(y, z) + k2Gk(y, z) = δ(y − z),
+(3.1)
+with Dirichlet boundary conditions Gk(0, z) = Gk(1, z) = 0, z ∈ (0, 1). Gk has the explicit
+formula
+Gk(y, z) =
+1
+k sinh k
+�
+sinh(k(1 − z)) sinh(ky)
+if y ≤ z,
+sinh(kz) sinh(k(1 − y))
+if y ≥ z,
+(3.2)
+and the symmetry
+Gk(y, z) = Gk(z, y),
+for k ∈ Z\{0}, y, z ∈ [0, 1].
+(3.3)
+We note the following bounds for Gk
+sup
+y∈[0,1],|A|≤10
+�
+|k|2��Gk(y, z)(log |z − A|)m��
+L1(z∈[0,1]) + |k|
+��∂y,zGk(y, z)(log |z − A|)m��
+L1(z∈[0,1])
+�
++
+sup
+y∈[0,1],α∈{0,1}
+�
+|k|3/2−α ��∂α
+y,zGk(y, z)
+��
+L2(z∈[0,1])
+�
+≲ | log ⟨k⟩|m,
+for m ∈ {0, 1, 2, 3}.
+(3.4)
+Define
+Fk(y, z) =
+1
+sinh k
+�
+−k cosh (k(1 − z)) cosh (ky),
+0 ≤ y ≤ z ≤ 1;
+−k cosh (kz) cosh (k(1 − y)),
+1 ≥ y > z ≥ 0.
+(3.5)
+We note that
+∂y∂zGk(y, z) = ∂z∂yGk(y, z) = δ(y − z) + Fk(y, z),
+for y, z ∈ [0, 1].
+(3.6)
+By direct computation, we see Fk satisfies the bounds
+sup
+y∈[0,1],|A|≤10
+���Fk(y, z)(log |z − A|)m��
+L1(z∈[0,1]) + |k|−1��∂y,zFk(y, z)(log |z − A|)m��
+L1(z∈[0,1])
+�
++
+sup
+y∈[0,1],α∈{0,1}
+�
+|k|−1/2−α ��∂α
+y,zFk(y, z)
+��
+L2(z∈[0,1])
+�
+≲ | log ⟨k⟩|m,
+for m ∈ {0, 1, 2, 3}.
+(3.7)
+The bounds (3.4) and (3.7) can be proved by explicit calculations and are useful in the proof
+of Lemma 4.1 below.
+3.2. Bounds on the modified Green’s function. It follows from Assumption 1.1 that there
+exists a δ0 ∈ (0, 1/8) such that
+inf{|y∗|, |y∗ − 1|} > 10δ0
+and
+sup
+y∈(y∗−4δ0,y∗+4δ0)
+|b′′′(y)|δ0 < |b′′(y∗)|/10.
+(3.8)
+Define the set
+Σδ0 := {b(y) : y ∈ [y∗ − δ0, y∗ + δ0]},
+(3.9)
+
+8
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+and fix a standard smooth cutoff function ϕ ∈ C∞
+c (−2, 2) satisfying ϕ ≡ 1 on [−3/2, 3/2]. For
+simplicity of notations, we denote
+I := (0, 1).
+(3.10)
+To simplify notations we define also for d ∈ (0, 1/10),
+Sd := [y∗ − d, y∗ + d].
+(3.11)
+For applications below, we also need to study the “modified Green’s function” Gk(y, z; λ+iǫ)
+for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0}, which satisfies for y, z ∈ (0, 1),
+(k2−∂2
+y)Gk(y, z; λ+iǫ)+
+b′′(y)
+b(y) − λ + iǫ
+�
+ϕ
+�y − y∗
+δ0
+�
+−ϕ
+�y − y∗
+δ(λ)
+��
+Gk(y, z; λ+iǫ) = δ(y−z), (3.12)
+with the boundary condition
+Gk(y, z; λ + iǫ)|y∈{0,1} = 0.
+(3.13)
+In the above, we have used the notation that
+δ(λ) := 8
+�
+|λ − b(y∗)|/b′′(y∗).
+(3.14)
+Define the weight ̺(y; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0} as
+̺(y; λ + iǫ) :=|λ − b(y∗)|1/2 + |ǫ|1/2 + |y − y∗|.
+(3.15)
+The crucial bounds we need for the modified Green’s function Gk(y, z; λ + iǫ) is the following.
+Lemma 3.1. Let Gk(y, z; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0} be defined as
+in (3.12). Then we have the identity for y, z ∈ [0, 1],
+Gk(y, z; λ + iǫ) = Gk(z, y; λ + iǫ),
+(3.16)
+and the following statements hold.
+(i) We have the bounds
+sup
+y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|}
+|Gk(y, z; λ + iǫ)| ≲ min{̺(z; λ + iǫ), 1/|k|},
+sup
+y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|}
+|∂yGk(y, z; λ + iǫ)| ≲ 1;
+(3.17)
+(ii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≳ 1/|k|, we have
+the bounds with α ∈ {0, 1}
+|∂α
+y Gk(y1, z; λ + iǫ)|
+≲
+�
+|k| + ̺−1(y1; λ + iǫ)
+�α
+e−|k||y1−y2|
+�
+|k|
+�
+[y2−1/|k|,y2+1/|k|]∩I
+|Gk(y, z; λ + iǫ)|2 dy
+�1/2
+.
+(3.18)
+(iii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≪ 1/|k|, we have
+the bounds with α ∈ {0, 1}
+|∂α
+y Gk(y1, z; λ + iǫ)| ≲
+�
+|k| + ̺−1(y1; λ + iǫ)
+���
+min
+�̺2(y1; λ + iǫ)
+̺2(y2; λ + iǫ), ̺(y2; λ + iǫ)
+̺(y1; λ + iǫ)
+�
+M,
+(3.19)
+where
+M :=
+�
+1
+̺(y2; λ + iǫ)
+�
+[y2−̺(y2;λ+iǫ),y2+̺(y2;λ+iǫ)]∩I
+|Gk(y, z; λ + iǫ)|2 dy
+�1/2
+.
+(3.20)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS
+9
+Proof. The proof is based on energy estimates and “entanglement inequalities”, as in [15]. See
+also the earlier work [33] where this type of inequality was used. We divide the proof into
+several steps.
+Step 1:
+the proof of (3.17).
+We first establish the bounds (3.17).
+For simplicity of
+notation, we suppress the dependence on z, λ + iǫ and set for y ∈ [0, 1],
+h(y) := Gk(y, z; λ + iǫ),
+V (y) :=
+b′′(y)
+b(y) − λ + iǫ
+�
+ϕ
+�y − y∗
+δ0
+�
+− ϕ
+�y − y∗
+δ
+��
+.
+(3.21)
+Multiplying h to (3.12) and integrating over [0, 1], we obtain that
+� 1
+0
+|∂yh(y)|2 + |k|2|h(y)|2 dy +
+� 1
+0
+b′′(y)
+b(y) − λ + iǫ
+�
+ϕ
+�y − y∗
+δ0
+�
+− ϕ
+�y − y∗
+δ
+��
+|h(y)|2 dy = h(z).
+(3.22)
+Note that for y ∈ [0, 1], ℜV (y) ≥ 0, and in addition, for y ∈ Sδ0 and
+|y − y∗| > C0
+�
+|λ − b(y∗)|1/2 + |ǫ|1/2�
+with sufficiently large C0 ≫ 1,
+1 + ℜV (y) ≳
+1
+̺2(y; λ + iǫ).
+(3.23)
+It follows from (3.22) that
+� 1
+0
+|∂yh(y)|2 + |k|2|h(y)|2 dy +
+�
+y∈Sδ0, |y−y∗|>C0(δ+|ǫ|1/2)
+1
+�
+̺(y; λ + iǫ)
+�2 |h(y)|2 dy
+≲ |h(z)|.
+(3.24)
+Using the Sobolev type inequality
+∥h∥L∞(J) ≲ ∥h∥L2(J∗)|J|−1/2 + ∥∂yh∥L2(J)|J|1/2,
+(3.25)
+for any interval J, J∗ with J∗ ⊆ J and |J∗| ≳ |J|, and choosing the interval J ⊂ I as an interval
+containing z with length of the size C1 min{1/|k|, ̺(z; λ + iǫ)}, we obtain from (3.24) that
+� 1
+0
+|∂yh(y)|2 + |k|2|h(y)|2 dy +
+�
+y∈Sδ0, |y−y∗|>C0(δ+|ǫ|1/2)
+1
+�
+̺(y; λ + iǫ)
+�2 |h(y)|2 dy
+≲ min{1/|k|, ̺(z; λ + iǫ)}.
+(3.26)
+The desired bound (3.17) follows from (3.26), (3.25), and equation (3.12).
+Step 2: the proof of (3.18). Denote
+M1 :=
+�
+|k|
+�
+[y2−1/|k|,y2+1/|k|]∩I
+|Gk(y, z; λ + iǫ)|2 dy
+�1/2
+.
+(3.27)
+For the sake of concreteness, we assume that y1 > z (so y2 ∈ [z, y1]). We shall also assume
+that y1 − y2 ≫ 1/|k| as the other case is analogous but easier. For ϕ ∈ C1
+p([y2, 1]), the space of
+piecewise C1 functions, with ϕ(y2) = 0, we multiply ϕ2h to equation (3.12) and integrate over
+
+10
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+[y2, 1] to obtain that
+� 1
+y2
+|∂yh(y)|2ϕ2(y) + 2∂yh(y)h(y)ϕ(y)∂yϕ(y) + |k|2ϕ2(y)|h(y)|2 + V (y)|h(y)|2ϕ2(y) dy = 0.
+(3.28)
+Taking the real part of (3.28) and using Cauchy-Schwarz inequality, we get that
+� 1
+y2
+�
+|∂yϕ(y)|2 − |k|2|ϕ(y)|2�
+|h(y)|2 dy ≥ 0.
+(3.29)
+We now choose ϕ more specifically as follows. We require that
+ϕ(y2) = 0, ϕ′′(y) = 0 for y ∈ [y2, y2 + 1/|k|], ϕ(y2 + 1/|k|) = 1,
+ϕ′(y) = |k|ϕ(y) for y ∈ [y2 + 1/|k|, y1 − 1/|k|], ϕ′(y) = 0 for y ∈ [y1 − 1/|k|, 1].
+(3.30)
+It follows from (3.29)-(3.30) that
+� 1
+y1−1/|k|
+|k|2ϕ2(y)|h(y)|2 dy ≲ |k|M2
+1 ,
+ϕ(y) ≈ e|k||y1−y2| for y ∈ [y1 − 1/|k|, y1 + 1/|k|] ∩ I.
+(3.31)
+The desired bounds (3.18) follow from (3.31) and equation (3.12).
+Step 3: the the proof of (3.19). For the sake of concreteness, we assume that y1 > z (and
+so y2 ∈ [z, y1]). We shall also assume that y1 − y2 ≫ ̺(y2; λ + iǫ) and that y2 > y∗ + δ + |ǫ|1/2
+as the other cases are analogous.
+For ϕ ∈ C1
+p([y2, 1]) with ϕ(y2) = 0, we multiply ϕ2h to equation (3.12) and integrate over
+[y2, 1] to obtain that
+� 1
+y2
+|∂yh(y)|2ϕ2(y) + 2∂yh(y)h(y)ϕ(y)∂yϕ(y) + |k|2ϕ2(y)|h(y)|2 + V (y)|h(y)|2ϕ2(y) dy = 0.
+(3.32)
+Write for y ∈ [y2, 1]
+h(y) = (y − y∗)1/2h∗(y).
+(3.33)
+Simple calculations show that
+� 1
+y2
+(y − y∗)|∂yh∗(y)|2ϕ2(y) + 2(y − y∗)∂yϕ(y)ϕ(y)∂yh∗(y)h∗(y) +
+1
+4(y − y∗)|h∗(y)|2ϕ2(y)
++ |k|2|h(y)|2ϕ2(y) + (y − y∗)V (y)ϕ2(y)|h∗(y)|2 dy = 0.
+(3.34)
+Therefore
+� 1
+y2
+�
+1
+4(y − y∗) + (y − y∗)ℜVy∗(y)
+�
+ϕ2(y)|h∗(y)|2 dy ≤
+� 1
+y2
+(y − y∗)(∂yϕ)2(y)|h∗(y)|2 dy, (3.35)
+which implies that
+� 1
+y2
+1
+y − y∗
+��
+(y − y∗)∂yϕ
+�2(y) −
+�
+1/4 + (y − y∗)2ℜV (y)
+�
+ϕ2(y)
+�
+|h∗(y)|2 dy ≥ 0.
+(3.36)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 11
+We notice the pointwise bounds for y ∈ [y2, 1],
+1/4 + (y − y∗)2ℜV (y) ≥ max
+�
+0, 9/4 − C2
+̺2(y2; λ + iǫ)
+(y − y∗)2
+− C2|y − y∗|
+�
+.
+(3.37)
+Now we choose ϕ ∈ C1
+p([y2, 1]) more precisely as follows. We require that
+ϕ(y2) = 0, ϕ′′(y) = 0 for y ∈ [y2, y2 + ̺(y2; λ + iǫ)], ϕ(y2 + ̺(y2; λ + iǫ)) = 1,
+(y − y∗)ϕ′(y) =
+�
+1/4 + (y − y∗)2ℜV (y)
+�1/2ϕ(y)
+for y ∈ [y2 + ̺(y2; λ + iǫ), y1 − ̺(y1; λ + iǫ)], and ϕ′(y) = 0 for y ∈ [y1 − ̺(y1; λ + iǫ), 1].
+(3.38)
+It follows from (3.36)-(3.38) that
+� y1
+y1���̺(y1;λ+iǫ)
+1
+̺(y1; λ + iǫ)ϕ2(y)|h∗(y)|2 dy ≲ M2/̺(y2; λ + iǫ),
+ϕ(y) ≈
+(y1 − y∗)3/2
+̺3/2(y2; λ + iǫ) for y ∈ [y1 − ̺(y1; λ + iǫ), y1].
+(3.39)
+The desired bounds (3.19) follow from the change of variable (3.33), the bound (3.36), (3.39)
+and equation (3.12).
+□
+As a corollary of Lemma 3.1, we have the following additional bounds on the modified
+Green’s function.
+Lemma 3.2. Let Gk(y, z; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0, k ∈ Z\{0} and ǫ ∈ [−1/8, 1/8]\{0}
+be defined as in (3.12). Recall the definition (3.14) for δ = δ(λ) > 0. Define
+h := 10(δ + |ǫ|1/2),
+(3.40)
+and also for y, z ∈ [0, 1],
+Hk(y, z; λ + iǫ) :=
+�
+∂z + ϕ
+�y − y∗
+h
+�
+∂y
+�
+Gk(y, z; λ + iǫ).
+(3.41)
+Then the following statements hold for z ∈ S4δ.
+(i) We have the bounds
+sup
+y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|}
+|Hk(y, z; λ + iǫ)| ≲ 1,
+sup
+y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|}
+|∂yHk(y, z; λ + iǫ)| ≲ 1/ min{̺(z; λ + iǫ), 1/|k|};
+(3.42)
+(ii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≳ 1/|k|, we have
+the bounds with α ∈ {0, 1}
+�
+min{̺(y1; λ + iǫ), 1/|k|}
+�α|∂α
+y Hk(y1, z; λ + iǫ)|
+≲
+e−|k||y1−y2|
+min{̺(z; λ + iǫ), 1/|k|}
+�
+|k|
+�
+[y2−1/|k|,y2+1/|k|]∩I
+|Gk(y, z; λ + iǫ)|2 dy
+�1/2
+.
+(3.43)
+
+12
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+(iii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≪ 1/|k|, we have
+the bounds with α ∈ {0, 1}
+�
+min{̺(y1; λ + iǫ), 1/|k|}
+�α|∂α
+y Hk(y1, z; λ + iǫ)|
+≲
+1
+min{̺(z; λ + iǫ), 1/|k|} min
+�̺2(y1; λ + iǫ)
+̺2(y2; λ + iǫ), ̺(y2; λ + iǫ)
+̺(y1; λ + iǫ)
+�
+M,
+(3.44)
+where
+M :=
+�
+1
+̺(y2; λ + iǫ)
+�
+[y2−̺(y2;λ+iǫ),y2+̺(y2;λ+iǫ)]∩I
+|Gk(y, z; λ + iǫ)|2 dy
+�1/2
+.
+(3.45)
+Proof. Denote with a slight abuse of notation for y ∈ [0, 1],
+ϕ†(y) := ϕ
+�y − y∗
+h
+�
+,
+V (y) :=
+b′′(y)
+b(y) − λ + iǫ
+�
+ϕ
+�y − y∗
+δ0
+�
+− ϕ
+�y − y∗
+δ(λ)
+��
+.
+(3.46)
+Then Hk,j(y, z; λ + iǫ) satisfies for y ∈ [0, 1], z ∈ S4δ,
+(k2 − ∂2
+y)Hk(y, z; λ + iǫ) + V (y)Hk(y, z; λ + iǫ)
+= −∂2
+yϕ†(y)∂yGk(y, z; λ + iǫ) − ∂yV (y)ϕ†(y)Gk(y, z; λ + iǫ) − 2∂yϕ†(y)∂2
+yGk(y, z; λ + iǫ).
+(3.47)
+The desired bounds then follow from equation (3.47), Lemma 3.1 and standard elliptic regu-
+larity theory.
+□
+The bounds in Lemma 3.1 and Lemma 3.2 are quite sharp, since we can exploit the decay
+coming from both k2 and
+b′′(y)
+b(y)−λ+iǫ
+�
+ϕ
+� y−y∗
+δ0
+�
+− ϕ
+� y−y∗
+δ(λ)
+��
+. It is however somewhat complicated
+to formulate a concrete bound that is easy to use. Instead, the following simple bounds are
+more often used.
+Corollary 3.3. Let Gk(y, z; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0} be defined
+as in (3.12). Then we have the following bounds.
+(i) For y, z ∈ [0, 1], we have the bounds with α ∈ {0, 1}
+�
+|k| + ̺−1(y; λ + iǫ)
+�−α
+|∂α
+y Gk(y, z; λ + iǫ)|
+≲
+1
+|k| + ̺−1(z; λ + iǫ) min
+�
+e−|k||y−z|, ̺2(y; λ + iǫ)
+̺2(z; λ + iǫ), ̺(z; λ + iǫ)
+̺(y; λ + iǫ)
+�
+.
+(3.48)
+(iii) For y ∈ [0, 1], z ∈ S4δ, we have the bounds with α ∈ {0, 1, 2}
+�
+|k| + ̺−1(y; λ + iǫ)
+�−α
+|∂α
+y Hk(y, z; λ + iǫ)| ≲ min
+�
+e−|k||y−z|, ̺2(y; λ + iǫ)
+̺2(z; λ + iǫ), ̺(z; λ + iǫ)
+̺(y; λ + iǫ)
+�
+.
+(3.49)
+Proof. The desired bounds (3.48)-(3.49) follow directly from Lemma 3.1 and Lemma 3.2, by
+choosing, if necessary, another point y′ between y and z such that ̺(y′; λ + iǫ) ≈ 1/|k|, and
+applying (3.48)-(3.49) on intervals [min{z, y′}, max{z, y′}] and [min{y′, y}, max{y′, y}] succes-
+sively.
+□
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 13
+4. The limiting absorption principle
+In this section we study the solvability of the main Rayleigh equations (2.8). It turns out
+that the situation is very different for the spectral range λ ∈ Σ\Σδ0/2 (the non-degenerate case)
+and λ ∈ Σδ0 (the degenerate case). We first consider the non-degenerate case.
+4.1. The non-degenerate case. Fix ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σ\Σδ0/2, k ∈ Z\{0}. Define for
+each g ∈ L2(0, 1) the operator
+Tk,λ,ǫg(y) :=
+� 1
+0
+Gk(y, z)
+b′′(z)g(z)
+b(z) − λ + iǫdz.
+(4.1)
+For applications below, we fix a smooth cutoff function Φ ∈ C∞
+0 (y∗ − δ0/3, y∗ + δ0/3) with
+Φ ≡ 1 on [y∗ − δ0/4, y∗ + δ0/4]. To obtain the optimal dependence on the frequency variable
+k, we define
+∥g∥H1
+k(I) := ∥g∥L2(I) + |k|−1∥g′∥L2(I).
+(4.2)
+Lemma 4.1. For ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σ\Σδ0/2, k ∈ Z\{0}, the operator Tk,λ,ǫ satisfies the
+bound
+∥Tk,λ,ǫg∥H1
+k(I) ≲ |k|−1/3∥g∥H1
+k(I),
+for all g ∈ H1
+k(I).
+(4.3)
+In addition, we have the more precise regularity structure
+�����∂yTk,λ,ǫg(y) + b′′(y)(1 − Φ(y))g(y)
+b′(y)
+log (b(y) − λ + iǫ)
+����
+W 1,1(R)
+≲ ⟨k⟩4/3∥g∥H1
+k(I).
+(4.4)
+Proof. We can decompose for y ∈ [0, 1],
+Tk,λ,ǫg(y) := T 1
+k,λ,ǫg(y) + T 2
+k,λ,ǫg(y),
+(4.5)
+where
+T 1
+k,λ,ǫg(y) :=
+� 1
+0
+Gk(y, z)Φ(z)b′′(z)g(z)
+b(z) − λ − iǫ dz,
+T 2
+k,λ,ǫg(y) :=
+� 1
+0
+Gk(y, z)(1 − Φ(z))b′′(z)g(z)
+b(z) − λ + iǫ
+dz.
+(4.6)
+It follows from the definition of Φ that T 1
+k,λ,ǫg(y) satisfies the bound
+∥T 1
+k,λ,ǫg(y)∥H1
+k(I) ≲ |k|−1/3∥g∥H1
+k(I),
+∥∂yT 1
+k,λ,ǫg(y)∥W 1,1(I) ≲ ⟨k⟩4/3∥g∥H1
+k(I).
+(4.7)
+To bound T 2
+k,λ,ǫg(y), we follow the approach in [14]. Using integration by parts, we obtain that
+T 2
+k,λ,ǫg(y) =
+� 1
+0
+Gk(y, z)(1 − Φ(z))b′′(z)g(z)
+b′(z)
+∂z log(b(z) − λ + iǫ) dz
+= −
+� 1
+0
+∂zGk(y, z)(1 − Φ(z))b′′(z)g(z)
+b′(z)
+log(b(z) − λ + iǫ) dz
+−
+� 1
+0
+Gk(y, z)∂z
+�(1 − Φ(z))b′′(z)g(z)
+b′(z)
+�
+log(b(z) − λ + iǫ) dz.
+(4.8)
+The desired bounds follow from the bound (3.4), the formula (3.6) and (3.7).
+□
+We now prove the limiting absorption principle, using the assumption that there is no discrete
+or generalized embedded eigenvalues.
+
+14
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+Lemma 4.2. There exist ǫ0, κ > 0, such that the following statement holds.
+For all λ ∈
+Σ\Σδ0/2, k ∈ Z\{0}, 0 < |ǫ| < ǫ0 and any g ∈ H1
+k(I), we have the bound
+∥g + Tk,λ,ǫg∥H1
+k(I) ≥ κ∥g∥H1
+k(I).
+(4.9)
+Proof. We prove (4.9) by contradiction. Assume that there exist for j ≥ 1, a sequence of num-
+bers kj ∈ Z\{0}, λj ∈ Σ\Σδ0/2, ǫj ∈ R\{0} → 0 and functions gj ∈ H1
+kj(I) with ∥gj∥H1
+kj (I) = 1,
+satisfying kj → k∗ ∈ (Z\{0}) ∪ {±∞}, λj → λ∗ ∈ Σ\Σδ0 as j → ∞, such that
+��gj + Tkj,λj,ǫjgj
+��
+H1
+kj (I) → 0,
+as j → ∞.
+(4.10)
+The bounds (4.3) and (4.10) imply that |kj| ≲ 1. Thus k∗ ∈ Z\{0}. Using ∥gj∥H1
+kj (I) = 1, the
+bounds (4.4) and the compact embedding W 1,1(I) → L2(I), we conclude that by passing to a
+subsequence, Tkj,λj,ǫjgj converges in H1(I). In view of (4.10) we can assume that gj → g in
+H1(I), where ∥g∥H1
+k∗ = 1.
+Using formula (4.1), we obtain from (4.10) that for y ∈ I,
+g(y) + lim
+j→∞
+� 1
+0
+Gk∗(y, z)
+b′′(z)g(z)
+b(z) − λ + iǫj
+dz = 0.
+(4.11)
+Applying k2
+∗ − d2
+dy2 to (4.11), we get that for y ∈ I,
+k2
+∗g(y) − g′′(y) + lim
+j→∞
+(b(y) − λ∗)b′′(y)g(y)
+(b(y) − λ∗)2 + ǫ2
+j
++ iπ
+�
+z∈[0,1],b(z)=λ
+b′′(z)g(z)
+|b′(z)|
+δ(y − z) = 0,
+(4.12)
+in the sense of distributions for y ∈ (0, 1), which contradicts our spectral assumption that λ∗
+is not a generalized embedded eigenvalue for Lk. The lemma is then proved.
+□
+4.2. The degenerate case λ ∈ Σδ0. Recall the definition (3.14) for δ = δ(λ). For λ ∈ Σδ0, k ∈
+Z\{0}, y ∈ I and ǫ ∈ [−1/8, 1/8]\{0}, we denote
+dk(λ, ǫ) :=
+�
+|λ − b(y∗)|1/2 + |ǫ|1/2�
+∧ 1
+|k|,
+̺k(y; λ + iǫ) := ̺(y; λ + iǫ) ∧ 1
+|k|.
+(4.13)
+Define the weight and the associated weighted Sobolev spaces XN,̺k and XL,̺k as
+∥g∥XN,̺k (I) :=
+�
+α∈{0,1}
+(δ + |ǫ|1/2)−1/2���
+�
+dk(λ, ǫ)
+�(−7/4+α)∂α
+y g
+���
+L2(S3(δ+|ǫ|1/2))
++
+�
+α∈{0,1}
+∥̺−7/4+α
+k
+(·; λ + iǫ)∂α
+y g∥L∞(I\S3(δ+|ǫ|1/2)),
+(4.14)
+and
+∥g∥XL,̺k (I) :=
+�
+α∈{0,1}
+(δ + |ǫ|1/2)−1/2��dα
+k(λ, ǫ)∂α
+y g
+��
+L2(S3(δ+|ǫ|1/2))
++
+�
+α∈{0,1}
+��dk(λ, ǫ)−1̺α+1
+k
+(·; λ + iǫ)∂α
+y g
+��
+L∞(I\S3(δ+|ǫ|1/2)),
+(4.15)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 15
+Fix ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σδ0, k ∈ Z\{0}. Recall the definition (3.14) for δ = δ(λ) > 0.
+Define for each g ∈ L2(0, 1) the operator
+T ∗
+k (λ + iǫ)g(y) :=
+� 1
+0
+Gk(y, z; λ + iǫ)
+�
+1 − ϕ
+�y − y∗
+δ0
+�
++ ϕ
+�y − y∗
+δ
+��
+b′′(z)g(z)
+b(z) − λ + iǫdz.
+(4.16)
+Then we have the following bounds for T ∗
+k (λ + iǫ).
+Lemma 4.3. For ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σδ0, k ∈ Z\{0}, the operator T ∗
+k (λ + iǫ) satisfies the
+bound for X ∈ {XN,̺k(I), XL,̺k(I)}
+∥T ∗
+k (λ + iǫ)g∥X ≲ (1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2))−1/4∥g∥X,
+for all g ∈ H1
+k(I).
+(4.17)
+Proof. We provide the detailed proof only for the case X = XN,̺k(I) as the other case is
+analogous. Since k, λ, ǫ are fixed, for simplicity of notations, we suppress the dependence on
+k, λ, ǫ to write T ∗ as T ∗
+k (λ + iǫ), and decompose for y ∈ I,
+T ∗g(y) := T ∗
+1 g(y) + T ∗
+2 g(y),
+(4.18)
+where
+T ∗
+1 g(y) :=
+� 1
+0
+Gk(y, z; λ + iǫ)
+�
+1 − ϕ
+�z − y∗
+δ0
+��
+b′′(z)g(z)
+b(z) − λ + iǫdz,
+T ∗
+2 g(y) :=
+� 1
+0
+Gk(y, z; λ + iǫ)ϕ
+�z − y∗
+δ
+�
+b′′(z)g(z)
+b(z) − λ + iǫdz.
+(4.19)
+It follows from the bounds on modified Green’s function Gk(y, z; λ + iǫ), see Lemma 3.1, that
+��T ∗
+1 g
+��
+XN,̺k(I) ≲ |k|−1/2��g
+��
+XN,̺k (I).
+(4.20)
+To prove (4.17), it suffices to prove
+∥T ∗
+2 g∥XN,̺k (I) ≲
+�
+1 + |k|(δ + |ǫ|1/2)
+�−1/4∥g∥XN,̺k (I).
+(4.21)
+We assume momentarily that |ǫ| ≲ |λ − b(y∗)| and explain how to remove this assumption
+at the end of the proof. We decompose further for y ∈ I,
+T ∗
+2 g(y) =
+� 1
+0
+Gk(y, z; λ + iǫ)ϕ
+�z − y∗
+δ′
+�
+ϕ
+�z − y∗
+δ
+�
+b′′(z)g(z)
+b(z) − λ + iǫdz
++
+� 1
+0
+Gk(y, z; λ + iǫ)
+�
+1 − ϕ
+�z − y∗
+δ′
+��
+ϕ
+�z − y∗
+δ
+�
+b′′(z)g(z)
+b(z) − λ + iǫdz
+:= T ∗
+2,Rg(y) + T ∗
+2,Sg(y),
+(4.22)
+where we have chosen δ′ = δ/C3 with a large constant C3 so that |b(y) − λ| ≈ |λ − b(y∗)| for
+|y − y∗| < δ′.
+It suffices to prove for ⋄ ∈ {R, S}
+∥T ∗
+2,⋄g∥XN,̺k (I) ≲
+�
+1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2)
+�−1/4∥g∥XN,̺k (I).
+(4.23)
+Step 1. We first prove (4.23) with ⋄ = R.
+Case I: 1/|k| > |λ − b(y∗)|1/2 + |ǫ|1/2. In this case for |z − y∗| ≲ δ and |y − y∗| ≲ 1 we have
+the bound
+��Gk(y, z; λ + iǫ)
+�� ≲
+δ2 + |ǫ|
+|y − y∗| + δ + |ǫ|1/2 ,
+��∂yGk(y, z; λ + iǫ)
+�� ≲
+δ2 + |ǫ|
+(|y − y∗| + δ + |ǫ|1/2)2 . (4.24)
+
+16
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+It follows from the bound (4.24) that
+∥T ∗
+2,Rg∥XN,̺k (I) ≲
+�
+1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2)
+�−1/4∥g∥XN,̺k (I)
+(4.25)
+Case II: 1/|k| ≪ |λ − b(y∗)|1/2 + |ǫ|1/2. In this case, we have for |z − y∗| ≲ δ and |y − y∗| ≲ 1
+that
+��Gk(y, z; λ + iǫ)
+�� + |k|−1��∂yGk(y, z; λ + iǫ)
+�� ≲ |k|−1e−|k||y−z|.
+(4.26)
+The desired bound
+∥T ∗
+2,Rg∥XN,̺k (I) ≲
+�
+1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2)
+�−1/4∥g∥XN,̺k (I)
+(4.27)
+follows from (4.26).
+Step 2. We now turn to the proof of (4.23) with ⋄ = S and still consider two cases.
+Case I: 1/|k| > |λ − b(y∗)|1/2 + |ǫ|1/2. Denoting for y ∈ I,
+ϕ∗�y − y∗
+δ
+�
+:=
+�
+1 − ϕ
+�z − y∗
+δ′
+��
+ϕ
+�z − y∗
+δ
+�
+,
+(4.28)
+we can rewrite
+T ∗
+2,Sg(y) =
+� 1
+0
+Gk(y, z; λ + iǫ)ϕ∗�z − y∗
+δ
+�b′′(z)g(z)
+b′(z)
+∂z log b(z) − λ + iǫ
+δ2
+= −
+� 1
+0
+∂z
+�
+Gk(y, z; λ + iǫ)ϕ∗�z − y∗
+δ
+�b′′(z)g(z)
+b′(z)
+�
+log b(z) − λ + iǫ
+δ2
+dz.
+(4.29)
+As a consequence of (4.29), we also have
+∂y
+�
+T ∗
+2,Sg(y)
+�
+= ∂y
+� 1
+0
+Gk(y, z; λ + iǫ)ϕ∗�z − y∗
+δ
+�b′′(z)g(z)
+b′(z)
+∂z log b(z) − λ + iǫ
+δ2
+dz
+= −
+� 1
+0
+�
+∂y(∂z + ∂y)Gk(y, z; λ, ǫ)ϕ∗�z − y∗
+δ
+�b′′(z)g(z)
+b′(z)
+�
+log b(z) − λ + iǫ
+δ2
+dz
++
+� 1
+0
+�
+∂2
+yGk(y, z; λ + iǫ)ϕ∗�z − y∗
+δ
+�b′′(z)g(z)
+b′(z)
+�
+log b(z) − λ + iǫ
+δ2
+dz
+−
+� 1
+0
+∂yGk(y, z; λ + iǫ)∂z
+�
+ϕ∗�z − y∗
+δ
+�b′′(z)g(z)
+b′(z)
+�
+log b(z) − λ + iǫ
+δ2
+dz.
+(4.30)
+Note that on the support of ϕ∗(z−y∗
+δ
+), we have
+|b′(z)| ≈ δ,
+̺(z; λ + iǫ) ≈ δ.
+(4.31)
+The desired bound (4.23) for ⋄ = S follows from (4.29)-(4.30), Lemma 3.1 and 3.2, and we
+have, in addition,
+(δ + |ǫ|1/2)−1/2
+����∂y
+�
+∂yT ∗
+2,Sg(y) + ϕ∗�y − y∗
+δ
+�b′′(y)g(y)
+b′(y)
+log b(y) − λ + iǫ
+δ2
+�����
+L2(S3(δ+|ǫ|1/2),j)
+≲ δ−1/4�
+1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2)
+�−1/4
+∥g∥XN,̺k (I).
+(4.32)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 17
+Case II: 1/|k| ≪ |λ − b(y∗)|1/2 + |ǫ|1/2. This case is analogous to Case I, using Lemma 3.1
+and Lemma 3.2.
+Finally we turn to the assumption that |ǫ|1/2 ≲ δ.
+Suppose |ǫ|1/2 ≫ δ, then the factor
+1
+b(z)−λ+iǫ is not truly singular, and the desired bounds (4.21) follow directly from the bounds
+on the modified Green’s function Gk(y, z; λ + iǫ) from Lemma 3.1 and Lemma 3.2. Indeed, we
+have the stronger bound
+∥T ∗
+2 g∥XN,̺k (I) ≲
+δ
+�
+|ǫ|
+∥g∥XN,̺k (I),
+(4.33)
+which will be useful below.
+□
+The following limiting absorption principle plays an essential role in establishing the vorticity
+depletion phenomenon.
+Lemma 4.4. There exist positive numbers ǫ0, κ such that the following statement holds.
+For ǫ ∈ [−ǫ0, ǫ0]\{0}, λ ∈ Σδ0, k ∈ Z\{0}, and X ∈ {XN,̺k(I), XL,̺k(I)},
+∥(I + T ∗
+k (λ + iǫ))g∥XN,̺k (I) ≥ κ∥g∥XN,̺k (I),
+for all g ∈ H1
+k(I).
+(4.34)
+Proof. We only consider the case X = XN,̺k(I) as the other case is analogous. We prove (4.34)
+by a contradiction argument. Assume (4.34) does not hold for any ǫ0 > 0. Then there exist
+for ℓ ∈ Z ∩ [1, ∞),
+λℓ → λ∗ ∈ Σδ0, ǫℓ ̸= 0 with ǫℓ → 0, kℓ → k∗ ∈ (Z\{0}) ∪ {±∞},
+(4.35)
+and functions gℓ satisfying
+∥gℓ∥XN,��kℓ (I) = 1
+(4.36)
+such that
+��(I + T ∗
+kℓ(λℓ + iǫℓ))gℓ
+��
+XN,̺kℓ (I) → 0.
+(4.37)
+We can assume that λ∗ = b(y∗), otherwise the proof follows from the argument in the non-
+degenerate case. We consider several cases.
+Case I: lim supℓ→∞ ∥gℓ∥H1(I\Sδ0) > 0. By the bound (4.20), we can assume that k∗ ∈ Z\{0}.
+By the bounds (4.36) and (4.37), we can assume (passing to a subsequence if necessary) that
+gℓ → g, in H1
+loc(I\{y∗}) as ℓ → ∞,
+g(0) = g(1) = 0.
+(4.38)
+Then it follows from (4.36) and (4.37) that g satisfies
+|g(y)| ≲ |y − y∗|7/4,
+(4.39)
+and for y ∈ (0, 1),
+(k2
+∗ − ∂2
+y)g(y) +
+b′′(y)
+b(y) − b(y∗)g(y) = 0,
+(4.40)
+which imply that b(y∗) is an embedded eigenvalue for Lk, a contradiction to the spectral
+assumption.
+Case II: lim supℓ→∞ ∥gℓ∥H1(I\Sδ0) = 0. By the bound (4.17) we can assume that |kℓ|(δℓ +
+|ǫℓ|1/2) ≲ 1. In this case, using (4.37), we obtain that (passing to a subsequence if necessary)
+��(|λℓ − b(y∗)| + |ǫ|)−9/8gℓ
+��
+L2([y∗−δℓ−|ǫℓ|1/2, y∗+δℓ+|ǫℓ|1/2])
++
+��(|λℓ − b(y∗)| + |ǫ|)−5/8∂ygℓ
+��
+L2([y∗−δℓ−|ǫℓ|1/2, y∗+δℓ+|ǫℓ|1/2]) ≥ σ > 0,
+(4.41)
+
+18
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+where we recall from (3.14) that
+δℓ ≈ |λℓ − b(y∗)|1/2.
+(4.42)
+We divide into several subcases.
+Subcase II.1: |ǫℓ|1/2 ≈ δℓ for a subsequence.
+Define the change of variables for ℓ ≥ 1, y ∈ I,
+y − y∗ = δℓY,
+gℓ(y) := (|λℓ − b(y∗)| + |ǫℓ|)−7/8Hℓ(Y ).
+(4.43)
+It follows from (4.32) that we can extract a nontrivial limit H ∈ H1(R) of Hℓ satisfying for
+Y ∈ R,
+(β2 − ∂2
+Y )H(Y ) +
+b′′(y∗)
+b′′(y∗)Y 2/2 + γ + iαH(Y ) = 0,
+(4.44)
+where β ∈ R, α, γ ∈ R\{0}. This is impossible since the shear flow (b′′(y∗)Y 2/2, 0), Y ∈ R is
+spectrally stable.
+Subcase II.2: |ǫℓ|1/2 = o(δℓ) for a subsequence.
+Passing to a subsequence and using rescaling
+as in (4.43) we can extract a nontrivial limit H ∈ H1(R), such that
+(β2 − ∂2
+Y )H(Y ) + lim
+ǫ→0
+b′′(y∗)
+b′′(y∗)Y 2/2 + γ + iǫH(Y ) = 0.
+(4.45)
+This is again impossible since the shear flow (b′′(y∗)Y 2/2, 0), Y ∈ R is spectrally stable.
+Subcase II.3: δℓ = o(|ǫℓ|1/2) for a subsequence.
+This case is not possible thanks to the
+bound (4.33). The lemma is now proved.
+□
+5. Bounds on ψι
+k,ǫ: the non-degenerate case
+In this section we obtain bounds on ψι
+k,ǫ(y, λ) in the non-degenerate case, i.e. when λ ∈
+Σ\Σδ0/2. Since the arguments are analogous to those in [14], we will be brief in the proofs, and
+provide only comments on the main ideas involved.
+We begin with the following preliminary bounds.
+Lemma 5.1. For λ ∈ Σ\Σδ0/2, k ∈ Z\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, we have the bounds
+∥ψι
+k,ǫ(·, λ)∥H1
+k(I) ≲ |k|−1/2∥ω0k∥H1
+k(I).
+(5.1)
+Proof. The desired bounds (5.1) follow directly from the Rayleigh equation (2.8) and Lemma
+4.2, once we use the Green’s function Gk to invert k2 − ∂2
+y and formulate (2.8) as an integral
+equation.
+□
+To obtain control on ∂λψι
+k,ǫ(·, λ) for λ ∈ Σ\Σδ0/2, we take derivative in (2.8), and obtain
+that
+(k2 − ∂2
+y)∂λψι
+k,ǫ(y, λ) +
+b′′(y)∂λψι
+k,ǫ(y, λ)
+b(y) − λ + iιǫ
+=
+ωk
+0(y)
+(b(y) − λ + iιǫ)2 −
+b′′(y)ψι
+k,ǫ(z, λ)
+(b(y) − λ + iιǫ)2 ,
+(5.2)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 19
+for y ∈ I with zero boundary value at y ∈ {0, 1}. Reformulating (5.2) as an integral equation,
+we obtain that
+∂λψι
+k,ǫ(y, λ) +
+� 1
+0
+Gk(y, z)
+b′′(z)∂λψι
+k,ǫ(z, λ)
+b(z) − λ + iιǫ
+dz
+=
+� 1
+0
+Gk(y, z)
+ωk
+0(z)
+(b(z) − λ + iιǫ)2 dz −
+� 1
+0
+Gk(y, z)
+b′′(z)ψι
+k,ǫ(z, λ)
+(b(z) − λ + iιǫ)2 dz.
+(5.3)
+Recall the definition of the smooth cutoff function Φ below (4.1). We have the following bounds
+for ∂λψι
+k,ǫ(y, λ) when λ ∈ Σ\Σδ0.
+Lemma 5.2. For λ ∈ Σ\Σδ0/2, k ∈ Z\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, ∂λψι
+k,ǫ(y, λ) satisfies the
+following decomposition
+∂λψι
+k,ǫ(y, λ) =
+�b′(y0)ωk
+0(y)
+|b′(y)|2
+−
+b′′(y)ψι
+k,ǫ(y, λ)
+|b′(y)|2
+�
+(1 − Φ(y)) log (b(y) − λ + iιǫ)
++
+�
+σ=0,1
+ωk
+0(σ)Ψι
+k,σ,ǫ(y, λ) log (b(σ) − λ + iιǫ) + Rι
+σ,k,y0,ǫ(y).
+(5.4)
+In the above for σ ∈ {0, 1}, ι ∈ {±}, 0 < ǫ < ǫ0, and λ ∈ Σ\Σδ0/2,
+��Rι
+σ,k,y0,ǫ
+��
+H1
+k(I) ≲ |k|1/2∥ω0k∥H2
+k(I),
+��Ψι
+k,σ,ǫ(·, λ)
+��
+H1
+k(I) ≲ |k|−1/2.
+(5.5)
+Proof. The basic idea is to expand the right hand side of (5.3) using integration by parts, and
+apply Lemma 4.2 after removing the most singular parts. Indeed, denoting schematically,
+U :=
+� 1
+0
+Gk(y, z)
+ωk
+0(z)
+(b(z) − λ + iιǫ)2 dz −
+� 1
+0
+Gk(y, z)
+b′′(z)ψι
+k,ιǫ(z, λ)
+(b(z) − λ + iιǫ)2 dz,
+(5.6)
+we note that ∂λψι
+k,ǫ(y, λ) − U satisfies the equation (recalling (4.1) for the definition of Tk,λ,ιǫ),
+(I + Tk,λ,ιǫ)
+�
+∂λψι
+k,ǫ(y, λ)
+�
+= −Tk,λ,ιǫU.
+(5.7)
+The term Tk,λ,ιǫU ∈ H1
+k(I) (noting however that for the boundary terms we need to track the
+singular coefficient log (b(σ) − λ + iιǫ), σ ∈ {0, 1}), and we can apply Lemma 4.2 to (5.7) in
+order to obtain the desired conclusions. We refer to [14] for the detailed proof.
+□
+To obtain bounds on ∂2
+λψι
+k,ǫ(y, λ) for λ ∈ Σ\Σδ0/2, we take two derivatives in (2.8) and obtain
+that
+(k2 − ∂2
+y)∂2
+λψι
+k,ǫ(y, λ) +
+b′′(y)∂2
+λψι
+k,ǫ(y, λ)
+b(y) − λ + iιǫ
+= 2
+ωk
+0(y)
+(b(y) − λ + iιǫ)3 − 2
+b′′(y)ψι
+k,ǫ(z, λ)
+(b(y) − λ + iιǫ)3 +
+b′′(y)∂λψι
+k,ǫ(z, λ)
+(b(y) − λ + iιǫ)2 ,
+(5.8)
+
+20
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+for y ∈ I with zero boundary value at y ∈ {0, 1}. We can reformulate (5.8) in the integral form
+for y ∈ I, as
+∂2
+λψι
+k,ǫ(y, λ) +
+� 1
+0
+Gk(y, z)
+b′′(z)∂2
+λψι
+k,ǫ(z, λ)
+b(z) − λ + iιǫ
+dz
+=
+� 1
+0
+Gk(y, z)
+�
+2
+ωk
+0(z)
+(b(z) − λ + iιǫ)3 − 2
+b′′(z)ψι
+k,ǫ(z, λ)
+(b(z) − λ + iιǫ)3 +
+b′′(z)∂λψι
+k,ǫ(z, λ)
+(b(z) − λ + iιǫ)2
+�
+dz.
+(5.9)
+We have the following bounds on ∂2
+λψι
+k,ǫ(y, λ) for λ ∈ Σ\Σδ0/2.
+Lemma 5.3. For k ∈ Z\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, we have the following bound
+����∂2
+λψι
+k,ǫ(y, λ) −
+ωk
+0(1)Φ1ι
+k,ǫ(y, λ)
+b(1) − λ + iιǫ −
+ωk
+0(0)Φ0ι
+k,ǫ(y, λ)
+b(0) − λ + iιǫ −
+b′′(y)ψι
+k,ǫ(y, λ) − ωk
+0(y)
+|b′(y)|2(b(y) − λ + iιǫ)
+����
+L2(y∈I,λ∈Σ\Σδ0/2)
+≲ |k|3/2∥ω0k∥H3
+k(I)
+(5.10)
+In the above the functions Φσι
+k,ǫ, σ ∈ {0, 1} satisfy the equation for y ∈ I
+(I + Tk,λ,ιǫ)Φ1ι
+k,ǫ =
+sinh (ky)
+|b′(1)|2 sinh k,
+(I + Tk,λ,ιǫ)Φ0ι
+k,ǫ = sinh (k(1 − y))
+|b′(0)|2 sinh k .
+(5.11)
+Proof. The main idea of the proof is to expand the right side of (5.9) and apply Lemma 4.2
+after removing the most singular terms. Indeed, denoting schematically,
+U∗ :=
+� 1
+0
+Gk(y, z)
+�
+2
+ωk
+0(z)
+(b(z) − λ + iιǫ)3 − 2
+b′′(z)ψι
+k,ιǫ(z, λ)
+(b(z) − λ + iιǫ)3 +
+b′′(z)∂λψι
+k,ιǫ(z, λ)
+(b(z) − λ + iιǫ)2
+�
+dz,
+(5.12)
+we have
+(I + Tk,λ,ιǫ)
+�
+∂2
+λψι
+k,ǫ(y, λ) − U∗ + Tk,λ,ιǫU∗�
+=
+�
+Tk,λ,ιǫ
+�2U∗.
+(5.13)
+We note that ∂2
+λψι
+k,ǫ(y, λ) − U∗ + Tk,λ,ιǫU∗ ∈ H1
+k(I) (however we again need to track the
+singularities in λ in the boundary terms, involving log(b(σ) − λ + iιǫ) and 1/(b(σ) − λ + iιǫ)
+for σ ∈ {0, 1}), and we can apply Lemma (4.2) in order to obtain the desired conclusions. We
+refer to [14] for the detailed proof.
+□
+6. Bounds on ψι
+k,ǫ: the degenerate case
+In this section we use the limiting absorption principle to study the Rayleigh equation (2.8)
+for λ ∈ Σδ0. More precisely, write for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0, 0 < ǫ < ǫ0, (recall the
+definition of ǫ0 from Lemma 4.4)
+ψι
+k,ǫ(y, λ) = φι
+k,ǫ(y, λ) + Ψ(y)
+1
+b′′(y)ω0k(y),
+(6.1)
+where Ψ ∈ C∞
+c (S3δ0) and Ψ ≡ 1 on S2δ0. Recall that Sd = [y∗ −d, y∗ +d] for d > 0 from (3.11).
+Then φι
+k,ǫ(y, λ) satisfies for y ∈ I,
+(k2 − ∂2
+y)φι
+k,ǫ(y, λ) +
+b′′(y)
+b(y) − λ + iιǫφι
+k,ǫ(y, λ) = gι
+k,ǫ(y, λ),
+(6.2)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 21
+where for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0, 0 < ǫ < ǫ0
+gι
+k,ǫ(y, λ) :=
+1 − Ψ(y)
+b(y) − λ + iιǫω0k(y) − (k2 − ∂2
+y)
+� Ψ(y)
+b′′(y)ω0k(y)
+�
+.
+(6.3)
+Our main results are bounds for the functions φι
+k,ǫ(y, λ). We begin with the following pre-
+liminary bounds.
+Lemma 6.1. Assume that k ∈ Z\{0}, λ ∈ Σδ0 and let φι
+k,ǫ(y, λ) with ι ∈ {±}, ǫ ∈ (0, ǫ0) be as
+defined in (6.1)-(6.2). Recall from (3.14) and (4.13) that
+δ := δ(λ) = 8
+�
+|λ − b(y∗)|/|b′′(y∗)|,
+dk = dk(λ, ǫ) :=
+�
+|λ − b(y∗)|1/2 + |ǫ|1/2�
+∧ 1
+|k|.
+(6.4)
+We have the bounds for k ∈ Z\{0}, ǫ ∈ (0, ǫ0), ι ∈ {±}, λ ∈ Σδ0,
+�
+α∈{0,1}
+��d−7/4+α
+k
+∂α
+y φι
+k,ǫ(y, λ)
+��
+L2�
+[y∗−3(δ+|ǫ|1/2),y∗+3(δ+|ǫ|1/2)]
+�(δ + |ǫ|1/2)−1/2
++
+�
+α∈{0,1}
+��(|y − y∗| ∧ dk)−7/4+α∂α
+y φι
+k,ǫ(y, λ)
+��
+L∞�
+[0,1]\[y∗−3(δ+|ǫ|1/2),y∗+3(δ+|ǫ|1/2)]
+�
+≲ |k|5/2��ω0k
+��
+H3
+k(I).
+(6.5)
+Define for y ∈ [0, 1], k ∈ Z\{0}, λ ∈ Σδ0\{b(y∗)},
+ψk(y, λ) := lim
+ǫ→0+
+�
+ψ+
+k,ǫ(y, λ) − ψ−
+k,ǫ(y, λ)
+�
+= lim
+ǫ→0+
+�
+φ+
+k,ǫ(y, λ) − φ−
+k,ǫ(y, λ)
+�
+.
+(6.6)
+Then we have the bounds for λ ∈ Σδ0\{b(y∗)},
+�
+α∈{0,1}
+��(δ ∧ |k|−1)−7/4+α∂α
+y ψk(y, λ)
+��
+L2([y∗−3δ,y∗+3δ])δ−1/2
++
+�
+��{0,1}
+��(δ ∧ |k|−1)−11/4(|y − y∗| ∧ 1
+|k|)1+α∂α
+y ψk(y, λ)
+��
+L∞([0,1]\[y∗−3δ,y∗+3δ]))
+≲ |k|5/2��ω0k
+��
+H3
+k(I).
+(6.7)
+Proof. It follows from (6.3) and our assumptions on the initial data ω0k that we have the bound
+for k ∈ Z\{0}, ι ∈ {±}, 0 < ǫ < ǫ0 and λ ∈ Σδ0,
+��gι
+k,ǫ(y, λ)
+��
+C2
+k(I) ≲ |k|1/2∥ω0k∥H3
+k(I).
+(6.8)
+We can reformulate equation (6.2) in the integral form as (recall the definition of T ∗(λ + iǫ)
+from (4.16))
+φι
+k,ǫ(y, λ) + T ∗
+k (λ + iιǫ)φι
+k,ǫ(y, λ) =
+� 1
+0
+Gk(y, z; λ + iιǫ)gι
+k,ǫ(z, λ)dz,
+(6.9)
+for y ∈ I. By Lemma 4.4, we obtain the bound
+��φι
+k,ǫ(·, λ)
+��
+XN,̺k(I) ≲
+���
+� 1
+0
+Gk(y, z; λ + iιǫ)gι
+k,ǫ(z, λ)dz
+���
+XN,̺k
+≲ |k|5/2∥ω0k∥H3
+k(I),
+(6.10)
+which, by the definition of the space XN,̺k, see (4.14), implies the desired bounds (6.5).
+
+22
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+For applications below on isolating the singularity at λ = b(y), we fix ϕδ(y) ∈ C∞
+c (S2δ) as
+ϕδ(y) := ϕ(y
+δ )
+�
+1 − ϕ( y
+δ′ )
+�
+,
+(6.11)
+for y ∈ I, with δ′ := δ/M and an M ≫ 1 sufficiently large such that |b(y) − λ| ≈ |λ − b(y∗)| for
+|y − y∗| < δ/M.
+To prove (6.7), we note from (6.2) that φ+
+k,ǫ(y, λ) − φ−
+k,ǫ(y, λ) satisfies the equation for y ∈ I.
+(k2 − ∂2
+y)
+�
+φ+
+k,ǫ(y, λ) − φ−
+k,ǫ(y, λ)
+�
++
+b′′(y)
+b(y) − λ + iǫ
+�
+φ+
+k,ǫ(y, λ) − φ−
+k,ǫ(y, λ)
+�
+= g+
+k,ǫ(y, λ) − g−
+k,ǫ(y, λ) −
+�
+b′′(y)
+b(y) − λ + iǫ −
+b′′(y)
+b(y) − λ − iǫ
+�
+φ−
+k,ǫ(y, λ).
+(6.12)
+Denote for λ ∈ Σδ0\{b(y∗)}, ǫ ∈ (0, ǫ0) and y ∈ I the function hk,ǫ(y, λ) as the solution to
+(k2 − ∂2
+y)hk,ǫ(y, λ) +
+b′′(y)
+b(y) − λ + iǫhk,ǫ(y, λ) = ϕδ(y)
+�
+b′′(y)
+b(y) − λ − iǫ −
+b′′(y)
+b(y) − λ + iǫ
+�
+φ−
+k,ǫ(y, λ),
+(6.13)
+with zero Dirichlet boundary condition. Then it is clear that for λ ∈ Σδ0\{b(y∗)}, y ∈ I,
+ψk(y, λ) = lim
+ǫ→0+ hk,ǫ(y, λ).
+(6.14)
+We can reformulate (6.13) as the following integral equation for λ ∈ Σδ0\{b(y∗)}, y ∈ I,
+hk,ǫ(y, λ) + T ∗
+k (λ + iǫ)hk,ǫ(y, λ)
+= −
+� 1
+0
+Gk(y, z; λ + iǫ)ϕδ(z)
+�
+b′′(z)
+b(z) − λ + iǫ −
+b′′(z)
+b(z) − λ − iǫ
+�
+φ−
+k,ǫ(z, λ) dz.
+(6.15)
+It follows from the bound (6.5) that for |ǫ| ≲ (δ ∧ 1
+|k|)4,
+����
+� 1
+0
+Gk(y, z; λ + iǫ)ϕδ(z)
+�
+b′′(z)
+b(z) − λ + iǫ −
+b′′(z)
+b(z) − λ − iǫ
+�
+φ−
+k,ǫ(z, λ) dz
+����
+XL,̺k
+≲ (δ ∧ 1
+|k|)7/4.
+(6.16)
+The desired bound (6.7) then follows from Lemma 4.4 with X = XL,̺k.
+□
+To obtain higher order regularity bounds (in λ) of φι
+k,ǫ(·, λ), we take the derivative ∂λ in
+(6.2). It follows that ∂λφι
+k,ǫ(y, λ) satisfies for y ∈ I,
+�
+k2 − ∂2
+y +
+b′′(y)
+b(y) − λ + iιǫ
+�
+∂λφι
+k,ǫ(y, λ) = −
+b′′(y)
+(b(y) − λ + iιǫ)2 φι
+k,ǫ(y, λ) + ∂λgι
+k,ǫ(y, λ),
+(6.17)
+with zero Dirichlet boundary condition.
+Recall the definition of ϕδ from (6.11). We have the following bounds on ∂λφι
+k,ǫ(y, λ).
+Lemma 6.2. Assume that k ∈ Z\{0}, λ ∈ Σδ0\{b(y∗)}.
+Let ψι
+k,ǫ(y, λ) and φι
+k,ǫ(y, λ) with
+ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} be as defined in (2.8) and (6.1) respectively. Recall from
+(3.14) that
+δ := δ(λ) = 8
+�
+|λ − b(y∗)|/b′′(y∗).
+(6.18)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 23
+Denote for y ∈ [0, 1], ι ∈ {±}, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0},
+Λι
+1,ǫ(y, λ) := φι
+k,ǫ(y, λ)ϕδ(y) b′′(y)
+(b′(y))2 log b(y) − λ + iιǫ
+δ2
+,
+Λ1(y, λ) := ψk(y, λ)ϕδ(y) b′′(y)
+(b′(y))2 log b(y) − λ
+δ2
+.
+(6.19)
+We have the bounds for 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, ι ∈ {±}, and λ ∈ Σδ0 that
+�
+α∈{0,1}
+���(δ ∧ |k|−1)1/4+α∂α
+y
+�
+∂λφι
+k,ǫ(y, λ) − Λι
+1,ǫ(y, λ)
+����
+L2([y∗−3δ,y∗+3δ])δ−1/2
++
+�
+α∈{0,1}
+���(δ ∧ |k|−1)2(|y − y∗| ∧ 1
+|k|)−7/4+α∂α
+y ∂λφι
+k,ǫ(y, λ)
+���
+L∞([0,1]\[y∗−3δ,y∗+3δ]))
+≲ |k|5/2��ω0k
+��
+H3
+k(I).
+(6.20)
+In addition, we have the bounds for λ ∈ Σδ0\{b(y∗)} and k ∈ Z\{0},
+�
+α∈{0,1}
+��(δ ∧ |k|−1)1/4+α∂α
+y
+�
+∂λψk(y, λ) − Λ1(y, λ)
+����
+L2([y∗−3δ,y∗+3δ])δ−1/2
++
+�
+α∈{0,1}
+��(δ ∧ |k|−1)−3/4(|y − y∗| ∧ 1
+|k|)1+α∂α
+y ∂λψk(y, λ)
+��
+L∞([0,1]\[y∗−3δ,y∗+3δ]))
+≲ |k|5/2��ω0k
+��
+H3
+k(I).
+(6.21)
+Proof. Define for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, y ∈ I,
+∂λφι
+k,ǫ(y, λ) := φι
+k,ǫ(y, λ; 1) +
+� 1
+0
+Gk(y, z; λ + iιǫ)
+�
+−b′′(z)
+(b(z) − λ + iιǫ)2 φι
+k,ǫ(z, λ) + ∂λgι
+k,ǫ(z, λ)
+�
+dz.
+(6.22)
+It follows from (6.17) that φι
+k,ǫ(y, λ; 1) satisfies for y ∈ I,
+φι
+k,ǫ(y, λ; 1) + T ∗
+k (λ + iιǫ)φι
+k,ǫ(y, λ; 1)
+= −T ∗
+k (λ + iιǫ)
+� 1
+0
+Gk(y, z; λ + iιǫ)
+�
+−
+b′′(z)
+(b(z) − λ + iιǫ)2 φι
+k,ǫ(z, λ) + ∂λgι
+k,ǫ(z, λ)
+�
+dz.
+(6.23)
+Denote for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, z ∈ I,
+hι
+k,ǫ(z, λ; 1) :=
+b′′(z)
+(b(z) − λ + iιǫ)2 ϕδ(z)φι
+k,ǫ(z, λ),
+hι
+k,ǫ(z, λ; 2) :=
+b′′(z)
+(b(z) − λ + iιǫ)2 (1 − ϕδ(z))φι
+k,ǫ(z, λ),
+hι
+k,ǫ(z, λ; 3) := ∂λgι
+k,ǫ(z, λ).
+(6.24)
+It follows from the bound (6.5) and Lemma 3.1 that for j ∈ {2, 3}
+��T ∗
+k (λ + iιǫ)
+� 1
+0
+Gk(y, z; λ + iιǫ)hι
+k,ǫ(z, λ; j) dz
+��
+XN,̺k ≲ (δ ∧ |k|−1)−2|k|5/2∥ω0k∥H3
+k(I).
+(6.25)
+
+24
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+Using integration by parts argument similar to (4.29)-(4.30), we have also
+����T ∗
+k (λ + iιǫ)
+� 1
+0
+Gk(y, z; λ + iιǫ)hι
+k,ǫ(z, λ; 1) dz
+����
+XN,̺k
+≲ (δ ∧ |k|−1)−2|k|5/2��ω0k
+��
+H3
+k(I).
+(6.26)
+It follows from (6.25)-(6.26) and Lemma 4.4 that for λ\{b(y∗)},
+��φι
+k,ǫ(y, λ; 1)
+��
+XN,̺k ≲ (δ ∧ |k|−1)−2|k|5/2��ω0k
+��
+H3
+k(I).
+(6.27)
+The desired bound (6.20) follows, as a consequence of (6.27) and (6.22).
+Using (6.17), we get that for y ∈ I,
+�
+k2 − ∂2
+y +
+b′′(y)
+b(y) − λ + iǫ
+��
+∂λφ+
+k,ǫ(y, λ) − ∂λφ−
+k,ǫ(y, λ)
+�
+= −
+�
+b′′(y)
+(b(y) − λ + iǫ)2 φ+
+k,ǫ(y, λ) −
+b′′(y)
+(b(y) − λ − iǫ)2 φ−
+k,ǫ(y, λ)
+�
++
+�
+∂λg+
+k,ǫ(y, λ) − ∂λg−
+k,ǫ(y, λ)
+�
+−
+�
+b′′(y)
+b(y) − λ + iǫ −
+b′′(y)
+b(y) − λ − iǫ
+�
+∂λφ−
+k,ǫ(y, λ),
+(6.28)
+with zero Dirichlet boundary condition.
+Denoting for λ ∈ Σδ0\{b(y∗)} and y ∈ I, Dφk,ǫ(y, λ) as the solution to
+�
+k2 − ∂2
+y +
+b′′(y)
+b(y) − λ + iιǫ
+�
+Dφk,ǫ(y, λ)
+= −ϕδ(y)
+�
+b′′(y)
+(b(y) − λ + iǫ)2 φ+
+k,ǫ(y, λ) −
+b′′(y)
+(b(y) − λ − iǫ)2 φ−
+k,ǫ(y, λ)
+�
+− ϕδ(y)
+�
+b′′(y)
+b(y) − λ + iιǫ −
+b′′(y)
+b(y) − λ − iιǫ
+�
+∂λφ−
+k,ǫ(y, λ),
+(6.29)
+for y ∈ I with zero Dirichlet boundary condition.
+We notice the identity that for y ∈ I, λ ∈ Σδ0\{b(y∗)},
+∂λψk(y, λ) = lim
+ǫ→0+ Dφk,ǫ(y, λ).
+(6.30)
+We can reformulate (6.29) as the integral equation for y ∈ I,
+Dφk,ǫ(y, λ) + T ∗
+k (λ + iǫ)Dφk,ǫ(y, λ)
+= −
+� 1
+0
+Gk(y, z; λ + iǫ)ϕδ(z)
+�
+b′′(z)
+(b(z) − λ + iǫ)2 φ+
+k,ǫ(z, λ) −
+b′′(z)
+(b(z) − λ − iǫ)2 φ−
+k,ǫ(z, λ)
+�
+dz
+−
+� 1
+0
+Gk(y, z; λ + iǫ)ϕδ(z)
+�
+b′′(z)
+b(z) − λ + iǫ −
+b′′(z)
+b(z) − λ − iιǫ
+�
+∂λφ−
+k,ǫ(z, λ) dz
+:= Rk,ǫ(y, λ).
+(6.31)
+We can write for y ∈ I, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0},
+Dφk,ǫ(y, λ) := Rk,ǫ(y, λ) + Dφk,ǫ(y, λ; 1).
+(6.32)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 25
+Then Dφk,ǫ(y, λ; 1) satisfies for y ∈ I, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, the
+equation
+Dφk,ǫ(y, λ; 1) + T ∗
+k (λ + iǫ)Dφk,ǫ(y, λ; 1) = −T ∗
+k (λ + iǫ)Rk,ǫ(y, λ).
+(6.33)
+The desired bounds (6.37) follow from (6.31)-(6.33), and Lemma 3.2 with X = XL,̺k.
+□
+Lastly we turn to the highest order derivative ∂2
+λψι
+k,ǫ(y, λ) that we need to control. To study
+∂2
+λψι
+k,ǫ(y, λ), we take the derivative ∂λ in (6.17) and obtain that
+�
+k2 − ∂2
+y +
+b′′(y)
+b(y) − λ + iιǫ
+�
+∂2
+λφι
+k,ǫ(·, λ) = −
+2b′′(y)
+(b(y) − λ + iιǫ)2 ∂λφι
+k,ǫ(·, λ)
+−
+2b′′(y)
+(b(y) − λ + iιǫ)3 φι
+k,ǫ(y, λ) + ∂2
+λgι
+k,ǫ(y, λ).
+(6.34)
+Lemma 6.3. Assume that k ∈ Z\{0}, λ ∈ Λδ0\{b(y∗)} and let φι
+k,ǫ(y, λ) with ι ∈ {±}, 0 < ǫ <
+min{|λ − b(y∗)|, ǫ0} be as defined in (6.2). Recall that
+δ := δ(λ) = 8
+�
+|λ − b(y∗)|/b′′(y∗).
+(6.35)
+Denoting for y ∈ [0, 1], λ ∈ Λδ0\{b(y∗)},
+Λ2(y, λ) := − ψk(y, λ)ϕδ(y) b′′(y)
+(b′(y))2 lim
+ǫ→0+
+1
+b(y) − λ + iǫ
+− ϕδ(y) b′′(y)
+(b′(y))2 lim
+ǫ→0+
+�
+1
+b(y) − λ + iǫ −
+1
+b(y) − λ − iǫ
+�
+φ−
+k,ǫ(y, λ),
+(6.36)
+then we have the bounds for λ ∈ Λδ0\{b(y∗)},
+�
+α∈{0,1}
+���(δ ∧ |k|−1)9/4�
+∂2
+λψk(y, λ) − Λ2(y, λ)
+����
+L2([y∗−3δ,y∗+3δ])δ−1/2
++
+�
+α∈{0,1}
+���(δ ∧ |k|−1)5/4(|y − y∗| ∧ 1
+|k|)∂2
+λψk(y, λ)
+���
+L∞([0,1]\[y∗−3δ,y∗+3δ])) ≲ |k|5/2��ω0k
+��
+H3
+k(I).
+(6.37)
+Proof. Denote for k ∈ Z\{0}, λ ∈ Λδ0\{b(y∗)}, ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} and y ∈ I,
+hι
+k,ǫ(z, λ; 4) := −
+2b′′(z)
+(b(z) − λ − iιǫ)2 ϕδ(z)∂λφι
+k,ǫ(z, λ),
+hι
+k,ǫ(z, λ; 5) = −
+2b′′(z)
+(b(z) − λ − iιǫ)3 ϕδ(z)φι
+k,ǫ(z, λ)
+hι
+k,ǫ(z, λ; 6) := −
+b′′(z)
+(b(z) − λ − iιǫ)2 (1 − ϕδ(z))∂λφι
+k,ǫ(z, λ),
+hι
+k,ǫ(z, λ; 7) = −
+2b′′(z)
+(b(z) − λ − iιǫ)3 (1 − ϕδ(z))φι
+k,ǫ(z, λ),
+hι
+k,ǫ(z, λ; 8) := ∂2
+λgι
+k,ǫ(z, λ).
+(6.38)
+
+26
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+Define for k ∈ Z\{0}, λ ∈ Λδ0\{b(y∗)}, ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} and z ∈ I,
+∂2
+λφι
+k,ǫ(y, λ) := φι
+k,ǫ(y, λ; 2) +
+8
+�
+j=4
+� 1
+0
+Gk(y, z; λ + iιǫ)hι
+k,ǫ(z, λ; j) dz
+−
+8
+�
+j=4
+T ∗
+k (λ + iιǫ)
+� 1
+0
+Gk(y, z; λ + iιǫ)hι
+k,ǫ(z, λ; j) dz
+(6.39)
+It follows from (6.34) that φι
+k,ǫ(y, λ; 2) satisfies for y ∈ I,
+φι
+k,ǫ(y, λ; 2) + T ∗
+k (λ + iιǫ)φι
+k,ǫ(y, λ; 2) =
+8
+�
+j=4
+�
+T ∗
+k (λ + iιǫ)
+�2 � 1
+0
+Gk(y, z; λ + iιǫ)hι
+k,ǫ(z, λ; j) dz.
+(6.40)
+It follows from Lemma 6.2 and Lemma 3.1 that for j ∈ {6, 7, 8}
+���
+�
+T ∗
+k (λ + iǫ)
+�2
+� 1
+0
+Gk(y, z; λ + iιǫ)hι
+k,ǫ(z, λ; j) dz
+���
+XN,̺k
+≲ (δ ∧ |k|−1)−4|k|5/2∥ω0k∥H3
+k(I). (6.41)
+Using integration by parts argument similar to (4.29)-(4.30), we have also for j ∈ {4, 5},
+����
+�
+T ∗
+k (λ + iǫ)
+�2 � 1
+0
+Gk(y, z; λ + iιǫ)hι
+k,ǫ(z, λ; j) dz
+����
+XN,̺k
+≲ (δ ∧ |k|−1)−4|k|5/2��ω0k
+��
+H3
+k(I).
+(6.42)
+It follows from (6.38)-(6.42) and Lemma 4.4 that for λ ∈ Λδ0\{b(y∗)}, ι ∈ {±}, 0 < ǫ <
+min{|λ − b(y∗)|, ǫ0},
+��φι
+k,ǫ(y, λ; 2)
+��
+XN,̺k ≲ (δ ∧ |k|−1)−4|k|5/2��ω0k
+��
+H1
+k(I).
+(6.43)
+Using (6.34), we get that for y ∈ I,
+�
+k2 − ∂2
+y +
+b′′(y)
+b(y) − λ + iǫ
+��
+∂2
+λφ+
+k,ǫ(y, λ) − ∂2
+λφ−
+k,ǫ(y, λ)
+�
+=
+8
+�
+j=4
+�
+h+
+k,ǫ(y, λ; j) − h−
+k,ǫ(y, λ; j)
+�
+.
+(6.44)
+Denoting D2φk,ǫ(y, λ), h ∈ I, λ ∈ Λδ0\{b(y∗)}, as the solution to
+�
+k2 − ∂2
+y +
+b′′(y)
+b(y) − λ + iιǫ
+�
+D2φk,ǫ(y, λ) =
+5
+�
+j=4
+�
+h+
+k,ǫ(y, λ; j) − h−
+k,ǫ(y, λ; j)
+�
+,
+(6.45)
+for y ∈ I with zero Dirichlet boundary condition.
+We note the identity that for y ∈ I, λ ∈ Σδ0\{b(y∗)},
+∂2
+λψk(y, λ) = lim
+ǫ→0+ D2φk,ǫ(y, λ).
+(6.46)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 27
+We can reformulate (6.45) as the integral equation for y ∈ I
+D2φk,ǫ(y, λ) + T ∗
+k (λ + iǫ)D2φk,ǫ(y, λ)
+=
+� 1
+0
+Gk(y, z; λ + iǫ)ϕδ(z)
+5
+�
+j=4
+�
+h+
+k,ǫ(z, λ; j) − h−
+k,ǫ(z, λ; j)
+�
+dz := R∗
+k,ǫ(y, λ).
+(6.47)
+We can write for λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, y ∈ I,
+D2φk,ǫ(y, λ) := D2φk,ǫ(y, λ; 2) + R∗
+k,ǫ(y, λ) − T ∗
+k (λ + iǫ)R∗
+k,ǫ(y, λ).
+(6.48)
+Then D2φk,ǫ(y, λ; 2) satisfies for y ∈ I, λ ∈ Σδ0\{b(y∗)},
+D2φk,ǫ(y, λ; 2) + T ∗
+k (λ + iǫ)D2φk,ǫ(y, λ; 2) =
+�
+T ∗
+k (λ + iǫ)
+�2R∗
+k,ǫ(y, λ).
+(6.49)
+The desired bounds (6.37) follow from (6.47)-(6.49), and Lemma 3.2 with X = XL,̺k, using
+also the bound
+���
+T ∗
+k (λ + iǫ)
+�2R∗
+k,ǫ(·, λ)
+��
+XL,̺k ≲
+�
+δ ∧ 1
+|k|
+�−4
+|k|5/2∥ω0k∥H3
+k(I).
+(6.50)
+□
+7. Proof of Theorem 1.2
+In this section, we prove Theorem 1.2. We can assume that t ≥ 1. We first give the proof of
+(1.8)-(1.9). Using the representation formula (2.7), we have
+ψk(t, y) =
+1
+2πi lim
+ǫ→0+
+�
+Σ
+e−ikλt�
+ψ+
+k,ǫ(y, λ) − ψ−
+k,ǫ(y, λ)
+�
+dλ
+= −
+1
+2πik2t2 lim
+ǫ→0+
+�
+Σ
+e−ikλt�
+∂2
+λψ+
+k,ǫ(y, λ) − ∂2
+λψ−
+k,ǫ(y, λ)
+�
+dλ.
+(7.1)
+Fix Φ∗ ∈ C∞
+0 (Σδ0) with Φ∗ ≡ 1 on Σ2δ0/3. We can decompose for t ≥ 1, y ∈ [0, 1],
+ψk(t, y) := ψ1
+k(t, y) + ψ2
+k(t, y),
+(7.2)
+where
+ψ1
+k(t, y) := −
+1
+2πik2t2 lim
+ǫ→0+
+�
+Σ
+e−ikλt(1 − Φ∗(λ))
+�
+∂2
+λψι
+k,ǫ(y, λ) − ∂2
+λψ−
+k,ǫ(y, λ)
+�
+dλ,
+ψ2
+k(t, y) := −
+1
+2πik2t2 lim
+ǫ→0+
+�
+Σ
+e−ikλtΦ∗(λ)
+�
+∂2
+λψι
+k,ǫ(y, λ) − ∂2
+λψ−
+k,ǫ(y, λ)
+�
+dλ.
+(7.3)
+For (1.8), it suffices to prove that for σ ∈ {1, 2}, k ∈ Z\{0} and t ≥ 1,
+��ψσ
+k(t, ·)
+��
+L2([0,1]) ≲ |k|3
+t2 ∥ω0k∥H3
+k([0,1]).
+(7.4)
+The case σ = 1 in (7.4) corresponding to the non-degenerate case is analogous to the case of
+monotonic shear flows, see [14], and follow from Lemma 5.1-Lemma 5.3. We focus on the main
+new case σ = 2 in (7.4). Denote for k ∈ Z\{0},
+Mk := |k|5/2∥ω0k∥H3
+k([0,1]).
+(7.5)
+Our main tools are Lemmas 6.1, Lemma 6.2 and Lemma 6.3, which imply the following bounds
+for y ∈ [0, 1], λ ∈ Σδ0.
+
+28
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+• If |λ − b(y∗)|1/2 < |y − y∗|/20, then
+|ψk(y, λ)| ≲
+�
+min
+�
+|λ − b(y∗)|1/2, |k|��1��11/4(|y − y∗|−1 + |k|)Mk,
+|∂2
+λψk(y, λ) ≲
+�
+min
+�
+|λ − b(y∗)|1/2, |k|−1��−5/4(|y − y∗|−1 + |k|)Mk;
+(7.6)
+• If |y − y∗|/20 < |λ − b(y∗)|1/2 < 20|y − y∗|, then
+|ψk(y, λ)| ≲
+�
+min
+�
+|λ − b(y∗)|1/2, |k|−1��5/4|λ − b(y∗)|1/4Mk,
+|ψk(y, λ) − ψk(y, b(y))| ≲ |λ − b(y)|1/2|λ − b(y∗)|3/8Mk,
+��∂2
+λψk(·, λ) − Λ2(·, λ)
+��
+L2(|y−y∗|≈|λ−b(y∗)|1/2) ≲ (|λ − b(y∗)|−1/2 + |k|)9/4|λ − b(y∗)|1/4Mk;
+(7.7)
+• If |λ − b(y∗)|1/2 > 20|y − y∗|, then
+|ψk(y, λ)| ≲ |λ − b(y∗)|1/4�
+min
+�
+|λ − b(y∗)|1/2, |k|−1��5/4Mk,
+��∂2
+λψk(·, λ) − Λ2(·, λ)
+��
+L2(|y−y∗|<|λ−b(y∗)|1/2/20) ≲ (|λ − b(y∗)|−1/2 + |k|)9/4|λ − b(y∗)|1/4Mk.
+(7.8)
+It follows from (7.6)-(7.8) that for y ∈ [0, 1], t ≥ 1,
+���
+�
+R
+e−ikλtΦ∗(λ)Λ2(y, λ)dλ
+��� ≲ |y − y∗|−1/4 max
+�
+1, |k|1/2|y − y∗|1/2�
+Mk,
+(7.9)
+and, by considering the cases |λ − b(y∗)| ≪ |y − y∗|2, |λ − b(y∗)| ≈ |y − y∗|2 and |λ − b(y∗)| ≫
+|y − y∗|2, also that for y ∈ [0, 1], t ≥ 1,
+���
+�
+R
+e−ikλtΦ∗(λ)
+�
+∂2
+λψk(y, λ) − Λ2(y, λ)
+�
+dλ
+���
+L2([0,1]) ≲ |k|9/4Mk.
+(7.10)
+The desired bound (7.4) for σ = 2 follows from (7.9)-(7.10).
+The proof of (1.9) is similar to the proof of (1.8), using Lemma 6.1 and Lemma 6.2.
+We now turn to the proof of the depletion bounds (1.11). Assume that k ∈ Z\{0}. Applying
+−k2 + ∂2
+y to ψk(t, y) in (2.7), and using (2.8), we get that for y ∈ [0, 1], t ≥ 1,
+ωk(t, y) = ω∗
+k(t, y) + ω∗∗
+k (t, y),
+(7.11)
+where
+ω∗
+k(t, y)
+:=
+1
+2πi lim
+ǫ→0+
+�
+Σ
+e−ikλt(1 − Φ∗(y))
+�b′′(y)ψ+
+k,ǫ(y, λ) − ω0k(y)
+b(y) − λ + iǫ
+−
+b′′(y)ψ−
+k,ǫ(y, λ) − ω0k(y)
+b(y) − λ − iǫ
+�
+dλ,
+ω∗∗
+k (t, y) :=
+1
+2πi lim
+ǫ→0+
+�
+Σ
+e−ikλtΦ∗(y)
+�b′′(y)ψ+
+k,ǫ(y, λ) − ω0k(y)
+b(y) − λ + iǫ
+−
+b′′(y)ψ−
+k,ǫ(y, λ) − ω0k(y)
+b(y) − λ − iǫ
+�
+dλ.
+(7.12)
+We have the bound for t ≥ 1,
+��ω∗
+k(t, y)
+��
+L∞([0,1]) ≲ |k|2Mk.
+(7.13)
+
+LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 29
+For |y − y∗| < δ0/10, t ≥ 1, since (b(y) − λ + iιǫ) with ι ∈ {±} is not singular in this case, we
+have in addition by integration by parts that
+|ω∗
+k(t, y)| ≲ |k|2 1
+t Mk.
+(7.14)
+We now turn to ω∗∗
+k (t, y). Using (6.1), we can write for y ∈ [0, 1], t ≥ 1,
+2πi ω∗∗
+k (t, y)
+= lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(λ)
+�φ+
+k,ǫ(y, λ) − (1 − Ψ(y))ω0k(y)
+b(y) − λ + iǫ
+−
+φ−
+k,ǫ(y, λ) − (1 − Ψ(y))ω0k(y)
+b(y) − λ − iǫ
+�
+dλ
+= lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(λ)
+�
+φ+
+k,ǫ(y, λ)
+b(y) − λ + iǫ −
+φ−
+k,ǫ(y, λ)
+b(y) − λ − iǫ
+�
+dλ + Wk(t, y),
+(7.15)
+where Wk(t, y) satisfies the bound for t ≥ 1,
+∥Wk(t, ·)∥L∞([0,1]) ≲ t−1Mk,
+(7.16)
+which follows from simple integration by parts argument. We decompose for y ∈ [0, 1]\{y∗},
+ω∗∗
+k (t, y) − Wk(t, y)
+2πi
+=
+1
+2πi lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(λ)
+�
+ψk(y, λ)
+b(y) − λ + iǫ
+�
+dλ
++
+1
+2πi lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(λ)φ−
+k,ǫ(y, λ)
+�
+1
+b(y) − λ + iǫ −
+1
+b(y) − λ − iǫ
+�
+dλ.
+(7.17)
+It follows from (7.6)-(7.8) that
+����
+1
+2πi lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(λ)φ−
+k,ǫ(y, λ)
+�
+1
+b(y) − λ + iǫ −
+1
+b(y) − λ − iǫ
+�
+dλ
+���� ≲ |y − y∗|7/4Mk.
+(7.18)
+For γ ∈ (1, ∞) to be fixed below, by considering the three ranges (I) |λ − b(y∗)| ≲ |y − y∗|2,
+(II) |λ − b(y∗)| ≥ γ|y − y∗|2, and (III) |y − y∗|2 ≪ |λ − b(y∗)| < γ|y − y∗|2, and using Lemma
+6.2 and Lemma 6.39, we get that
+����
+1
+2πi lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(y)
+�
+ψk(y, λ)
+b(y) − λ + iǫ
+�
+dλ
+����
+≲
+�
+|y − y∗|7/4�
+1 + |k|1/2|y − y∗|1/2�
++
+1
+|k|t(|k|1/2 + γ−1/8|y − y∗|−1/4) + γ7/8|y − y∗|7/4�
+Mk.
+(7.19)
+In the above, we used integration by part to get decay in t in range (II). Optimizing in γ, we
+get that for t ≥ 1,
+(i) if t|y − y∗|2 ≲ 1,
+����
+1
+2πi lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(y)
+�
+ψk(y, λ)
+b(y) − λ + iǫ
+�
+dλ
+����
+≲
+�
+t−1 + |k|1/2|y − y∗|7/4 + t−7/8�
+(7.20)
+
+30
+ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA
+(ii) if t|y − y∗|2 ≫ 1,
+����
+1
+2πi lim
+ǫ→0+
+�
+R
+e−ikλtΦ∗(y)
+�
+ψk(y, λ)
+b(y) − λ + iǫ
+�
+dλ
+����
+≲
+�
+|y − y∗|7/4�
+1 + |k|1/2|y − y∗|1/2�
++
+1
+|k|1/2t7/8 + |y − y∗|7/4�
+Mk.
+(7.21)
+The desired bounds (7.16), (7.18), (7.20)-(7.21). Theorem 1.2 is now proved.
+References
+[1] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998.
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+Princeton University
+Email address: [email protected]
+University of California, Davis
+Email address: [email protected]
+University of Minnesota
+Email address: [email protected]
+

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+arXiv:2301.11430v1  [math.AP]  26 Jan 2023
+Vortex sheet solutions for the Ginzburg-Landau system
+in cylinders: symmetry and global minimality
+Radu Ignat∗
+Mircea Rus†
+January 30, 2023
+Abstract
+We consider the Ginzburg-Landau energy Eε for RM-valued maps defined in a
+cylinder shape domain BN ×(0, 1)n satisfying a degree-one vortex boundary condition
+on ∂BN × (0, 1)n in dimensions M ≥ N ≥ 2 and n ≥ 1.
+The aim is to study
+the radial symmetry of global minimizers of this variational problem. We prove the
+following: if N ≥ 7, then for every ε > 0, there exists a unique global minimizer
+which is given by the non-escaping radially symmetric vortex sheet solution uε(x, z) =
+(fε(|x|) x
+|x|, 0RM−N), ∀x ∈ BN that is invariant in z ∈ (0, 1)n. If 2 ≤ N ≤ 6 and M ≥
+N + 1, the following dichotomy occurs between escaping and non-escaping solutions:
+there exists εN > 0 such that
+• if ε ∈ (0, εN), then every global minimizer is an escaping radially symmetric
+vortex sheet solution of the form R˜uε where ˜uε(x, z) = ( ˜fε(|x|) x
+|x|, 0RM−N−1, gε(|x|))
+is invariant in z-direction with gε > 0 in (0, 1) and R ∈ O(M) is an orthogonal
+transformation keeping invariant the space RN × {0RM−N};
+• if ε ≥ εN, then the non-escaping radially symmetric vortex sheet solution
+uε(x, z) = (fε(|x|) x
+|x|, 0RM−N), ∀x ∈ BN, z ∈ (0, 1)n is the unique global minimizer;
+moreover, there are no bounded escaping solutions in this case.
+We also discuss the problem of vortex sheet SM−1-valued harmonic maps.
+Keywords:
+vortex, uniqueness, symmetry, minimizers, Ginzburg-Landau equation,
+harmonic maps.
+MSC: 35A02, 35B06, 35J50.
+Contents
+1
+Introduction and main results
+2
+1.1
+Minimality of the RN-valued vortex sheet solution
+. . . . . . . . . . . . . .
+2
+1.2
+Escaping RM-valued vortex sheet solutions when M ≥ N + 1 . . . . . . . .
+5
+∗Institut de Math´ematiques de Toulouse & Institut Universitaire de France, UMR 5219, Universit´e de
+Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. Email: [email protected]
+†Department of Mathematics, Technical University of Cluj-Napoca, 400027 Cluj-Napoca, Romania.
+Email: [email protected]
+1
+
+2
+The non-escaping vortex sheet solution. Proof of Theorems 1 and 3
+7
+3
+Properties of escaping vortex sheet solutions when M ≥ N + 1
+11
+3.1
+Minimality of escaping vortex sheet solutions . . . . . . . . . . . . . . . . .
+11
+3.2
+Escaping radial profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+3.3
+Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+A Appendix. Vortex sheet SM−1-valued harmonic maps in cylinders
+17
+1
+Introduction and main results
+In this paper, we consider the following Ginzburg-Landau type energy functional
+Eε(u) =
+�
+Ω
+�1
+2|∇u|2 + 1
+2ε2 W(1 − |u|2)
+�
+dX,
+(1)
+where ε > 0, X = (x, z) ∈ Ω = BN × (0, 1)n is a cylinder shape domain with BN the unit
+ball in RN, n ≥ 1, N ≥ 2 and the potential W ∈ C2((−∞, 1]; R) satisfies
+W(0) = 0, W(t) > 0 for all t ∈ (−∞, 1] \ {0} and W is convex.
+(2)
+(The prototype potential is W(t) = t2
+2 for t ≤ 1.) We investigate the global minimizers of
+the energy Eε in the set of RN-valued maps:
+AN := {u ∈ H1(Ω; RN) : u(x, z) = x for every x ∈ ∂BN = SN−1, z ∈ (0, 1)n}.
+The boundary assumption u(x, z) = x for every x ∈ SN−1 and every z ∈ (0, 1)n is referred
+in the literature as the degree-one vortex boundary condition.
+The direct method in the calculus of variations yields the existence of a global minimizer
+uε of Eε over AN for all range of ε > 0. Moreover, any minimizer uε satisfies |uε| ≤ 1 in
+Ω, uε belongs to C1(Ω; RN) and solves the system of PDEs (in the sense of distributions)
+with mixed Dirichlet-Neumann boundary conditions:
+
+
+
+−∆uε = 1
+ε2uε W ′(1 − |uε|2)
+in Ω,
+∂uε
+∂z = 0
+on BN × ∂(0, 1)n,
+u(x, z) = x
+on ∂BN × (0, 1)n.
+(3)
+1.1
+Minimality of the RN-valued vortex sheet solution
+The first goal of this paper is to prove the uniqueness and radial symmetry of the global
+minimizer of Eε in AN for all ε > 0 in dimensions N ≥ 7 and n ≥ 1. In fact, in these
+dimensions, we show that the global minimizer of Eε in AN is unique and given by the
+following radially symmetric critical point of Eε that is invariant in z: 1
+uε(x, z) = fε(|x|) x
+|x|
+for all x ∈ BN and z ∈ (0, 1)n,
+(4)
+1If n = 0 and N ≥ 2, then SO(N) induces a group action on AN given by u(x) �→ R−1u(Rx) for every
+x ∈ BN, R ∈ SO(N) and u ∈ AN under which the energy Eε and the vortex boundary condition are
+invariant. Then every bounded critical point of Eε in AN that is invariant under this SO(N) group action
+has the form (4), see e.g. [8, Lemma A.4].
+2
+
+where the radial profile fε : [0, 1] → R in r = |x| is the unique solution to the ODE:
+� −f ′′
+ε − N−1
+r f ′
+ε + N−1
+r2 fε = 1
+ε2fε W ′(1 − f 2
+ε )
+for r ∈ (0, 1),
+fε(0) = 0, fε(1) = 1.
+(5)
+We recall that the unique radial profile fε satisfies fε > 0 and f ′
+ε > 0 in (0, 1) (see
+e.g. [7, 9, 8]). Note that the zero set of uε is given by the n-dimensional vortex sheet
+{0RN } × (0, 1)n in Ω (in particular, if n = 0, it is a vortex point, while for n = 1, it is a
+vortex filament); therefore, uε in (4) is called (radially symmetric) vortex sheet solution to
+the Ginzburg-Landau system (3).
+Theorem 1. Assume that W satisfies (2) and n ≥ 1. If N ≥ 7, then uε given in (4) is
+the unique global minimizer of Eε in AN for every ε > 0.
+The proof is reminiscent of the works of Ignat-Nguyen-Slastikov-Zarnescu [12, 11]
+studying uniqueness and symmetry of minimizers of the Ginzburg-Landau functionals for
+RM-valued maps defined on smooth N-dimensional domains, where M is not necessarily
+equal to N. The idea is to analyze Eε(u) for an arbitrary map u and to exploit the convex-
+ity of W to lower estimate the excess energy w.r.t. Eε(uε) by a suitable quadratic energy
+functional depending on u − uε. This quadratic functional comes from the linearized PDE
+at uε and can be handled by a factorization argument. The positivity of the excess energy
+then follows by a Hardy-type inequality holding true only in high dimensions N ≥ 7. This
+is similar to the result of J¨ager and Kaul [14] on the minimality of the equator map for
+the harmonic map problem in dimension N ≥ 7 that is proved using a certain inequality
+involving the sharp constant in the Hardy inequality.
+We expect that our result remains valid in dimensions 2 ≤ N ≤ 6:
+Open problem 2. Assume that W satisfies (2), n ≥ 1 and 2 ≤ N ≤ 6. Is it true that
+for every ε > 0, uε given in (4) is the unique global minimizer of Eε in AN?
+It is well known that the uniqueness of uε holds true for large enough ε > 0 in any
+dimension N ≥ 2. Indeed, denoting by λ1 the first eigenvalue of −∆x in BN with zero
+Dirichlet boundary condition, then for any ε >
+�
+W ′(1)/λ1, Eε is strictly convex in AN
+(see e.g., [1, Theorem VIII.7], [12, Remark 3.3]) and thus has a unique critical point in
+AN that is the global minimizer of our problem. We improve this result as follows: for
+the radial profile fε in (5), we denote by ℓ(ε) the first eigenvalue of the operator
+Lε = −∆x − 1
+ε2 W ′(1 − f 2
+ε )
+(6)
+acting on maps defined in BN with zero Dirichlet boundary condition. It is proved in [8,
+Lemma 2.3] that if 2 ≤ N ≤ 6 and W ∈ C2((−∞, 1]) satisfies (2), then the first eigenvalue
+ℓ(ε) is a continuous function in ε and there exists εN ∈ (0, ∞) such that
+ℓ(ε) < 0 in (0, εN),
+ℓ(εN) = 0
+and
+ℓ(ε) > 0 in (εN, ∞).
+(7)
+3
+
+Note that2 0 = ℓ(εN) > λ1 −
+1
+ε2
+N W ′(1) yielding
+εN <
+�
+W ′(1)/λ1.
+Theorem 3. Assume that W satisfies (2), n ≥ 1 and 2 ≤ N ≤ 6. If ε ≥ εN, then uε
+given in (4) is a global minimizer of Eε in AN. Moreover, if either ε > εN, or (ε = εN
+and W is in addition strictly convex), then uε is the unique global minimizer of Eε in AN.
+The case ε < εN is still not solved as stated in Open Problem 2. Let us summarize
+some known results:
+I. The case of n = 0 and Ω = BN (we also discuss here the problem for Ω = RN). In
+this case, the above question was raised in dimension N = 2 for the disk Ω = B2 in
+the seminal book of Bethuel, Brezis and H´elein [1, Problem 10, page 139], and in general
+dimensions N ≥ 2 and also for the blow-up limiting problem around the vortex point
+(when the domain Ω is the whole space RN and by rescaling, ε can be assumed equal to 1)
+in an article of Brezis [3, Section 2]. For sufficiently small ε > 0 and for the disk domain
+Ω = B2, Pacard and Rivi`ere [20, Theorem 10.2] showed that Eε has a unique critical point
+in A2 and so, it is given by the radially symmetric solution uε in (4) (for n = 0). For
+N ≥ 7, Ω = BN and any ε > 0, it is proved in [11] that Eε has a unique minimizer in AN
+which is given by the radially symmetric solution uε in (4) (for n = 0). For 2 ≤ N ≤ 6
+and Ω = BN, Ignat-Nguyen [8] proved that for any ε > 0, uε is a local minimizer of Eε
+in A (which is an extension of the result of Mironescu [18] in dimension N = 2). Also,
+Mironescu [19] showed in dimension N = 2 that, when B2 is replaced by R2 and ε = 1, a
+local minimizer of Eε satisfying a degree-one boundary condition at infinity is unique (up
+to translation and suitable rotation). This was extended in dimension N = 3 by Millot and
+Pisante [17] and in dimensions N ≥ 4 by Pisante [21] in the case of the blow-up limiting
+problem on RN and ε = 1. All these results (holding for n = 0) are related to the study of
+the limit problem obtained by sending ε → 0 when the Ginzburg-Landau problem on the
+unit ball ‘converges’ to the harmonic map problem from BN into the unit sphere SN−1.
+For that harmonic map problem, the vortex boundary condition yields uniqueness of the
+minimizing harmonic SN−1-valued map x �→
+x
+|x| if N ≥ 3; this is proved by Brezis, Coron
+and Lieb [4] in dimension N = 3 and by Lin [15] in any dimension N ≥ 3; we also mention
+J¨ager and Kaul [14] in dimension N ≥ 7 for the equator map x ∈ BN �→ ( x
+|x|, 0) ∈ SN.
+II. The case of n ≥ 1 and Ω = BN × (0, 1)n. As we explain in Remark 6 below, for some
+ε > 0, if the minimality of the radially symmetric solution uε in (4) holds in the case n = 0
+(so, for Ω = BN), then this implies the minimality of uε in Ω = BN ×(0, 1)n also for every
+dimension n ≥ 1. In particular, the result of Pacard-Rivi`ere [20, Theorem 10.2] for n = 0
+and N = 2 yields the minimality of uε in (4) defined in B2×(0, 1)n for every n ≥ 1 if ε > 0
+is sufficiently small. Also, the result of Ignat-Nguyen-Slastikov-Zarnescu [11, Theorem 1]
+2Indeed, if v ∈ H1
+0(BN) is a first eigenfunction of LεN in BN such that ∥v∥L2(BN ) = 1 then
+λ1 ≤
+�
+BN |∇xv|2 dx = 1
+ε2
+N
+�
+BN W ′(1 − f 2
+εN )v2 dx < W ′(1)
+ε2
+N
+because ℓ(εN) = 0, 0 < fεN < 1 in (0, 1) and (2) implies W ′(0) = 0 and W ′(t) > 0 for t ∈ (0, 1].
+4
+
+for n = 0, N ≥ 7 and any ε > 0 generalizes to dimension n ≥ 1 for Ω = BN × (0, 1)n (see
+the proof of Theorem 1). We also mention the work of Sandier-Shafrir [24] where they
+treat the case of topologically trivial R2-valued solutions in the domain Ω = R3 (see also
+[5, 22] for vortex filament solutions).
+1.2
+Escaping RM-valued vortex sheet solutions when M ≥ N + 1
+In dimension 2 ≤ N ≤ 6 and for ε < εN given in (7), a different type of radially symmetric
+vortex sheet solution appears provided that the target space has dimension M ≥ N + 1.
+More precisely, we consider the energy functional Eε in (1) over the set of RM-valued maps
+A := {u ∈ H1(Ω; RM) : u(x, z) = (x, 0RM−N ) on ∂BN = SN−1 ⊂ RM, z ∈ (0, 1)n}.
+(8)
+If M ≥ N + 1, the prototype of radially symmetric critical points of Eε in A has the
+following form (invariant in z-direction): 3
+˜uε(x, z) = ( ˜fε(r) x
+|x|, 0RM−N−1, gε(r)) ∈ A ,
+x ∈ BN, z ∈ (0, 1)n, r = |x|,
+(9)
+where ( ˜fε, gε) satisfies the system of ODEs
+− ˜f ′′
+ε − N − 1
+r
+˜f ′
+ε + N − 1
+r2
+˜fε = 1
+ε2 W ′(1 − ˜f 2
+ε − g2
+ε) ˜fε
+in (0, 1),
+(10)
+−g′′
+ε − N − 1
+r
+g′
+ε = 1
+ε2 W ′(1 − ˜f 2
+ε − g2
+ε)gε
+in (0, 1),
+(11)
+˜fε(1) = 1 and gε(1) = 0.
+(12)
+We distinguish two type of radial profiles:
+• the non-escaping radial profile ( ˜fε = fε, gε = 0) with the unique radial profile fε given
+in (5); in this case, we say that ˜uε = (uε, 0RM−N ) is a non-escaping (radially symmetric)
+vortex sheet solution where uε is given in (4).
+• the escaping radial profile ( ˜fε, gε) with gε > 0 in (0, 1); in this case, we call an
+escaping (radially symmetric) vortex sheet solution ˜uε in (9). In this case, ˜fε ̸= fε and
+obviously, ( ˜fε, −gε) is another radial profile to (9)-(12).
+The properties of such radial profiles (e.g., existence, uniqueness, minimality, mono-
+tonicity) are analyzed in Theorem 9 below and are based on ideas developed by Ignat-
+Nguyen [8].
+Our main result proves the radial symmetry of global minimizers of Eε in A . More
+precisely, the following dichotomy occurs at εN defined in (7): if ε < εN, then escaping
+radially symmetric vortex sheet solutions exist and determine (up to certain orthogonal
+transformations) the full set of global minimizers of Eε in A ; if instead ε ≥ εN, then the
+non-escaping radially symmetric vortex sheet solution is the unique global minimizer of
+Eε in A and no escaping radially symmetric vortex sheet solutions exist in this case.
+3If M = N + 1, then ˜uε(x, z) = ( ˜fε(r) x
+|x|, gε(r)) for every x ∈ BN and z ∈ (0, 1)n. In fact, if n = 0 (so,
+for Ω = BN), every bounded critical point of Eε in A that is invariant under the action of a special group
+(isomorphic to SO(N)) has the form of ˜uε, see [8, Definition A.1, Lemma A.5].
+5
+
+Theorem 4. Let n ≥ 1, 2 ≤ N ≤ 6, M ≥ N + 1, W ∈ C2((−∞, 1]) satisfy (2) and be
+strictly convex. Consider εN ∈ (0, ∞) such that ℓ(εN) = 0 in (7). Then there exists an
+escaping radially symmetric vortex sheet solution ˜uε in (9) with gε > 0 in (0, 1) if and
+only if 0 < ε < εN. Moreover,
+1. if 0 < ε < εN, the escaping radially symmetric vortex sheet solution ˜uε is a global
+minimizer of Eε in A and all global minimizers of Eε in A are radially symmetric
+given by R˜uε where R ∈ O(M) is an orthogonal transformation of RM satisfying
+Rp = p for all p ∈ RN × {0RM−N }.
+In this case, the non-escaping vortex sheet
+solution (uε, 0RM−N ) in (4) is an unstable critical point of Eε in A .
+2. if ε ≥ εN, the non-escaping vortex sheet solution (uε, 0RM−N ) in (4) is the unique
+global minimizer of Eε in A . Furthermore, there are no bounded critical points wε
+of Eε in A that escape in some direction e ∈ SM−1 (i.e., wε · e > 0 a.e. in Ω).
+The result above holds also if n = 0, i.e., Ω = BN and the vortex sheets corresponding
+to the above solutions become vortex points (see Theorem 10). It generalizes [12, Theorem
+1.1] that was proved in the case N = 2 and M = 3 (without identifying the meaning of
+the dichotomy parameter εN in (7)). The dichotomy in Theorem 4 happens in dimensions
+2 ≤ N ≤ 6 because of the phenomenology occurring for the limit problem ε → 0. More
+precisely, if M ≥ N + 1, then minimizing SM−1-valued harmonic maps in A are smooth
+and escaping in a direction of SM−1 provided that N ≤ 6; if N ≥ 7, then there is a unique
+minimizing SM−1-valued harmonic maps in A , non-escaping and singular, the singular
+set being given by a vortex sheet of dimension n in Ω (see Theorem 11 in Appendix
+below). This suggests why in dimension N ≥ 7 and for any ε > 0, there is no escaping
+radially symmetric vortex sheet critical point ˜uε of Eε in A while the non-escaping vortex
+sheet solution (uε, 0RM−N ) is the unique global minimizer of Eε in A (see Theorem 5 and
+Remark 8 below).
+The paper is meant to be self-contained and it is organized as follows. In Section 2, we
+prove the minimality and the uniqueness results for the non-escaping radially symmetric
+solution in Theorems 1 and 3; this is done in a more general setting by considering the
+target dimension M ≥ N for the set of configurations A instead of AN. Section 3 is
+devoted to characterize escaping vortex sheet solutions. First, we prove the minimality
+of such bounded solutions stated in Theorem 7. Second, we prove existence, minimality
+and uniqueness results for the escaping radial profile in Theorem 9. Finally, we prove our
+main result on the dichotomy between escaping / non-escaping radially symmetric vortex
+sheet solutions in Theorem 4. In Appendix, we prove the corresponding dichotomy result
+for SM−1-valued harmonic maps in Theorem 11 which again is based on the minimality of
+escaping SM−1-valued harmonic maps in Theorem 12.
+Acknowledgment. R.I. is partially supported by the ANR projects ANR-21-CE40-0004
+and ANR-22-CE40-0006-01. He also thanks for the hospitality of the Hausdorff Research
+Institute for Mathematics in Bonn during the trimester “Mathematics for Complex Ma-
+terials”.
+6
+
+2
+The non-escaping vortex sheet solution. Proof of Theo-
+rems 1 and 3
+Theorem 1 will be obtained as a consequence of a stronger result on the uniqueness of
+global minimizers of the RM-valued Ginzburg-Landau functional with M ≥ N ≥ 7. For
+that, we consider the energy functional Eε in (1) over the set A defined in (8). The aim
+is to prove the minimality and uniqueness of the vortex sheet solution (uε, 0RM−N ) where
+uε given in (4) with the obvious identification uε ≡ (uε, 0RM−N ) if M = N, following the
+ideas of Ignat-Nguyen-Slastikov-Zarnescu [12, 11].
+Theorem 5. Assume that W satisfies (2) and n ≥ 1. If M ≥ N ≥ 7, then for every
+ε > 0, (uε, 0RM−N ) given in (4) is the unique global minimizer of Eε in A .
+Proof. To simplify notation, we identify
+uε ≡ (uε, 0RM−N )
+when
+M ≥ N.
+(13)
+The proof will be done in several steps following the strategy in [12, Theorem 1.7], [11,
+Theorem 1].
+First, for an arbitrary competitor uε + v, we consider the excess energy
+Eε(uε + v) − Eε(uε) for the critical point uε defined in (4) and show a lower estimate
+by a quadratic energy functional Fε(v) coming from the operator Lε in (6). Second, we
+show that Fε(v) ≥ 0 using the properties of the radial profile fε in (5) and a Hardy
+decomposition method; this proves in particular that uε is a global minimizer of Eε over
+A . Finally, by analyzing the zero excess energy states, we conclude to the uniqueness of
+the global minimizer uε.
+Step 1: Excess energy. For any v ∈ H1
+0(BN × Rn; RM), we have
+Eε(uε + v) − Eε(uε) =
+�
+Ω
+�
+∇uε · ∇v + 1
+2|∇v|2�
+dxdz
++ 1
+2ε2
+�
+Ω
+�
+W(1 − |uε + v|2) − W(1 − |uε|2)
+�
+dxdz.
+Note that for every u ∈ A , uε−u can be extended to v ∈ H1
+0(BN ×Rn; RM). In particular,
+v(·, z) ∈ H1
+0(BN, RM) for a.e. z ∈ (0, 1)n. The convexity of W yields
+W(1 − |uε + v|2) − W(1 − |uε|2) ≥ −W ′(1 − |uε|2)(|uε + v|2 − |uε|2).
+(14)
+Combining the above relations, we obtain the following lower bound for the excess energy:
+Eε(uε + v) − Eε(uε) ≥
+�
+Ω
+�
+∇uε · ∇v − 1
+ε2 W ′(1 − f 2
+ε )uε · v
+�
+dxdz
++
+�
+Ω
+�1
+2|∇v|2 − 1
+2ε2 W ′(1 − f 2
+ε )|v|2�
+dxdz
+=
+�
+Ω
+1
+2|∇zv|2 dxdz +
+�
+(0,1)n
+1
+2Fε(v(·, z)) dz,
+(15)
+7
+
+where we used the PDE (3) and introduced the quadratic functional
+Fε(Ψ) =
+�
+BN
+�
+|∇xΨ|2 − 1
+ε2 W ′(1 − f 2
+ε )|Ψ|2�
+dx,
+for all Ψ ∈ H1
+0(BN; RM). Note that the L2-gradient of Fε represents a part of the lin-
+earization of the PDE (3) at uε and it is given by the operator Lε in (6). The rest of the
+proof is devoted to show that for N ≥ 3:
+Fε(ψ) ≥
+�(N − 2)2
+4
+− (N − 1)
+� �
+BN
+ψ2
+r2 dx,
+∀ψ ∈ H1
+0(BN)
+yielding the conclusion for N ≥ 7 and also the inequality for the first eigenvalue ℓ(ε) of
+the operator Lε in (6) in BN: 4
+ℓ(ε) ≥ (N − 2)2
+4
+− (N − 1) > 0,
+∀ε > 0
+and
+N ≥ 7.
+To keep the paper self-contained, we explain in the following the simple idea used in
+[12, 11].
+Step 2: A factorization argument. As fε > 0 is a smooth positive radial profile in (0, 1),
+we decompose every scalar test function ψ ∈ C∞
+c (BN \ {0}; R) as follows
+ψ(x) = fε(r)w(x),
+∀x ∈ BN \ {0}, r = |x|,
+where w ∈ C∞
+c (BN \ {0}; R). Integrating by parts (see e.g. [10, Lemma A.1]), we deduce:
+Fε(ψ) =
+�
+BN Lεψ · ψ dx =
+�
+BN w2(Lεfε · fε) dx +
+�
+BN f 2
+ε |∇xw|2 dx
+=
+�
+BN f 2
+ε
+�
+|∇xw|2 − N − 1
+r2
+w2
+�
+dx,
+because Lεfε · fε = − N−1
+r2 f 2
+ε in BN by (5). Furthermore, we decompose
+w = ϕg
+in
+BN \ {0}
+with ϕ = |x|− N−2
+2
+satisfying
+−∆xϕ = (N − 2)2
+4|x|2
+ϕ
+in RN \ {0}
+and g ∈ C∞
+c (BN \ {0}; R). Then
+|∇xw|2 = |∇xg|2ϕ2 + |∇xϕ|2g2 + 1
+2∇x(ϕ2) · ∇x(g2).
+4Observe the difference between dimension N ≥ 7 and the case of dimension 2 ≤ N ≤ 6 where we have
+ℓ(ε) < 0 for ε < εN in (7); moreover, if N ≤ 6, then ℓ(ε) blows up as − 1
+ε2 as ε → 0 (see [8, Lemma 2.3]).
+8
+
+As |∇xϕ|2 = (N−2)2
+4|x|2 ϕ2 and ϕ2 is harmonic in BN \ {0} (recall that N ≥ 7), integration by
+parts yields
+Fε(ψ) =
+�
+BN f 2
+ε
+�
+|∇xg|2ϕ2 + (N − 2)2
+4r2
+ϕ2g2 − N − 1
+r2
+ϕ2g2
+�
+dx − 1
+2
+�
+BN ∇x(ϕ2) · ∇x(f 2
+ε )g2 dx
+≥
+�
+BN f 2
+ε |∇xg|2ϕ2 dx +
+�(N − 2)2
+4
+− (N − 1)
+� �
+BN
+f 2
+ε
+r2 ϕ2g2 dx
+≥
+�(N − 2)2
+4
+− (N − 1)
+� �
+BN
+ψ2
+r2 dx ≥ 0,
+(16)
+where we used N ≥ 7 and 1
+2∇x(ϕ2)·∇x(f 2
+ε ) = 2ϕϕ′fεf ′
+ε ≤ 0 in BN \{0} because ϕ, fε, f ′
+ε >
+0 and ϕ′ < 0 in (0, 1) (see e.g. [7, 9, 8]).
+Step 3: We prove that Fε(Ψ) ≥ 0 for every Ψ ∈ H1
+0(BN; RM); moreover, Fε(Ψ) = 0 if and
+only if Ψ = 0. Let Ψ ∈ H1
+0(BN; RM). As a point in RN has zero H1 capacity, a standard
+density argument implies the existence of a sequence Ψk ∈ C∞
+c (BN \ {0}; RM) such that
+Ψk → Ψ in H1(BN, RM) and a.e. in BN. On the one hand, by definition of Fε, since
+W ′(1 − f 2
+ε ) ∈ L∞, we deduce that Fε(Ψk) → Fε(Ψ) as k → ∞. On the other hand, by
+(16) and Fatou’s lemma, we deduce
+lim inf
+k→∞ Fε(Ψk) ≥
+�(N − 2)2
+4
+− (N − 1)
+�
+lim inf
+k→∞
+�
+BN
+|Ψk|2
+r2
+dx
+≥
+�(N − 2)2
+4
+− (N − 1)
+� �
+BN
+|Ψ|2
+r2 dx.
+Therefore, we conclude that
+Fε(Ψ) ≥
+�(N − 2)2
+4
+− (N − 1)
+� �
+BN
+|Ψ|2
+r2 dx ≥ 0,
+∀Ψ ∈ H1
+0(BN; RM).
+Moreover, Fε(Ψ) = 0 if and only if Ψ = 0.
+Step 4: Conclusion. By (15) and Step 3, we deduce that uε is a global minimizer of Eε
+over A . For uniqueness, assume that ˆuε is another global minimizer of Eε over A . If
+v := ˆuε − uε, then v can be extended in H1
+0(BN × Rn; RM) and by Steps 1 and 3, we have
+that
+0 = Eε(ˆuε) − Eε(uε) ≥
+�
+Ω
+1
+2|∇zv|2 dxdz +
+�
+(0,1)n
+1
+2Fε(v(·, z)) dz ≥ 0,
+which yields ∇zv = 0 a.e. in Ω and Fε(v(·, z)) = 0 for a.e. z ∈ (0, 1)n. In other words,
+v = v(x) and Step 3 implies that v = 0, i.e., ˆuε = uε in Ω.
+Remark 6. Theorem 5 reveals the following fact: if for n = 0 (i.e., Ω = BN) and some
+ε > 0, a (radially symmetric) critical point ˆuε : BN → RM of Eε in A is proved to be a
+global minimizer (and additionally, if one proves that it is the unique global minimizer),
+then for any dimensions n ≥ 1 (i.e., Ω = BN ×(0, 1)n), this z-invariant solution ˆuε of (3)
+9
+
+in BN × (0, 1)n is also a global minimizer (and additionally, it is the unique minimizer)
+of Eε in A . This is because for every u : BN × (0, 1)n → RM with u ∈ A , then u(·, z)
+satisfies the degree-one vortex boundary condition on ∂BN for every z ∈ (0, 1)n yielding
+Eε(u) =
+�
+Ω
+1
+2|∇zu|2 dxdz +
+�
+(0,1)n Eε(u(·, z)) dz
+≥
+�
+(0,1)n Eε(ˆuε) dz = Eε(ˆuε);
+the equality occurs only when u is z-invariant. Thus, if the uniqueness of the global mini-
+mizer ˆuε holds in BN (i.e., n = 0), then this yields uniqueness of the global minimizer ˆuε
+in Ω = BN × (0, 1)n (as a map independent of z-variable) for every n ≥ 1.
+Proof of Theorem 3. We prove the result in the more general setting of RM-valued maps
+u belonging to A for M ≥ N using the same identification (13). By Step 1 in the proof
+of Theorem 5 (see (15)), the excess energy is estimated for every v ∈ H1
+0(BN × Rn; RM):
+Eε(uε + v) − Eε(uε) ≥
+�
+Ω
+1
+2|∇zv|2 dxdz + 1
+2
+�
+(0,1)n < Lεv(·, z), v(·, z) > dz,
+where Lε is the operator in (6) and < ·, · > denotes the duality pairing (H−1, H1
+0) in BN.
+If ε ≥ εN, then ℓ(ε) ≥ 0 (by [8, Lemma 2.3]) and therefore, 5
+< Lεv(·, z), v(·, z) > ≥ ℓ(ε)∥v(·, z)∥2
+L2(BN ) ≥ 0
+for a.e. z ∈ (0, 1)n,
+(17)
+where we used that v(·, z) ∈ H1
+0(BN; RM) for a.e. z ∈ (0, 1)n. Thus, uε is a minimizer
+of Eε over A . It remains to prove uniqueness of the global minimizer. For that, if ˆuε is
+another global minimizer of Eε over A , setting v := ˆuε − uε, then v can be extended in
+H1
+0(BN × Rn; RM) and
+0 = Eε(ˆuε) − Eε(uε) ≥
+�
+Ω
+1
+2|∇zv|2 dxdz + ℓ(ε)
+2
+�
+(0,1)n
+�
+BN |v(x, z)|2 dxdz ≥ 0
+(18)
+because ℓ(ε) ≥ 0 for ε ≥ εN. Thus, equality holds in the above inequalities.
+Case 1: ε > εN. In this case, ℓ(ε) > 0 and we conclude that v = 0 in Ω, i.e., ˆuε = uε in Ω.
+5 Indeed, for a scalar function v ∈ C∞
+c (BN \ {0}, R), if ψ = ψ(r) > 0 is a radial first eigenfunction of
+Lε in BN with zero Dirichlet data, i.e., Lεψ = ℓ(ε)ψ in BN, then the duality pairing (H−1, H1
+0) term in
+BN writes (see e.g. [10, Lemma A.1]):
+< Lεv, v > =
+�
+BN ψ2|∇( v
+ψ )|2 dx +
+�
+BN ( v
+ψ )2Lεψ · ψ dx =
+�
+BN ψ2|∇( v
+ψ )|2 dx + ℓ(ε)∥v∥2
+L2(BN ).
+By a density argument, Fatou’s lemma yields for every scalar function v ∈ H1
+0(BN, R),
+< Lεv, v > ≥
+�
+BN ψ2|∇( v
+ψ )|2 dx + ℓ(ε)∥v∥2
+L2(BN ).
+10
+
+Case 2: ε = εN and W is in addition strictly convex. In this case, ℓ(ε) = 0 and by (18), v
+is invariant in z, i.e., v = v(x) and equality holds in (17) and in (15), thus, equality holds
+in (14). Note that by footnote 5 the equality in (17) holds if and only if v = λψ for some
+λ ∈ RM, where ψ = ψ(r) is a radial first eigenfunction of Lε in BN with zero Dirichlet
+data, in particular ψ > 0 in [0, 1) and ψ(1) = 0. Also, by the strict convexity of W, the
+equality (14) is achieved if and only if |uε + v| = |uε| a.e. in Ω, that is, |v|2 + 2v · uε = 0
+a.e. in BN. It yields
+|λ|2ψ2 + 2fε(|x|)( x
+|x|, 0RM−N ) · λψ = 0
+for every x ∈ BN.
+(19)
+Dividing by ψ in BN, the continuity up to the boundary ∂BN leads to 2fε(|x|)(x, 0RM−N )·
+λ = 0 for every x ∈ ∂BN since ψ = 0 on ∂BN. As fε(1) = 1, it follows that the first N
+components of λ vanish. Coming back to (19), we conclude that |λ|2ψ2 = 0 in BN, i.e.,
+λ = 0 and so, v = 0 and ˆuε = uε in Ω.
+3
+Properties of escaping vortex sheet solutions when M ≥
+N + 1
+3.1
+Minimality of escaping vortex sheet solutions
+In this section, we require the additional assumption of strict convexity of W in order to
+determine the set of global minimizers of Eε over A in (8). However, W is assumed to be
+only C1 not C2. We prove that every bounded solution to (3) escaping in some direction
+is a global minimizer of Eε over A ; moreover, such global minimizer is unique up to an
+orthogonal transformation of RM keeping invariant the space RN × {0RM−N }.
+Theorem 7. We consider the dimensions n ≥ 1 and M > N ≥ 2, the potential W ∈
+C1((−∞, 1], R) satisfying (2) and an escaping direction e ∈ SM−1. Fix any ε > 0 and let
+wε ∈ H1 ∩ L∞(Ω, RM) be a critical point of the energy Eε in the set A which is positive
+in the direction e inside Ω:
+wε · e > 0 a.e. in Ω.
+(20)
+Then wε is a global minimizer of Eε in A . If in addition W is strictly convex, then all
+minimizers of Eε in A are given by Rwε where R ∈ O(M) is an orthogonal transformation
+of RM satisfying Rp = p for all p ∈ RN × {0RM−N }.
+This result is reminiscent from [12, Theorem 1.3]. However, it doesn’t apply directly
+as the domain Ω is not smooth here and the boundary condition is a mixed Dirichlet-
+Neumann condition (w.r.t. Dirichlet boundary condition in [12]).
+Proof. In the following, we denote the variable X = (x, z) ∈ Ω = BN × (0, 1)n. As a
+critical point of Eε in the set A , wε : Ω → RM satisfies
+
+
+
+−∆wε = 1
+ε2wε W ′(1 − |wε|2)
+in Ω,
+∂wε
+∂z = 0
+on BN × ∂(0, 1)n,
+wε(x, z) = (x, 0RM−N )
+on ∂BN × (0, 1)n.
+(21)
+11
+
+In particular, ∆wε ∈ L∞(Ω) (as W ′ is continuous and wε ∈ L∞(Ω)); then standard elliptic
+regularity for the mixed boundary conditions in (21) yields wε ∈ C1(¯Ω, RM). Thus, (20)
+implies wε·e ≥ 0 in ¯Ω and the vortex boundary condition in A implies that e is orthogonal
+to RN × {0RM−N }. By the invariance of the energy and the vortex boundary condition
+under the transformation wε(X) �→ Rwε(X) for any R ∈ O(M) satisfying Rp = p for all
+p ∈ RN × {0RM−N }, we know that Rwε is also a critical point of Eε over A ; thus, we can
+assume that
+e := eM = (0, . . . , 0, 1) ∈ RM.
+(22)
+We prove the result in several steps.
+Step 1: Excess energy.
+By Step 1 in the proof of Theorem 5, we have for any v ∈
+H1
+0(BN × Rn, RM):
+Eε(wε + v) − Eε(wε) ≥
+�
+Ω
+�1
+2|∇v|2 − 1
+2ε2 W ′(1 − |wε|2)|v|2�
+dX =: 1
+2Gε(v)
+(23)
+(note that Gε(v) is larger than the integration of Fε(v) in (15) over (0, 1)n as it contains
+also the integration of |∇zv|2). If in addition W is strictly convex, then equality holds
+above if and only if |wε(X) + v(X)| = |wε(X)| a.e. X ∈ Ω (by (14)).
+Step 2: Global minimality of wε. It is enough to show that the quadratic energy Gε(v)
+defined in (23) is nonnegative for any v ∈ H1
+0(BN ×Rn, RM). Denoting the M-component
+of wε by φ := wε · eM, we know that φ ∈ C1(¯Ω), φ ≥ 0 in Ω (by (20)) and satisfies the
+Euler-Lagrange equation in the sense of distributions:
+
+
+
+−∆φ − 1
+ε2W ′(1 − |wε|2)φ = 0 in Ω,
+φ = 0 on ∂BN × (0, 1)n,
+∂φ
+∂z = 0 on BN × ∂(0, 1)n.
+(24)
+Note that by strong maximum principle, φ > 0 in Ω (as φ cannot be identically 0 in Ω
+by (20)). Moreover, Hopf’s lemma yields φ > 0 on BN × ∂(0, 1)n as ∂φ
+∂z vanishes there.
+Now, for any smooth map v ∈ C∞
+c (BN × Rn; RM), we can define Ψ = v
+φ ∈ C1(¯Ω; RM)
+with Ψ = 0 in a neighborhood of ∂BN × (0, 1)n and integration by parts yields for every
+component vj = φΨj with 1 ≤ j ≤ M (as in [10, Lemma A.1.]):
+Gε(vj) =
+�
+Ω
+�
+|∇vj|2 − 1
+ε2 W ′(1 − |wε|2)φ · φΨ2
+j
+�
+dX
+(24)
+=
+�
+Ω
+�
+|∇(φΨj)|2 − ∇φ · ∇(φ Ψ2
+j)
+�
+dX =
+�
+Ω
+φ2|∇Ψj|2 dX.
+As Gε is continuous in strong H1(Ω) topology (since W ′(1 − |wε|2) ∈ L∞(Ω)), by density
+of C∞
+c (BN × Rn; RM) in H1
+0(BN × Rn; RM), Fatou’s lemma yields
+Gε(v) ≥
+�
+Ω
+φ2|∇
+�v
+φ
+�
+|2 dX ≥ 0,
+∀v ∈ H1
+0(BN × Rn; RM).
+12
+
+As a consequence of (23), we deduce that wε is a minimizer of Eε over A . Moreover,
+Gε(v) = 0 if and only if there exists a (constant) vector λ ∈ RM such that v = λφ for a.e.
+x ∈ Ω.
+Step 3: Set of global minimizers. From now on, we assume that W is strictly convex and
+denote wε = (wε,1, . . . , wε,M). Note that the map
+˜wε := (wε,1, . . . , wε,N, 0RM−N−1,
+�
+w2
+ε,N+1 + · · · + w2
+ε,M)
+(25)
+belongs to A , | ˜wε| = |wε| and |∇ ˜wε| ≤ |∇wε| in Ω, so Eε(wε) ≥ Eε( ˜wε) and
+�
+w2
+ε,N+1 + · · · + w2
+ε,M ≥ wε,M = φ > 0
+in
+Ω.
+Hence, ˜wε is a minimizer of Eε on A (as wε minimizes Eε over A by Step 2). Therefore,
+up to interchanging wε and ˜wε, we may assume
+� wε,N+1 = · · · = wε,M−1 ≡ 0 in Ω
+wε,M = φ
+(20)
+> 0 in Ω.
+We now consider another minimizer Uε of Eε over A and denote v := Uε −wε ∈ H1
+0(BN ×
+Rn; RM) after a suitable extension. From Steps 1 and 2 we know that Eε(Uε) = Eε(v +
+wε) = Eε(wε), Gε(v) = 0, |v+wε| = |wε| a.e. in Ω and v = λφ for some λ = (λ1, . . . , λM) ∈
+RM where we recall that φ = wε·eM. By continuity of wε and φ, the relation |v+wε| = |wε|
+a.e. in Ω implies 2wε · v + |v|2 = 0 everywhere in Ω. Since v = λφ, dividing by φ > 0 in
+Ω, we obtain
+2λ · wε + φ|λ|2 = 0 in Ω
+(26)
+and by continuity, the equality holds also on ∂Ω. As for every (x, z) ∈ ∂BN × (0, 1)n,
+φ(x, z) = 0 and wε(x, z) = (x, 0RM−N ), we deduce that λ · (x, 0RM−N ) = 0 for every
+x ∈ ∂BN. It follows that λ1 = λ2 = · · · = λN = 0 and therefore, recalling that wε,N+1 =
+· · · = wε,M−1 = 0 in Ω, we have by (26):
+2λMφ + (λ2
+N+1 + · · · + λ2
+M)φ = 0 in Ω.
+As φ > 0 in Ω, we obtain
+λ2
+N+1 + · · · + λ2
+M−1 + (λM + 1)2 = 1;
+hence we can find R ∈ O(M) such that Rp = p for all p ∈ RN × {0RM−N } and
+ReM = (0, . . . , 0, λN+1, . . . , λM−1, λM + 1).
+This implies Uε = wε+v = wε+λφ = Rwε as required. The converse statement is obvious:
+if wε is a minimizer of Eε over A and R ∈ O(M) is a transformation fixing all points of
+RN ×{0RM−N }, then Rwε is also a minimizer of Eε over A (because Eε and the boundary
+condition in A are invariant under such orthogonal transformation R).
+13
+
+Remark 8. Note that if n ≥ 1, M > N ≥ 7 and W satisfies (2) (not necessarily strictly
+convex), then there are no bounded critical points of the energy Eε in the set A escaping in
+a direction e ∈ SM−1. Indeed, if such an escaping critical point of Eε in A exists, then by
+Theorem 7, this solution would be a global minimizer of Eε in A which is a contradiction
+with the uniqueness of the global minimizer (uε, 0RM−N ) in (4) (that is non-escaping)
+proved in Theorem 5.
+3.2
+Escaping radial profile
+Let M ≥ N + 1. We give a necessary and sufficient condition for the existence of an
+escaping radial profile ( ˜fε, gε > 0) in (0, 1) to the system (9)–(12); we also prove uniqueness,
+minimality and monotonicity of the escaping radial profile. For that, in the context of Eε
+defined over A , we introduce the functional
+Iε(f, g) =
+1
+|SN−1|Eε
+�
+(f(r) x
+|x|, 0RM−N−1, g(r))
+�
+= 1
+2
+� 1
+0
+�
+(f ′)2 + (g′)2 + N − 1
+r2
+f 2 + 1
+ε2 W(1 − f 2 − g2)
+�
+rN−1 dr
+where (f, g) belongs to
+B =
+�
+(f, g) : r
+N−1
+2 f ′, r
+N−3
+2 f, r
+N−1
+2 g′, r
+N−1
+2 g ∈ L2(0, 1), f(1) = 1, g(1) = 0
+�
+.
+(27)
+The following result is reminiscent from Ignat-Nguyen [8, Theorem 2.4] (for ˜W ≡ 0).
+The proof of [8, Theorem 2.4] is rather complicated (as it is proved for some general
+potentials ˜W). We present here a simple proof that works in our context:
+Theorem 9. Let 2 ≤ N ≤ 6, M ≥ N + 1, W ∈ C2((−∞, 1]) satisfy (2) and be strictly
+convex. Consider εN ∈ (0, ∞) in (7) such that ℓ(εN) = 0. Then the system (9)–(12) has
+an escaping radial profile ( ˜fε, gε) with gε > 0 in (0, 1) if and only if 0 < ε < εN. Moreover,
+in the case 0 < ε < εN,
+1. ( ˜fε, gε > 0) is the unique escaping radial profile of (9)–(12) and
+˜fε
+r , gε ∈ C2([0, 1]),
+˜f 2
+ε + g2
+ε < 1, ˜fε > 0, ˜f ′
+ε > 0, g′
+ε < 0 in (0, 1);
+2. there are exactly two minimizers of Iε in B given by ( ˜fε, ±gε);
+3. the non-escaping radial profile (fε, 0) is an unstable critical point of Iε in B where
+fε is the unique radial profile in (5).
+Recall that for ε ≥ εN, the non-escaping radial profile (fε, 0) is the unique global
+minimizer of Iε in B (by Theorem 3 whose proof yields the minimality of (uε, 0RM−N ) of
+Eε in A ).
+Proof of Theorem 9. First, we focus on the existence of escaping radial profiles of (9)–(12).
+Note that the direct method in calculus of variations implies that Iε admits a minimizer
+14
+
+( ˜fε, gε) ∈ B.
+Since ( ˜fε, gε) ∈ B, ( ˜fε, gε) ∈ C((0, 1]).
+It follows that ( ˜fε, gε) satisfies
+(10)–(12) in the weak sense, and so ˜fε, gε ∈ C2((0, 1]). Since (| ˜fε|, |gε|) is also a minimizer
+of Iε in B, the above argument also shows that | ˜fε|, |gε| ∈ C2((0, 1]) satisfies (10)–(12).
+Since | ˜fε|, |gε| ≥ 0 and ˜fε(1) = 1, the strong maximum principle yields | ˜fε| > 0 in (0, 1),
+and either |gε| > 0 in (0, 1) or gε ≡ 0 in (0, 1). It follows that ˜fε > 0 in (0, 1), and there
+are three alternatives: gε > 0 in (0, 1), gε < 0 in (0, 1) or gε ≡ 0 in (0, 1). Clearly, when
+gε ≡ 0, ˜fε is equal to the unique radial profile fε in (5). By considering ( ˜fε, −gε) instead
+of ( ˜fε, gε) if necessary, we assume in the sequel that gε ≥ 0.
+Claim: if 0 < ε < εN, then gε > 0 in (0, 1) and (fε, 0) is an unstable critical point of Iε in
+B.
+Proof of Claim: We define the second variation of Iε at (fε, 0) as
+Qε(α, β) = d2
+dt2
+����
+t=0
+Iε
+�
+(fε, 0) + t(α, β)
+�
+=
+�
+BN
+�
+Lεα · α + Lεβ · β + N − 1
+r2
+α2 + 2
+ε2 W ′′(1 − f 2
+ε )f 2
+ε α2�
+dx,
+for α, β ∈ C∞
+c ((0, 1)) which extends by density to the Hilbert space
+H = {(α, β) : (fε + α, β) ∈ B} with the norm
+∥(α, β)∥H := ∥(α x
+|x|, β)∥H1(BN,RN+1).
+As ε ∈ (0, εN), we have ℓ(ε) < 0 by (7). Taking β ∈ H1
+0(BN) to be any first eigenfunction
+of Lε in BN, which is radially symmetric, we have r
+N−1
+2 β′, r
+N−1
+2 β ∈ L2(0, 1), β(1) = 0 and
+Qε(0, β) =
+�
+BN Lεβ · β dx = ℓ(ε)
+�
+BN β2 dx < 0.
+So, (fε, 0) is an unstable critical point of Iε in B if ε < εN. In particular, (fε, 0) is not
+minimizing Iε in B and therefore, by the above construction of the minimizer ( ˜fε, gε) of
+Iε in B, we deduce that gε > 0. This proves the above Claim.
+Moreover, by [8, Lemmas 2.7 and A.5, Proposition 2.9] (for ˜W ≡ 0), we deduce that
+˜fε
+r , gε ∈ C2([0, 1]), ˜f 2
+ε + g2
+ε < 1, ˜f ′
+ε > 0 and g′
+ε < 0 in (0, 1).
+To conclude, we distinguish two cases:
+Case 1: if ε ∈ (0, εN), Claim yields the existence of an escaping radial profile ( ˜fε, gε > 0).
+By [8, Lemmas 2.7], every escaping radial profile ( ˜fε, gε > 0) is bounded (i.e., ˜f 2
+ε + g2
+ε < 1
+in (0, 1)) and therefore, by Theorem 7, the corresponding (bounded) escaping critical point
+˜uε in (9) is a global minimizer of Eε over A and the set of minimizers of Eε over A is then
+given by {R˜uε : R ∈ O(M), Rp = p, ∀p ∈ RN × {0RM−N }}. Therefore, ( ˜fε, ±gε) are the
+only two minimizers of Iε in B. In particular, this proves the uniqueness of the escaping
+radial profile ( ˜fε, gε > 0).
+Case 2: if ε ≥ εN, by the proof of Theorem 3, the non-escaping vortex sheet solution
+uε(x) ≡ (fε(|x|) x
+|x|, 0RM−N ) (by (13)) is the unique minimizer of Eε over A . In particular,
+(fε, 0) is the unique minimizer of Iε in B, i.e., in the above construction of the minimizer
+15
+
+( ˜fε, gε) of Iε in B, we have ˜fε = fε and gε = 0 in (0, 1). We claim that no escaping
+radial profile ( ˆfε, ˆgε > 0) exists if ε ≥ εN. Assume by contradiction that such an escaping
+radial profile ( ˆfε, ˆgε > 0) exists. The same argument presented in Case 1 would imply
+that ( ˆfε, ˆgε > 0) is a minimizer of Iε in B which contradicts the uniqueness of the global
+minimizer (fε, 0).
+3.3
+Proof of Theorem 4
+We now prove the main result:
+Proof of Theorem 4. By Theorem 9, the existence of an escaping radially symmetric so-
+lution ˜uε in (9) is equivalent to ε ∈ (0, εN). Moreover, in that case, the escaping radial
+profile ( ˜fε, gε > 0) is unique and bounded, i.e., ˜f 2
+ε + g2
+ε < 1 in (0, 1).
+Case 1: if ε ∈ (0, εN), Theorem 7 implies that the (bounded) escaping radially symmetric
+critical point ˜uε in (9) is a global minimizer of Eε over A and every minimizer of Eε over
+A has the form R˜uε for some orthogonal transformation R ∈ O(M) keeping invariant
+the space RN × {0RM−N }. Moreover, by Theorem 9, the non-escaping radial profile (fε, 0)
+is proved to be an unstable critical point of Iε in B, so the non-escaping vortex sheet
+solution (uε, 0RM−N ) is an unstable critical point of Eε in A .
+Case 2: if ε ≥ εN, the proof of Theorem 3 implies that the non-escaping radially symmetric
+vortex sheet solution uε(x) ≡ (fε(|x|) x
+|x|, 0RM−N ) (by (13)) is the unique minimizer of Eε
+over A . In this case, there is no bounded critical point wε of Eε over A that escapes in
+some direction e ∈ SM−1; indeed, if such (bounded) escaping solution wε satisfying (20)
+exists, then Theorem 7 would imply that wε is a global minimizer of Eε over A which
+contradicts that the non-escaping vortex sheet solution uε is the unique global minimizer
+of Eε over A .
+Theorem 4 holds also for the “degenerate” dimension n = 0. In this case, Ω = BN and
+vortex sheets are vortex points,
+Eε(u) =
+�
+BN
+�1
+2|∇u|2 + 1
+2ε2 W(1 − |u|2)
+�
+dx,
+A := {u ∈ H1(BN; RM) : u(x) = (x, 0RM−N ) on ∂BN = SN−1}
+and radially symmetric vortex critical points of Eε in A have the corresponding form in
+(9):
+˜uε(x) = ( ˜fε(r) x
+|x|, 0RM−N−1, gε(r)) ∈ A ,
+x ∈ BN, r = |x|,
+(28)
+where the radial profiles ( ˜fε, gε) satisfy the system (10)-(12) and are described in Theo-
+rem 9; the non-escaping radially symmetric vortex solution is given here by
+uε(x) = (fε(|x|) x
+|x|, 0RM−N )
+for all x ∈ BN,
+(29)
+where the radial profile fε is the unique solution to (5). We obtain the following result
+which generalizes [12, Theorem 1.1] that was proved in the case N = 2 and M = 3 (without
+identifying the meaning of the dichotomy parameter εN in (7)).
+16
+
+Theorem 10. Let 2 ≤ N ≤ 6, M ≥ N + 1, Ω = BN, W ∈ C2((−∞, 1]) satisfy (2) and
+be strictly convex. Consider εN ∈ (0, ∞) such that ℓ(εN) = 0 in (7). Then there exists an
+escaping radially symmetric vortex solution ˜uε in (28) with the radial profile ( ˜fε, gε > 0)
+given in Theorem 9 if and only if 0 < ε < εN. Moreover,
+1. if 0 < ε < εN, ˜uε is a global minimizer of Eε in A and all global minimizers of
+Eε in A are radially symmetric given by R˜uε where R ∈ O(M) is an orthogonal
+transformation of RM satisfying Rp = p for all p ∈ RN ×{0RM−N }. In this case, the
+non-escaping vortex solution uε in (29) is an unstable critical point of Eε in A .
+2. if ε ≥ εN, the non-escaping vortex solution uε in (29) is the unique global minimizer
+of Eε in A . Furthermore, there are no bounded critical points wε of Eε in A that
+escape in a direction e ∈ SM−1, i.e., wε · e > 0 a.e. in Ω.
+The proof follows by the same argument used for Theorem 4, the main difference is
+that in the ball Ω = BN, a critical point wε of Eε in A satisfies the PDE system with
+Dirichlet boundary condition (instead of the mixed Dirichlet-Neumann condition in (21)):
+−∆wε = 1
+ε2 wε W ′(1 − |wε|2)
+in BN,
+wε(x) = (x, 0RM−N )
+on ∂BN.
+A
+Appendix. Vortex sheet SM−1-valued harmonic maps in
+cylinders
+In dimensions M > N ≥ 2 and n ≥ 1, for the cylinder shape domain Ω = BN × (0, 1)n,
+we consider the harmonic map problem for SM−1-valued maps u ∈ H1(Ω; SM−1) ∩ A
+associated to the Dirichlet energy
+E(u) = 1
+2
+�
+Ω
+|∇u|2 dxdz.
+Any critical point u : Ω → SM−1 of this problem satisfies
+
+
+
+−∆u = u |∇u|2
+in Ω,
+∂u
+∂z = 0
+on BN × ∂(0, 1)n,
+u(x, z) = (x, 0RM−N )
+on ∂BN × (0, 1)n.
+(30)
+We will focus on radially symmetric vortex sheet SM−1-valued harmonic maps having the
+following form (invariant in z-direction):
+u(x, z) = (f(r) x
+|x|, 0RM−N−1, g(r)) ∈ A ,
+x ∈ BN, z ∈ (0, 1)n, r = |x|,
+(31)
+where the radial profile (f, g) satisfies
+f 2 + g2 = 1
+in
+(0, 1),
+(32)
+17
+
+and the system of ODEs:
+−f ′′ − N − 1
+r
+f ′ + N − 1
+r2
+f = Γ(r)f
+in
+(0, 1),
+(33)
+−g′′ − N − 1
+r
+g′ = Γ(r)g
+in
+(0, 1),
+(34)
+f(1) = 1 and g(1) = 0,
+(35)
+where
+Γ(r) = (f ′)2 + N − 1
+r2
+f 2 + (g′)2
+is the Lagrange multiplier due to the unit length constraint in (32). As for the Ginzburg-
+Landau system, we distinguish two type of radial profiles:
+• the non-escaping radial profile ( ¯f ≡ 1, ¯g ≡ 0) yielding the non-escaping (radially
+symmetric) vortex sheet SM−1-valued harmonic map (also called “equator” map):
+¯u(x, z) = ( x
+|x|, 0RM−N )
+x ∈ BN, z ∈ (0, 1)n.
+(36)
+Note that ¯u is singular and the singular set of this map is the vortex sheet {0RM−N }×(0, 1)n
+of dimension n in Ω. Also, observe that ¯u ∈ H1(Ω, SM−1) if and only if N ≥ 3.
+• the escaping radial profile (f, g) with g > 0 in (0, 1); in this case, it holds f(0) = 0,
+g(0) = 1 and we say that u in (31) is an escaping (radially symmetric) vortex sheet SM−1-
+valued harmonic map. Note that u is smooth for every dimension M > N ≥ 2 and n ≥ 1
+and the zero set of (u1, . . . , uN) is the vortex sheet {0RM−N } × (0, 1)n of dimension n in
+Ω. Obviously, (f, −g < 0) is another radial profile satisfying (32)-(35).
+The properties of such radial profiles are proved in [14] (see also [8, Theorem 2.6] for
+˜W ≡ 0 in those notations). More precisely,
+(a) If N ≥ 7, the non-escaping radial profile ( ¯f ≡ 1, ¯g ≡ 0) is the unique minimizer of
+I(f, g) =
+1
+|SN−1|E
+�
+(f(r) x
+|x|, 0RM−N−1, g(r))
+�
+= 1
+2
+� 1
+0
+�
+(f ′)2 + (g′)2 + N − 1
+r2
+f 2�
+rN−1 dr,
+where (f, g) belongs to B ∩
+�
+(f, g) : f 2 + g2 = 1
+�
+with B defined in (27). Moreover,
+the system (32)–(35) has no escaping radial profile (f, g) with g > 0 in (0, 1).
+(b) If 2 ≤ N ≤ 6, then there exists a unique escaping radial profile (f, g) with g > 0
+satisfying (32)–(35). Moreover, (f, ±g) are the only two global minimizers of I in
+B ∩
+�
+(f, g) : f 2 + g2 = 1
+�
+, f
+r , g ∈ C∞([0, 1]), f(0) = 0, g(0) = 1, f > 0, f ′ > 0 and
+g′ < 0 in (0, 1). In addition, for 3 ≤ N ≤ 6, the non-escaping solution ( ¯f ≡ 1, ¯g ≡ 0)
+is an unstable critical point of I in B ∩
+�
+(f, g) : f 2 + g2 = 1
+�
+.6
+6For N = 2, (1, 0) /∈ B; however, we can define the second variation of I at (1, 0) along directions (0, q)
+compactly supported in (0, 1):
+Q(0, q) =
+� 1
+0
+�
+(q′)2 − N − 1
+r2
+q2�
+rN−1 dr,
+and one can prove the existence of q ∈ Lipc(0, 1) such that Q(0, q) < 0 (see e.g. [8, Remark 2.16]).
+18
+
+There is a large number of articles studying existence, uniqueness, regularity and
+stability of radially symmetric SM−1-valued harmonic maps (e.g., [13, 14, 25, 26, 23, 16,
+12]). We summarize here the main result for our problem in the cylinder shape domain
+Ω = BN × (0, 1)n: if N ≤ 6, then minimizing SM−1-valued harmonic maps in A are
+smooth, radially symmetric and escaping in one-direction; if N ≥ 7, then there is a unique
+minimizing SM−1-valued harmonic map in A which is singular and given by the equator
+map ¯u in (36). 7
+Theorem 11. Let n ≥ 1, N ≥ 2, M ≥ N + 1 and Ω = BN × (0, 1)n. Then
+1. if 2 ≤ N ≤ 6, then the escaping radially symmetric vortex sheet solution u in (31)
+with g > 0 is a minimizing SM−1-valued harmonic map in A and all minimizing
+SM−1-valued harmonic maps in A are smooth radially symmetric given by Ru where
+R ∈ O(M) satisfies Rp = p for all p ∈ RN ×{0RM−N }. In this case, the equator map
+¯u in (36) is an unstable SM−1-valued harmonic map in A .
+2. if N ≥ 7, the non-escaping vortex sheet solution ¯u in (36) is the unique minimizing
+SM−1-valued harmonic map in A . Moreover, there is no SM−1-valued harmonic
+map w in A escaping in a direction e ∈ SM−1, i.e., w · e > 0 a.e. in Ω.
+The main ingredient is the following result yielding minimality of escaping SM−1-valued
+harmonic maps. This is reminiscent from Sandier-Shafrir [23] (see also [12, Theorem 1.5]).
+Theorem 12. Let n ≥ 1, M > N ≥ 2 and Ω = BN × (0, 1)n.
+Assume that w ∈
+A ∩ H1(Ω, SM−1) is a SM−1-valued harmonic map satisfying (30) and
+w · e > 0 a.e. in Ω
+(37)
+in an escaping direction e ∈ SM−1. Then w is a minimizing SM−1-valued harmonic map
+in A and all minimizing SM−1-valued harmonic maps in A are of the form Rw where R ∈
+O(M) is an orthogonal transformation of RM satisfying Rp = p for all p ∈ RN ×{0RM−N }.
+Proof of Theorem 12. We give here a simple proof based on the argument in [12] that
+avoids the regularity results used in [23]. By the H1/2-trace theorem applied for w ∈
+H1(Ω, SM−1), (37) implies that w · e ≥ 0 on ∂BN × (0, 1)n. Combined with the vortex
+boundary condition in (30), we deduce that the escaping direction e has to be orthogonal
+to RN × {0RM−N } and up to a rotation, we can assume that e = eM (as in (22)). Then
+φ = w · eM > 0 a.e. in Ω satisfies
+− ∆φ = |∇w|2φ in Ω, ∂φ
+∂z = 0 on BN × ∂(0, 1)n, φ = 0 on ∂BN × (0, 1)n.
+(38)
+We consider configurations8 ˜w = w + v : Ω → SM−1 with v ∈ H1
+0(BN × Rn, RM) (in
+particular, |v| ≤ 2 in Ω). Then
+2w · v + |v|2 = 0
+a.e. in Ω.
+(39)
+7We mention the paper of Bethuel-Brezis-Coleman-H´elein [2] about a similar phenomenology in a do-
+main Ω = (B2 \ Bρ) × (0, 1) ⊂ R3 where Bρ ⊂ R2 is the disk centered at 0 of radius ρ.
+8Note that for any ˜w ∈ A ∩ H1(Ω, SM−1), the map ˜w − w has an extension in H1
+0(BN × Rn, RM).
+19
+
+Using (30) and (39), we obtain
+2
+�
+Ω
+∇w · ∇v = 2
+�
+Ω
+|∇w|2w · v dx = −
+�
+Ω
+|∇w|2|v|2 dx,
+yielding9
+�
+Ω
+|∇(w + v)|2 dx −
+�
+Ω
+|∇w|2 dx =
+�
+Ω
+|∇v|2 − |∇w|2|v|2 dx =: Q(v).
+(40)
+To show that w is minimizing, we prove that Q(v) ≥ 0 for all v ∈ H1
+0(BN × Rn, RM) ∩
+L∞(Ω; RM) (note that this is a class larger than what we need, as we do not require
+that v satisfy the pointwise constraint (39)). For that, we take an arbitrary map ˜v ∈
+C∞
+c (BN × Rn, RM) of support ω and decompose it as ˜v = φΨ in Ω. This decomposition
+makes sense as φ ≥ δ > 0 in ω ∩ Ω for some δ > 0 (which may depend on ω). Indeed, by
+(37) and (38), φ is a superharmonic function (i.e., −∆φ ≥ 0 in Ω) that belongs to H1(Ω).
+As ∂φ
+∂z = 0 on BN × ∂(0, 1)n, φ can be extended by even mirror symmetry to the domain
+˜Ω = BN × (−1, 2)n so that φ is superharmonic in ˜Ω. Thus, the weak Harnack inequality
+(see e.g. [6, Theorem 8.18]) implies that on the compact set ω ∩Ω in ˜Ω, we have φ ≥ δ > 0
+for some δ. So, ˜v = φΨ in Ω with Ψ = (Ψ1, . . . , ΨM) ∈ H1 ∩ L∞(Ω; RM) vanishing in a
+neighborhood of ∂BN × (0, 1)n. Then integration by parts yields for 1 ≤ j ≤ M:
+Q(˜vj) =
+�
+Ω
+|∇˜vj|2 − |∇w|2φ · φΨ2
+j dx
+(38)
+=
+�
+Ω
+|∇(φΨj)|2 − ∇φ · ∇(φ Ψ2
+j) dx =
+�
+Ω
+φ2|∇Ψj|2 dx ≥ 0
+for all ˜v ∈ C∞
+c (BN × Rn, RM). Then for every v ∈ H1
+0(BN × Rn, RM) ∩ L∞(Ω; RM), there
+exists a sequence ˜vk ∈ C∞
+c (BN × Rn, RM) such that ˜vk → v and ∇˜vk → ∇v in L2 and
+a.e. in BN × Rn and |˜vk| ≤ ∥v∥L∞(Ω) + 1 in Ω for every k. In particular, by dominated
+convergence theorem, we have Q(˜vk) → Q(v) thanks to (40). Thus, we deduce that for
+every compact ω ⊂ ˜Ω = BN × (−1, 2)n,
+Q(v) = lim
+k→∞ Q(˜vk) ≥ lim inf
+k→∞
+�
+ω∩Ω
+φ2|∇
+�˜vk
+φ
+�
+|2 dx ≥
+�
+ω∩Ω
+φ2|∇
+� v
+φ
+�
+|2 dx ≥ 0,
+where we used Fatou’s lemma. In particular, w is a minimizing SM−1-valued harmonic
+map by (40) and Q(v) = 0 yields the existence of a vector λ ∈ RM such that v = λφ a.e.
+in Ω. Then the classification of the minimizing SM−1-valued harmonic maps follows by
+(39) as in the Step 3 of the proof of Theorem 7.
+Proof of Theorem 11. 1. This part concerning the dimension 2 ≤ N ≤ 6 follows from
+Theorem 12 and the instability of the radial profile (1, 0) for I in B∩
+�
+(f, g) : f 2+g2 = 1
+�
+as explained above.
+9Note that the functional Q represents the second variation of E at w, but here the map v is not
+necessarily orthogonal to w.
+20
+
+2. This part for dimension N ≥ 7 follows the ideas in [14]. More precisely, calling
+X = (x, z) the variable in Ω, we have as in the proof of Theorem 12 for every v ∈
+H1
+0(BN × Rn, RM) with |v + ¯u| = 1 in Ω:
+�
+Ω
+|∇(¯u + v)|2 dX−
+�
+Ω
+|∇¯u|2 dX =
+�
+Ω
+�
+|∇v|2 − |∇¯u|2|v|2�
+dX
+=
+�
+Ω
+|∇zv|2 dX +
+�
+(0,1)n dz
+�
+BN
+�
+|∇xv|2 − N − 1
+|x|2 |v|2�
+dx
+≥
+�
+Ω
+|∇zv|2 dX +
+�(N − 2)2
+4
+− (N − 1)
+� �
+Ω
+|v|2
+|x|2 dX ≥ 0
+where we used the Hardy inequality for v(·, z) ∈ H1
+0(BN, RM) for a.e. z ∈ (0, 1)n. This
+proves that ¯u is the unique minimizing SM−1-valued harmonic map in A .
+Combined
+with Theorem 12, we conclude that there is no escaping SM−1-valued harmonic map w in
+A .
+References
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+Ginzburg-Landau vortex in the unit ball in dimensions N ≥ 7, C. R. Math. Acad.
+Sci. Paris 356 (2018), 922-926.
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+of Ginzburg-Landau functionals, Ann. Sci. ´Ec. Norm. Sup´er. 53 (2020), 589-613.
+[13] W. Jager and H. Kaul, Uniqueness and stability of harmonic maps and their Jacobi
+fields, Manuscripta Math. 28, 1-3 (1979), 269-291.
+[14] W. Jager and H. Kaul, Rotationally symmetric harmonic maps from a ball into a
+sphere and the regularity problem for weak solutions of elliptic systems, J. Reine
+Angew. Math. 343 (1983), 146-161.
+[15] F.-H., Lin, A remark on the map x/|x|, C. R. Acad. Sci. Paris S´er. I Math. 305, 12
+(1987), 529-531.
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+Ginzburg-Landau functional, J. Eur. Math. Soc. (JEMS) 12, 5 (2010), 1069-1096.
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+Funct. Anal. 130 (1995), 334-344.
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+sym´etrie radiale, C. R. Acad. Sci. Paris S´er. I Math. 323, 6 (1996), 593-598.
+[20] F. Pacard and T. Rivi`ere, Linear and nonlinear aspects of vortices, vol. 39 of Progress
+in Nonlinear Differential Equations and their Applications. Birkh¨auser Boston, Inc.,
+Boston, MA, 2000. The Ginzburg-Landau model.
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+mension, J. Funct. Anal. 260, 3 (2011), 892-905.
+[22] E. Sandier, Ginzburg-Landau minimizers from Rn+1 to Rn and minimal connections,
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+Anal. 272, 9 (2017), 3946-3964.
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+Geometry 17 (1982), 307-335.
+23
+

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+ESTIMATION OF USER’S WORLD MODEL USING GRAPH2VEC ∗
+Tatsuya Sakai,
+Takayuki Nagai
+Graduate School of Engineering Science, Osaka University
+1-3, Machikaneyama, Toyonaka, Osaka, Japan
+[email protected]
+ABSTRACT
+To obtain advanced interaction between autonomous robots and users, robots should be able to
+distinguish their state space representations (i.e., world models). Herein, a novel method was
+proposed for estimating the user’s world model based on queries. In this method, the agent learns
+the distributed representation of world models using graph2vec and generates concept activation
+vectors that represent the meaning of queries in the latent space. Experimental results revealed that
+the proposed method can estimate the user’s world model more efficiently than the simple method of
+using the “AND” search of queries.
+Keywords Autonomous robot · Explainability · Representation learning · User’s world model
+1
+Introduction
+Autonomous robots are increasingly being used in numerous applications. Currently, they assist humans in performing
+tasks by executing commands. For autonomous robots performing sophisticated decisions, the blind execution of
+commands is not always the best strategy. Moreover, in many situations, fully executing commands is difficult. These
+autonomous robots should be able to explain the reasons for their decisions to gain user trust. Explainable autonomous
+robots (XAR) are defined as robots that have these explanatory capabilities. The following four requirements have been
+identified for the XARs [1]:
+(1) Owning an interpretable decision-making space
+(2) Estimation of the model of others
+(3) Estimation of the information required for a user to estimate the policy of the robot
+(4) Presentation to the user of explanation
+This explanation mechanism is a mutual process between the robot and the user and is displayed in Fig. 1. The world
+model refers to the correspondence between actions and state changes, that is, the internal model [2] that represents the
+dynamics of the environment, and we do not distinguish between the world model and the environment in this paper.
+“Policy sharing” in Fig.1 is a spontaneous presentation of information of a policy (e.g., presentation of a sequence of
+actions to be taken), and an explanation is generated when a query to this information presentation is requested by
+others.
+Among the four requirements, the estimation of the other’s world model is particularly crucial for providing a user-
+specific explanation. In the context of human–robot interaction, the importance of estimating the user’s internal state
+has already been recognized. Gao et al. [3] and Clair et al. [4] proposed a framework for estimating plausible action
+strategies based on the actions and interaction history of users. Huang et al. [5] and Lage et al. [6] focused on restoring
+explanations to policies and advocated the importance of appropriately estimating the restoration algorithm of users
+requesting explanation. These studies have focused on internal states, particularly policies, and planning algorithms.
+∗This paper is an extended (English translated) version of “T.Sakai, T.Horii, T.Nagai, Representation Learning of World Models
+and Estimation of World Model of Others Using Graph2vec, Journal of RSJ, 40, 2, pp.166-169, 2022,” (in Japanese) ISSN 1884-7145,
+Print ISSN 0289-1824, https://doi.org/10.7210/jrsj.40.166
+arXiv:2301.03793v1  [cs.RO]  10 Jan 2023
+
+Preprint
+Figure 1: Explanation process as communication. To clarify elements of the explanation, observations/actions, and
+interactions between world models of others are segregated in the figure.
+Figure 2: Simplified explanation process considered in this paper. We only consider unidirectional explanation process
+from robot to user.
+However, real-world robots are designed to exhibit desired behavior in terms of algorithms and policies, and they
+share the same final objective with their users. In these situations, the results of action decisions typically stem from
+discrepancies in environmental awareness.
+In this study, a novel method was proposed for estimating the user’s world model based on the robot’s world model
+and the questions (queries) posed by the user. The proposed method can identify differences in the world models and
+generate an explanation that resolves the discrepancy between the perception of the environment by the robot and the
+user. In this study, we simplified the mechanism of Fig.1 as in Fig.2 2 and focused on estimating the world model of the
+explained person as the user’s world model.
+2
+Proposed method
+In the proposed method, the world model of the user is estimated by the following procedure (Fig. 3).
+(1) Acquisition of a distributed representation of the world model: A distributed representation of each world
+model is obtained using a graph-structured world model.
+(2) Acquisition of the query vector: Based on the query given by the user, the system acquires a direction vector
+that represents the meaning of the query in the representation space of the world model.
+(3) Estimation of user’s world model: The distributed representations of the world model and query vector are
+used to estimate the user’s world model based on cosine similarity.
+2When the model of others outputs an explanation, the content of the explanation is determined by considering information
+received from the world model.
+2
+
+Preprint
+Why donʼt you do 
+Agentʼs
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Userʼs
+Figure 3: Schematic of our proposed method.
+…
+…
+…
+=
+1
+2
+⋮
+=
+1
+2
+⋮
+Figure 4: Learning of a distributed representation of the world model. The experience on the environment is used to
+obtain graph-based world models. Then, after converting to WL labels, a distributed representation of each environment
+is obtained.
+The robot and user are assumed to share the same state space and measures; only the connection relation between states
+is assumed to be unshared. As a distributed representation of the world model, the parameters of a model representing a
+continuous state space, for example, [2], could be used. However, presenting the differences of the world model to the
+user requires the discretization of the state transition structure; the parameter space of the model does not necessarily
+represent the similarity of the state transition structure of the world model. Therefore, the world model is considered to
+be a discretized graph for obtaining a distributed representation. The generation of explanations is outside the scope of
+this study.
+2.1
+Acquisition of a distributed representation of the world model
+The learning process of a distributed representation of the world model is shown in Fig. 4. The robot learns a
+representation space representing the similarity of environments based on its experiences. First, the robot acquires an
+undirected graph representing the state transitions of each environment; simultaneously, it acquires policies through
+reinforcement learning. Specifically, the robot assumes that the states whose transitions were observed during the policy
+learning search are adjacent to each other and adds edges3.
+After acquiring the undirected graphs of all environments, graph2vec is applied to acquire the distributed representation
+of the graph of each environment [8]. Graph2vec is a method in which doc2vec [9] is used to acquire distributed
+representations of graphs, and instead of predicting the occurrence of words, the occurrence of labels that represent
+each subgraph is presented. Labels representing subgraphs are obtained using the Weisfeiler–Lehman (WL) relabeling
+process [10]. In this process, the label of the next layer is determined by considering the labels of neighboring nodes.
+Higher layer labels represent more global information regarding the graph.
+3This method is assumed to be used in a discrete state space; discretization of the state space is required for application to a real
+robot. However, this measure is beyond the scope of this study. For discretization, the method proposed in [7], wherein the policy
+implications of each state is considered, can be used.
+3
+
+------Preprint
+Graph2vec allows graphs in which the same subgraphs occur, that is, graphs in environments with similar state transition
+structures, to be embedded closer together in the representation space. Efficient search based on user queries becomes
+possible by acquiring the representation space of the world model in advance.
+When acquiring a world model, we explicitly provide an environment label to indicate the environment wherein the
+experience occurs. If no environment label is given, a world model is obtained using temporally continuous experiences.
+Furthermore, we consider that models with similar distributed representations represent the same environment and
+merging multiple models may be effective in obtaining a world model with higher accuracy.
+2.2
+Acquisition of the query vector
+Based on the query given by the user, the robot acquires a direction vector that represents the meaning of the query in
+the representation space of the world model. Kim et al. [11] focused on the middle layer of the neural network and
+generated concept vectors (CAV: concept activation vectors) by calculating the difference in latent representations when
+features that satisfy a specific concept and features that do not satisfy the concept are input. In this study, this method
+was applied to define a query vector vquery as Eq. (1) by considering the difference of distributed representations
+between environments that satisfy the query and those that do not. The query assumes the form “action aquery should
+be selected in state squery.”
+vquery = vpos − vneg,
+(1)
+where
+vpos =
+�
+i vi · P(aquery|vi, squery)
+�
+i P(aquery|vi, squery)
+,
+vneg =
+�
+i vi · (1 − P(aquery|vi, squery))
+�
+i(1 − P(aquery|vi, squery))
+.
+(2)
+Here, vi represents a distributed representation of the i-th environment. To correspond to a policy in which actions
+are selected probabilistically, the probability value of selecting an action is used as the coefficient of each distributed
+representation vi. If necessary, the coefficients can be expressed as the binary values of {0, 1}. Note that if the state
+Squery is not included in the undirected graph of the environment i, the CAV is calculated by excluding vi4.
+2.3
+Estimation of user’s world model
+Using the distributed representation of the world model and the query vector, the user’s world model was estimated
+using cosine similarity. The likelihood of an environment i as an estimated environment is expressed by Eq. (3).
+S(vi, vobs, Vq) =
+�
+j
+similarity(vj
+query, vi − vobs) − λ · distance(vi, vobs),
+(3)
+where vobs is a distributed representation of the environment currently observed by the agent, Vq is any number of query
+vectors, and vj
+query ∈ Vq is the j-th query vector. Furthermore, similarity(a, b) and distance(a, b) are functions that
+output the cosine similarity [−1, 1] and the distance between vectors a and b in the representation space, respectively.
+When reasoning in a real environment, the robot and the user observe almost the same environment. Therefore, because
+their world models are similar, the similarity between the direction of each environment and the direction of the query
+vector, as seen from vobs, is used as the estimation criterion. The coefficient λ is a hyperparameter that determines
+how much distance between vectors is considered and represents the strength of the assumption that the observed
+environment of the robot and the user are similar.
+Using the definitions, the world model of others to be estimated is expressed by Eq. (4).
+Env_est = arg max
+i
+S(vi, vobs, Vq)
+(4)
+In this study, we assume that the importance of all queries are equivalent and designed the evaluation function S as the
+sum of the similarities for each query vector vj
+query. However, in the real world, the importance of each query may
+differ, in which case the similarity should be multiplied by a coefficient ρj.
+4When estimating the user’s world model, the likelihood S(vi, vobs, Vq) is computed for all environments i, including those that
+do not contain the state squery.
+4
+
+Preprint
+2.3.1
+User vectors
+In addition to the query vector, the user vector that represents “what kind of environment with the distributed repre-
+sentation the user is likely to retain as a world model” can also be defined. The user vector is a vector that represents
+how the user tends to misunderstand the environment and plays a role in adding this tendency to the result of the user’s
+world model estimation.
+vuser = vu_pos − vu_neg,
+(5)
+where
+vu_pos =
+�
+i vi · P(vi)
+�
+i P(vi)
+,
+vu_neg =
+�
+i vi · (1 − P(vi))
+�
+i(1 − P(vi))
+.
+(6)
+P(vi) is the probability that the user estimates the environment corresponding to the distributed representation vi as
+the current world model when no information about the current environment is given to the user. In this paper, P(vi) is
+assumed to be known, and the estimation of P(vi) is outside the scope of this research.
+2.4
+Explanation by language
+Using a pre-trained language vector Eq. (7) in the representation space, the relationship between the world model
+maintained by the agent and the user’s world model can also be explained by language.
+vword(x) = average(vm − vn),
+(7)
+where vm and vn are distributed representations of environment pairs whose relation is represented by the language
+x. By averaging the differences of distributed representations of environment pairs vm and vn that satisfy a specific
+language description x, we can obtain a semantic vector represented by the language description x in the representation
+space.
+Using the language vector vword, the language describing the relationship between the world model maintained by the
+agent and the user’s world model is represented by Eq. (8).
+Explanation = arg max
+x
+similarity(vword(x), vEnv_est − vobs)).
+(8)
+By means of Eq. (8), the language x that represents the closest relationship between the current world models is selected
+as the explanation.
+3
+Experiment
+The proposed method was applied to an agent that acquires action strategies by proximal policy optimization (PPO) in a
+simulation environment, and its usefulness was evaluated. A partially modified version of the grid environment [12]
+with multiple objects was used for the experiments (Fig. 5). In this environment, the agent (triangle) obtains a reward by
+taking a key, opening a door, and reaching a goal in the lower right corner. The position of the goal remains unchanged,
+but the positions of the key and the door change every trial. The agent has five actions, namely go straight, turn left,
+turn right, take the key from the grid in front of it, and open the door. The agent observes the absolute position of the
+key (x, y coordinates), the absolute position of the door (x, y coordinates), its own absolute position (x, y coordinates)
+and orientation, holding/not holding the key, and opening/closing the door, for a total of nine dimensions.
+3.1
+Experiment 1: Acquisition of a distributed representation of the world model
+A graph representing the state transitions of each environment was obtained simultaneously with the learning of policies
+by PPO, and graph2vec was used to obtain a 16-dimensional distributed representation. An example of the acquired
+graph is shown in Fig.6. There are three edges that transition from the group of states before key acquisition to the
+group of states after key acquisition because keys can be acquired from three directions. On the other hand, there is
+only one edge that transitions from the group of states where the door is not open to the group of states where the door
+is open, because the door can be opened from only one state.
+5
+
+Preprint
+Why donʼt you pick up 
+Figure 5: Estimation results of the user’s world model.
+Table 1: Cumulative frequency and average value of the order in which the optimal environment appears. The number
+of trials is 100 for each method.
+Method
+Order
+1
+2
+3
+Order Average
+Our method
+35
+49
+60
+7.1
+Random
+6
+8
+11
+23.1
+The representation space was compressed to eight dimensions using independent component analysis, and the visualized
+results are displayed in Fig. 7. In this experiment, the absolute positions of the key and door were used as environment
+labels, and the five-dimensional observations excluding them were used as node features. The experimental results
+revealed that clusters are formed in the representation space based on the absolute positions of the key and door. In
+particular, clusters related to the position of the door are apparent, which can be attributed to the fact that the surrounding
+state transition relationship changes considerably compared with that of the key. In this experiment, the absolute
+positions of keys and doors are used only for identifying the environmental graph to be updated and are not embedded
+in the graph itself. Therefore, graph2vec created a representation space that appropriately reflects the differences in the
+location information of keys and doors expressed in the state transition structure.
+3.2
+Experiment 2: Estimation of the user’s world model
+The query vector was applied to the obtained representation space to estimate the world model of others. In this
+verification, we assume that questions were asked in the situations of acquiring a key and opening a door, and the query
+“In the state Squery, we should {take the key/open the door}” was considered. Figure 5 displays the results of estimating
+the others’ world model given the reference world model and query. The environment that satisfies the query has the
+highest evaluation value, and the environment in which the key is located on the opposite side of the grid environment
+has the lowest evaluation value. This result suggests that the obtained query vector is appropriate for estimating the
+world model.
+The optimal environment is defined as the agent’s world model with minimal modifications for satisfying the query. For
+example, given the query “In state Squery, the agent should take the key,” the optimal environment is the environment
+in which only the position of the key is changed to satisfy the query, whereas the position of the door is left unchanged.
+For each randomly selected agent world model/query pair, the evaluation values for each environment obtained using
+Eq. (3) were sorted in descending order to obtain the order of appearance of the optimal environment (Table 1). The
+order of appearance of the environments that satisfy the query when sorted randomly is displayed for comparison. The
+experimental results confirmed that the proposed method ranks the optimal environments higher.
+6
+
+FoFo
+--
+ooPreprint
+Figure 6: An example graph of the acquired world model. The blue nodes represent the state before the key acquisition.
+The orange nodes represent the state after the key acquisition when the door is not open, and the green nodes represent
+the state when the door is open.
+Although the direct manipulation of the positions of keys and doors can change the order of the appearance of the
+optimal environment to one, direct manipulation is not always possible in cases in which directly manipulatable
+information is not given as a query. The proposed method estimates the optimal environment at an early stage, although
+the state transition relations to be changed are not explicitly given. This property of the proposed method is crucial.
+3.3
+Experiment 3: Validation with multiple queries
+An explanatory agent A and an explained agent B are prepared, and the number of queries required for A to accurately
+estimate B’s world model is evaluated. This experiment assumes a situation in which the user is asked to confirm the
+correctness of the estimation results of the other’s world model through the presentation of an action sequence, which
+improves the estimation accuracy (Fig. 8). The outline of the experiment is as follows:
+(1) A and B share an initial state (absolute position and orientation of the agent, and the state of the key and door)
+and an environment-policy pair that specifies the strategy to be used in specific environments. They have
+arbitrary world models with different key and door positions.
+(2) Here, A sets its own world model as the initial value of the other’s world model and presents the optimal
+sequence of actions in that model in turn (policy sharing) 5.
+(3) B adds the query “In state squery, action aquery should be chosen” when the given action differs from the
+optimal action in its measure. The existing query is not deleted.
+(4) A updates the other’s world model based on the query and presents the action sequences in the updated other’s
+world model again in sequence with Squery as the initial state.
+(5) Repeat (3) and (4) to evaluate the number of environment updates required for A to obtain B’s world model as
+the other’s world model.
+The environment selected once is not selected, and the second or subsequent candidate is adopted. If the same
+environment has not been obtained after all the action sequences are presented, the environment is continuously updated
+5Policy sharing in this experiment (Fig. 8) is performed to confirm the estimation results of the other’s world model and differs
+from the presentation of information about one’s own policy in Fig. 1 and Fig. 2.
+7
+
+Preprint
+Figure 7: Results of latent space visualization. Each data point is illustrated according to (a) X-coordinate of the key,
+(b) Y-coordinate of the key, (c) X-coordinate of the door, and (d) Y-coordinate of the door.
+Figure 8: Schematic of experiment 3. The agent transmits information on its policy to the user and updates the user’s
+world model based on queries.
+without increasing the number of queries. In this verification, the proposed method was compared with the “AND”
+search of queries as a method of updating the world model of others. In practice, the following three methods are
+compared.
+Proposed method:
+Select a plausible environment using Eq. (4).
+AND search 1:
+Randomly selects an environment from among the environments that satisfy the query.
+AND search 2:
+The environment is randomly selected with the constraint that “the optimal behavior for the policy B
+is selected in all states from the initial state until reaching state squery”. Thus, it adds constraints and increases
+the information provided compared with the two update methods described.
+Experimental results revealed that the proposed method can estimate others’ world models with the fewest number
+of updates (Table 2). The results of the corresponding two-tailed t-test revealed that t(100) = 8.07 and p < .01 for
+the proposed method and AND search 1, and t(100) = 4.59 and p < .01 for the proposed method and AND search 2.
+Thus, both significant differences were confirmed.
+8
+
+V
+V
+M
+V
+V
+V
+V
+V
+会
+VI
+V
+V
+V
+V
+V
+V
+V
+V
+V
+V
+△
+W
+V
+V
+V
+W
+△
+V
+△
+V
+V
+V
+V
+V
+V
+V
+V
+V
+V
+V
+△
+1
+口 
+I
+I
+V
+△△
+口
+V
+口
+口
+口
+I
+口
+口
+I
+1
+口
+1
+口口
+口
+口口
+口
+口
+V
+V
+V
+V
+V
+V
+V
+V
+VV
+V口
+0VX
+W
+V
+V
+口
+口
+口
+口
+口
+■
+口
+口
+口Preprint
+Table 2: Number of updates required to estimate the user’s world model.
+Method
+Number of updates
+Standard deviation
+Our method
+5.53
+4.83
+AND search 1
+20.72
+17.43
+AND search 2
+8.69
+4.71
+Table 3: Comparison with the use of probabilistic evaluation.
+Method
+Nuber of updates
+Standard deviation
+Our method (λ = 0.05)
+3.93
+5.38
+Our method (λ = 0)
+5.75
+7.11
+Probabilistic value (λ = 0.05)
+7.13
+7.79
+The most common information given directly is “AND search2”. By contrast, the proposed method can reduce the
+number of updates by vectorizing queries and utilizing prior knowledge embedded in the representation space as
+additional information.
+3.4
+Experiment 4: Comparison with the use of probabilistic evaluation
+We compare the proposed method with the case where the probability value of each environment satisfying the query is
+used as the evaluation function instead of CAV. The proposed method uses Eq. (3) as the evaluation function, while the
+comparison method uses Eq. (9).
+S(vi, vobs, Vq) =
+�
+j
+P(aquery|vi, squery) − λ · distance(vi, vobs).
+(9)
+The procedure for this experiment is the same as in experiment 3; however, the initial world model is not completely
+random, and the coordinates of either the key or the door are assumed to be identical. This condition replicates the
+assumption that the agent’s world model and the user’s world model are similar6.
+Experimental results showed that the proposed method, which takes into account the distance in the representation
+space (λ = 0.05), was able to estimate the user’s world model with the fewest number of updates (Table3). The results
+of the corresponding two-tailed t test showed that the proposed method with λ = 0.05 compared to λ = 0 showed
+t(99) = 4.59 and p < .005, while the proposed method with λ = 0.05 compared to the method using probability values
+showed t(99) = 4.44 and p < .005, both of which are significantly different from each other7.
+The proposed method, which takes into account the distance in the representation space, was able to estimate the
+environment with a significantly smaller number of updates than the method using the same coordinates for either the
+key or the door. Compared to the method using probability value as the evaluation function, the proposed method
+was able to absorb small errors in probability values, resulting in a significantly smaller number of updates. When the
+distance in the representation space corresponds to the similarity of the state transition structure between environments,
+as in the present verification, it is effective to use CAV to obtain environments that are perpendicular to the virtual
+separation boundary as the user’s world models.
+3.5
+Experiment 5: Number of samples and accuracy of CAV
+We evaluate the number of queries required to correctly estimate the world model when the number of samples (number
+of environments) used to compute the CAV is reduced. The evaluation procedure is the same as in Experiment 4. In this
+experiment, the maximum number of samples is 300 because 300 environments are embedded in the representation
+space. We also set λ = 0.05.
+6Without this assumption, the world model with minimal modification to satisfy the query (the optimal environment) is not
+necessarily the user’s world model. However, theoretically, when the evaluation value is calculated using Eq. (3), other environments
+that satisfy the query will have a lower evaluation value compared to the optimal environment. Therefore, if the assumption that the
+world models of the agent and user are similar cannot be made, it is desirable to use Eq. (9). However, in a real environment, the
+world models of the agent and user are not completely independent, and similarity can be assumed.
+7Because the t test was applied twice in this verification, a significant difference was found at the significance level α = 0.01
+based on the Bonferroni method.
+9
+
+Preprint
+Table 4: Relationship between the number of CAV samples and the order of appearance of the optimal environment.
+Each value represents the cumulative frequency, and the number of trials is 100 for each.
+Nuber of samples
+Order of appearance
+1
+2
+3
+300
+40
+62
+69
+250
+44
+61
+64
+200
+42
+51
+62
+150
+40
+51
+55
+100
+35
+46
+54
+50
+9
+18
+26
+Prior distribution 1
+Prior distribution 2
+P(door_y = 1) = 0.4
+P(door_y = 2) = 0.3
+P(door_y = 3) = 0.2
+P(door_y = 4) = 0.1
+P(door_y = 5) = 0.0
+P(door_y = 6) = 0.0
+P(door_y = 1) = 0.0
+P(door_y = 2) = 0.1
+P(door_y = 3) = 0.4
+P(door_y = 4) = 0.4
+P(door_y = 5) = 0.1
+P(door_y = 6) = 0.0
+Figure 9: The prior distribution used in experiment 6. The prior distribution for the y-coordinate of the door (in the
+vertical direction) is defined.
+The experimental results show that the accuracy deteriorates much more slowly up to 100 samples than when the CAV
+is generated with 300 samples (Table 4). This suggests that a specific level of estimation accuracy can be maintained
+even when the number of samples is reduced. The fact that the user’s world model is estimated taking into account
+the distance in the representation space may also contribute to maintaining accuracy. On the other hand, the accuracy
+dropped drastically when the number of samples was 50. This may be because of an increase in the number of trials in
+which the number of positive data (the number of data satisfying the query) in the sample is very small.
+3.6
+Experiment 6: Use of the user vector
+We test whether the estimation accuracy of the other-world model can be improved by using the user vector defined
+by Eq. (5), as opposed to using only CAV. In this verification, two types of prior distributions are defined for the
+y-coordinate of the door (Fig. 9), and the user vector is calculated for each of them. The user vector is treated as one
+of the query vectors in the evaluation value calculation for each environment. Note that λ = 0 is assumed in this
+verification because there is no assumption that the world models held by agents A and B are similar8.
+The estimation results of the other-world model for the same reference environment and query are shown in Fig. 10.
+User vectors 1 and 2 correspond to prior distributions 1 and 2, respectively. It can be seen that while the results of the
+inference of the door location are unstable when only the query vector is applied, the results of the inference using the
+other vectors show that the door coordinates are concentrated in locations that have high probability values in the prior
+distribution.
+The same validation as in experiment 3 was conducted by applying the prior distribution shown in Fig. 9, and the
+number of queries required for both distributions 1 and 2 was lower when using the user vector (Tab. 5). The results
+8If this experiment is conducted under the assumption that the world models of agents A and B are similar, it is necessary to set
+the number of objects that determine the state transition structure of the environment to three or more and that they are placed at the
+same coordinates as the current environment. Under these conditions, it is desirable to set λ = 0.05.
+10
+
+Fo
+-Preprint
+Why don’t you 
+pick up the key
+at the state?
+Base environment
+Query vector
+Query vector
++User vector 1
+Query vector
++User vector 2
+Highest 
+Score
+Lower
+Figure 10: User vector and environments.
+Table 5: Change in estimation accuracy when using user vectors. The number of trials is 100 for each.
+Distribution 1
+Distribution 2
+Method
+Number of updates
+Standard deviation
+Number of updates
+Standard deviation
+Query + User
+5.31
+4.59
+4.69
+3.34
+Query
+6.39
+7.13
+5.11
+3.46
+of the corresponding two-tailed t test showed that t(99) = 2.34 and p < .05 for distribution 1 and t(99) = 1.45 and
+p = 0.15 for distribution 2. These results suggest that the number of queries required for estimation can be reduced by
+using the user vector, although the size of the effect depends on the shape of the prior distribution.
+3.7
+Experiment 7: Explanation by language
+We test whether the language vectors learned in advance can be used to correctly output language that describes the
+relationship between the agent’s world model and user’s world model. The language vectors used in this study are eight
+different explanations, such as “In the world model assumed by the user, {key, door} is located at {upper, lower, right,
+left} than in the world model maintained by the agent.” In the experiment, language explanations were first given to n
+pairs of world models with different coordinates of keys or doors, and the language vectors were obtained using Eq. (7).
+We then generated linguistic explanations for randomly selected pairs of world models using the same conditions as
+those used to generate the language vectors and evaluated the percentage of the explanations that correctly explained
+the relationships between the world models (i.e., the percentage of correct responses).
+The percentage of correct responses after 100 trials is shown in Table 6. Note that if more than one linguistic explanation
+correctly represented the relationship between the world models, both were considered as correct answers. For example,
+if the key is located on the upper right, “the key is on the right” and “the key is on the top” are treated as correct
+answers. The “1st” in the table indicates the percentage of correct explanations generated by Eq. (8). The “1st and 2nd”
+represents the percentage of correctness of the two languages that were the first and second most similar. Note that “1st
+and 2nd” was evaluated only in cases where more than one language explanation was considered to be a correct answer.
+Although the number of world model pairs considered in this experiment is approximately 9000, the experimental
+results show that language explanation generation is possible with high accuracy even when the number of data used
+for language vector acquisition is n = 1000. The accuracy was also maintained even when the number of data was
+extremely reduced to n = 100 and n = 50, suggesting that the learning in the representation space is effective. In this
+experiment, only eight language vectors were set, but it is expected that more accurate explanation generation will
+become possible by setting more detailed language vectors. However, it should be noted that in these cases, sufficient
+teacher data is required.
+11
+
+-------Fo
+-Preprint
+Table 6: Accuracy of language description generation.
+n
+1st
+1st and 2nd
+5000
+0.89
+0.67
+3000
+0.89
+0.73
+1000
+0.88
+0.68
+500
+0.84
+0.63
+300
+0.80
+0.57
+100
+0.69
+0.54
+50
+0.60
+0.37
+4
+Conclusion
+In this study, a novel method was proposed for estimating the user’s world model from the robot’s world model and the
+query given by the user to obtain XAR. The proposed method can estimate others’ world models more efficiently than
+using the “AND” search of queries. In the future, user vectors should be introduced, and the methods for generating
+explanations using differences in world models should be devised.
+Acknowledgments
+This study was supported by the New Energy and Industrial Technology Development Organization (NEDO).
+References
+[1] Sakai, Tatsuya and Nagai, Takayuki, "Explainable Autonomous Robots: A Survey and Perspective," Advanced
+Robotics, 36(5-6), pp.219-238, 2022.
+[2] Ha, David and Schmidhuber, Jürgen, "Recurrent World Models Facilitate Policy Evolution," In Advances in
+Neural Information Processing Systems 31, pp.2450-2462, 2018.
+[3] Xiaofeng Gao, Ran Gong, Yizhou Zhao, et al. "Joint Mind Modeling for Explanation Generation in Complex
+Human-Robot Collaborative Tasks," In Proceedings of 29th IEEE International Conference on Robot and Human
+Interactive Communication (RO-MAN), pp.1119-1126, 2020.
+[4] A. S. Clair and M. Matari ´c, "How Robot Verbal Feedback Can Improve Team Performance in Human-Robot Task
+Collaborations," In Proceedings of 10th ACM/IEEE International Conference on Human-Robot Interaction (HRI),
+pp.213-220, 2015.
+[5] Sandy H. Huang, David Held, Pieter Abbeel, et al. "Enabling Robots to Communicate Their Objectives," ArXiv,
+abs/1702.03465, 2017.
+[6] Isaac Lage, Daphna Lifschitz, Finale Doshi-Velez, et al. "Toward Robust Policy Summarization," In Proceedings of
+the 18th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ’19, pp.2081-2083,
+2019.
+[7] Lunjun Zhang, Gengcong Yang, and Bradly C. Stadie, "World Model as a Graph: Learning Latent Landmarks for
+Planning," ArXiv, abs/2011.12491, 2020.
+[8] A. Narayanan, Mahinthan Chandramohan, R. Venkatesan, et al. "graph2vec: Learning distributed representations
+of graphs," ArXiv, abs/1707.05005, 2017.
+[9] Quoc Le and Tomas Mikolov, "Distributed Representations of Sentences and Documents," In Proceedings of the
+31st International Conference on Machine Learning, volume 32 of Proceedings of Machine Learning Research,
+pp.1188-1196, 2014.
+[10] Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, et al. "Weisfeiler-Lehman Graph Kernels," Journal
+of Machine Learning Research, 12(77), pp.2539–2561, 2011.
+[11] Been Kim, M. Wattenberg, J. Gilmer, et al. "Interpretability Beyond Feature Attribution: Quantitative Testing
+with Concept Activation Vectors (TCAV)," In ICML, 2018.
+[12] Maxime Chevalier-Boisvert, Lucas Willems, and Suman Pal, "Minimalistic Gridworld Environment for OpenAI
+Gym," 2018.
+12
+

ENE2T4oBgHgl3EQfSgdx/content/tmp_files/load_file.txt

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+page_content='ESTIMATION OF USER’S WORLD MODEL USING GRAPH2VEC ∗  Tatsuya Sakai,  Takayuki Nagai  Graduate School of Engineering Science, Osaka University  1-3, Machikaneyama, Toyonaka, Osaka, Japan  nagai@sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='jp  ABSTRACT  To obtain advanced interaction between autonomous robots and users, robots should be able to  distinguish their state space representations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=', world models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Herein, a novel method was  proposed for estimating the user’s world model based on queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this method, the agent learns  the distributed representation of world models using graph2vec and generates concept activation  vectors that represent the meaning of queries in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Experimental results revealed that  the proposed method can estimate the user’s world model more efficiently than the simple method of  using the “AND” search of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Keywords Autonomous robot · Explainability · Representation learning · User’s world model  1  Introduction  Autonomous robots are increasingly being used in numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Currently, they assist humans in performing  tasks by executing commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' For autonomous robots performing sophisticated decisions, the blind execution of  commands is not always the best strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Moreover, in many situations, fully executing commands is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' These  autonomous robots should be able to explain the reasons for their decisions to gain user trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Explainable autonomous  robots (XAR) are defined as robots that have these explanatory capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The following four requirements have been  identified for the XARs [1]:  (1) Owning an interpretable decision-making space  (2) Estimation of the model of others  (3) Estimation of the information required for a user to estimate the policy of the robot  (4) Presentation to the user of explanation  This explanation mechanism is a mutual process between the robot and the user and is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The world  model refers to the correspondence between actions and state changes, that is, the internal model [2] that represents the  dynamics of the environment, and we do not distinguish between the world model and the environment in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  “Policy sharing” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1 is a spontaneous presentation of information of a policy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=', presentation of a sequence of  actions to be taken), and an explanation is generated when a query to this information presentation is requested by  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Among the four requirements, the estimation of the other’s world model is particularly crucial for providing a user-  specific explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In the context of human–robot interaction, the importance of estimating the user’s internal state  has already been recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' [3] and Clair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' [4] proposed a framework for estimating plausible action  strategies based on the actions and interaction history of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' [5] and Lage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' [6] focused on restoring  explanations to policies and advocated the importance of appropriately estimating the restoration algorithm of users  requesting explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' These studies have focused on internal states, particularly policies, and planning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  ∗This paper is an extended (English translated) version of “T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='Sakai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='Horii, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='Nagai, Representation Learning of World Models  and Estimation of World Model of Others Using Graph2vec, Journal of RSJ, 40, 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='166-169, 2022,” (in Japanese) ISSN 1884-7145,  Print ISSN 0289-1824, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='7210/jrsj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='166  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='03793v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='RO]  10 Jan 2023  Preprint  Figure 1: Explanation process as communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' To clarify elements of the explanation, observations/actions, and  interactions between world models of others are segregated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Figure 2: Simplified explanation process considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' We only consider unidirectional explanation process  from robot to user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  However, real-world robots are designed to exhibit desired behavior in terms of algorithms and policies, and they  share the same final objective with their users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In these situations, the results of action decisions typically stem from  discrepancies in environmental awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  In this study, a novel method was proposed for estimating the user’s world model based on the robot’s world model  and the questions (queries) posed by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The proposed method can identify differences in the world models and  generate an explanation that resolves the discrepancy between the perception of the environment by the robot and the  user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this study, we simplified the mechanism of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='2 2 and focused on estimating the world model of the  explained person as the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  2  Proposed method  In the proposed method, the world model of the user is estimated by the following procedure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (1) Acquisition of a distributed representation of the world model: A distributed representation of each world  model is obtained using a graph-structured world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (2) Acquisition of the query vector: Based on the query given by the user, the system acquires a direction vector  that represents the meaning of the query in the representation space of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (3) Estimation of user’s world model: The distributed representations of the world model and query vector are  used to estimate the user’s world model based on cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  2When the model of others outputs an explanation, the content of the explanation is determined by considering information  received from the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  2  Preprint  Why donʼt you do  Agentʼs  1  2  3  4  5  6  7  8  9  Userʼs  Figure 3: Schematic of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  …  …  …  =  1  2  ⋮  =  1  2  ⋮  Figure 4: Learning of a distributed representation of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The experience on the environment is used to  obtain graph-based world models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Then, after converting to WL labels, a distributed representation of each environment  is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The robot and user are assumed to share the same state space and measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' only the connection relation between states  is assumed to be unshared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' As a distributed representation of the world model, the parameters of a model representing a  continuous state space, for example, [2], could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' However, presenting the differences of the world model to the  user requires the discretization of the state transition structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' the parameter space of the model does not necessarily  represent the similarity of the state transition structure of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Therefore, the world model is considered to  be a discretized graph for obtaining a distributed representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The generation of explanations is outside the scope of  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  Acquisition of a distributed representation of the world model  The learning process of a distributed representation of the world model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The robot learns a  representation space representing the similarity of environments based on its experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' First, the robot acquires an  undirected graph representing the state transitions of each environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' simultaneously, it acquires policies through  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Specifically, the robot assumes that the states whose transitions were observed during the policy  learning search are adjacent to each other and adds edges3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  After acquiring the undirected graphs of all environments, graph2vec is applied to acquire the distributed representation  of the graph of each environment [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Graph2vec is a method in which doc2vec [9] is used to acquire distributed  representations of graphs, and instead of predicting the occurrence of words, the occurrence of labels that represent  each subgraph is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Labels representing subgraphs are obtained using the Weisfeiler–Lehman (WL) relabeling  process [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this process, the label of the next layer is determined by considering the labels of neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Higher layer labels represent more global information regarding the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3This method is assumed to be used in a discrete state space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' discretization of the state space is required for application to a real  robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' However, this measure is beyond the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' For discretization, the method proposed in [7], wherein the policy  implications of each state is considered, can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3  ------Preprint  Graph2vec allows graphs in which the same subgraphs occur, that is, graphs in environments with similar state transition  structures, to be embedded closer together in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Efficient search based on user queries becomes  possible by acquiring the representation space of the world model in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  When acquiring a world model, we explicitly provide an environment label to indicate the environment wherein the  experience occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' If no environment label is given, a world model is obtained using temporally continuous experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Furthermore, we consider that models with similar distributed representations represent the same environment and  merging multiple models may be effective in obtaining a world model with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='2  Acquisition of the query vector  Based on the query given by the user, the robot acquires a direction vector that represents the meaning of the query in  the representation space of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' [11] focused on the middle layer of the neural network and  generated concept vectors (CAV: concept activation vectors) by calculating the difference in latent representations when  features that satisfy a specific concept and features that do not satisfy the concept are input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this study, this method  was applied to define a query vector vquery as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (1) by considering the difference of distributed representations  between environments that satisfy the query and those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The query assumes the form “action aquery should  be selected in state squery.”  vquery = vpos − vneg,  (1)  where  vpos =  �  i vi · P(aquery|vi, squery)  �  i P(aquery|vi, squery)  ,  vneg =  �  i vi · (1 − P(aquery|vi, squery))  �  i(1 − P(aquery|vi, squery))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (2)  Here, vi represents a distributed representation of the i-th environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' To correspond to a policy in which actions  are selected probabilistically, the probability value of selecting an action is used as the coefficient of each distributed  representation vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' If necessary, the coefficients can be expressed as the binary values of {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Note that if the state  Squery is not included in the undirected graph of the environment i, the CAV is calculated by excluding vi4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='3  Estimation of user’s world model  Using the distributed representation of the world model and the query vector, the user’s world model was estimated  using cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The likelihood of an environment i as an estimated environment is expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  S(vi, vobs, Vq) =  �  j  similarity(vj  query, vi − vobs) − λ · distance(vi, vobs),  (3)  where vobs is a distributed representation of the environment currently observed by the agent, Vq is any number of query  vectors, and vj  query ∈ Vq is the j-th query vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Furthermore, similarity(a, b) and distance(a, b) are functions that  output the cosine similarity [−1, 1] and the distance between vectors a and b in the representation space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  When reasoning in a real environment, the robot and the user observe almost the same environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Therefore, because  their world models are similar, the similarity between the direction of each environment and the direction of the query  vector, as seen from vobs, is used as the estimation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The coefficient λ is a hyperparameter that determines  how much distance between vectors is considered and represents the strength of the assumption that the observed  environment of the robot and the user are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Using the definitions, the world model of others to be estimated is expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Env_est = arg max  i  S(vi, vobs, Vq)  (4)  In this study, we assume that the importance of all queries are equivalent and designed the evaluation function S as the  sum of the similarities for each query vector vj  query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' However, in the real world, the importance of each query may  differ, in which case the similarity should be multiplied by a coefficient ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  4When estimating the user’s world model, the likelihood S(vi, vobs, Vq) is computed for all environments i, including those that  do not contain the state squery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  4  Preprint  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  User vectors  In addition to the query vector, the user vector that represents “what kind of environment with the distributed repre-  sentation the user is likely to retain as a world model” can also be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The user vector is a vector that represents  how the user tends to misunderstand the environment and plays a role in adding this tendency to the result of the user’s  world model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  vuser = vu_pos − vu_neg,  (5)  where  vu_pos =  �  i vi · P(vi)  �  i P(vi)  ,  vu_neg =  �  i vi · (1 − P(vi))  �  i(1 − P(vi))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (6)  P(vi) is the probability that the user estimates the environment corresponding to the distributed representation vi as  the current world model when no information about the current environment is given to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this paper, P(vi) is  assumed to be known, and the estimation of P(vi) is outside the scope of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='4  Explanation by language  Using a pre-trained language vector Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (7) in the representation space, the relationship between the world model  maintained by the agent and the user’s world model can also be explained by language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  vword(x) = average(vm − vn),  (7)  where vm and vn are distributed representations of environment pairs whose relation is represented by the language  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' By averaging the differences of distributed representations of environment pairs vm and vn that satisfy a specific  language description x, we can obtain a semantic vector represented by the language description x in the representation  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Using the language vector vword, the language describing the relationship between the world model maintained by the  agent and the user’s world model is represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Explanation = arg max  x  similarity(vword(x), vEnv_est − vobs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (8)  By means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (8), the language x that represents the closest relationship between the current world models is selected  as the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3  Experiment  The proposed method was applied to an agent that acquires action strategies by proximal policy optimization (PPO) in a  simulation environment, and its usefulness was evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' A partially modified version of the grid environment [12]  with multiple objects was used for the experiments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this environment, the agent (triangle) obtains a reward by  taking a key, opening a door, and reaching a goal in the lower right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The position of the goal remains unchanged,  but the positions of the key and the door change every trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The agent has five actions, namely go straight, turn left,  turn right, take the key from the grid in front of it, and open the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The agent observes the absolute position of the  key (x, y coordinates), the absolute position of the door (x, y coordinates), its own absolute position (x, y coordinates)  and orientation, holding/not holding the key, and opening/closing the door, for a total of nine dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  Experiment 1: Acquisition of a distributed representation of the world model  A graph representing the state transitions of each environment was obtained simultaneously with the learning of policies  by PPO, and graph2vec was used to obtain a 16-dimensional distributed representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' An example of the acquired  graph is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' There are three edges that transition from the group of states before key acquisition to the  group of states after key acquisition because keys can be acquired from three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' On the other hand, there is  only one edge that transitions from the group of states where the door is not open to the group of states where the door  is open, because the door can be opened from only one state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  5  Preprint  Why donʼt you pick up  Figure 5: Estimation results of the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Table 1: Cumulative frequency and average value of the order in which the optimal environment appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The number  of trials is 100 for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Method  Order  1  2  3  Order Average  Our method  35  49  60  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  Random  6  8  11  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  The representation space was compressed to eight dimensions using independent component analysis, and the visualized  results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this experiment, the absolute positions of the key and door were used as environment  labels, and the five-dimensional observations excluding them were used as node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The experimental results  revealed that clusters are formed in the representation space based on the absolute positions of the key and door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In  particular, clusters related to the position of the door are apparent, which can be attributed to the fact that the surrounding  state transition relationship changes considerably compared with that of the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this experiment, the absolute  positions of keys and doors are used only for identifying the environmental graph to be updated and are not embedded  in the graph itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Therefore, graph2vec created a representation space that appropriately reflects the differences in the  location information of keys and doors expressed in the state transition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='2  Experiment 2: Estimation of the user’s world model  The query vector was applied to the obtained representation space to estimate the world model of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this  verification, we assume that questions were asked in the situations of acquiring a key and opening a door, and the query  “In the state Squery, we should {take the key/open the door}” was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Figure 5 displays the results of estimating  the others’ world model given the reference world model and query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The environment that satisfies the query has the  highest evaluation value, and the environment in which the key is located on the opposite side of the grid environment  has the lowest evaluation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' This result suggests that the obtained query vector is appropriate for estimating the  world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The optimal environment is defined as the agent’s world model with minimal modifications for satisfying the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' For  example, given the query “In state Squery, the agent should take the key,” the optimal environment is the environment  in which only the position of the key is changed to satisfy the query, whereas the position of the door is left unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  For each randomly selected agent world model/query pair, the evaluation values for each environment obtained using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (3) were sorted in descending order to obtain the order of appearance of the optimal environment (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The  order of appearance of the environments that satisfy the query when sorted randomly is displayed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The  experimental results confirmed that the proposed method ranks the optimal environments higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  6  FoFo  --  ooPreprint  Figure 6: An example graph of the acquired world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The blue nodes represent the state before the key acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The orange nodes represent the state after the key acquisition when the door is not open, and the green nodes represent  the state when the door is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Although the direct manipulation of the positions of keys and doors can change the order of the appearance of the  optimal environment to one, direct manipulation is not always possible in cases in which directly manipulatable  information is not given as a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The proposed method estimates the optimal environment at an early stage, although  the state transition relations to be changed are not explicitly given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' This property of the proposed method is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='3  Experiment 3: Validation with multiple queries  An explanatory agent A and an explained agent B are prepared, and the number of queries required for A to accurately  estimate B’s world model is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' This experiment assumes a situation in which the user is asked to confirm the  correctness of the estimation results of the other’s world model through the presentation of an action sequence, which  improves the estimation accuracy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The outline of the experiment is as follows:  (1) A and B share an initial state (absolute position and orientation of the agent, and the state of the key and door)  and an environment-policy pair that specifies the strategy to be used in specific environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' They have  arbitrary world models with different key and door positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (2) Here, A sets its own world model as the initial value of the other’s world model and presents the optimal  sequence of actions in that model in turn (policy sharing) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (3) B adds the query “In state squery, action aquery should be chosen” when the given action differs from the  optimal action in its measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The existing query is not deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (4) A updates the other’s world model based on the query and presents the action sequences in the updated other’s  world model again in sequence with Squery as the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (5) Repeat (3) and (4) to evaluate the number of environment updates required for A to obtain B’s world model as  the other’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The environment selected once is not selected, and the second or subsequent candidate is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' If the same  environment has not been obtained after all the action sequences are presented, the environment is continuously updated  5Policy sharing in this experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 8) is performed to confirm the estimation results of the other’s world model and differs  from the presentation of information about one’s own policy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  7  Preprint  Figure 7: Results of latent space visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Each data point is illustrated according to (a) X-coordinate of the key,  (b) Y-coordinate of the key, (c) X-coordinate of the door, and (d) Y-coordinate of the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Figure 8: Schematic of experiment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The agent transmits information on its policy to the user and updates the user’s  world model based on queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  without increasing the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this verification, the proposed method was compared with the “AND”  search of queries as a method of updating the world model of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In practice, the following three methods are  compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Proposed method:  Select a plausible environment using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  AND search 1:  Randomly selects an environment from among the environments that satisfy the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  AND search 2:  The environment is randomly selected with the constraint that “the optimal behavior for the policy B  is selected in all states from the initial state until reaching state squery”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Thus, it adds constraints and increases  the information provided compared with the two update methods described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Experimental results revealed that the proposed method can estimate others’ world models with the fewest number  of updates (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The results of the corresponding two-tailed t-test revealed that t(100) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='07 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='01 for  the proposed method and AND search 1, and t(100) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='59 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='01 for the proposed method and AND search 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Thus, both significant differences were confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  8  V  V  M  V  V  V  V  V  会  VI  V  V  V  V  V  V  V  V  V  V  △  W  V  V  V  W  △  V  △  V  V  V  V  V  V  V  V  V  V  V  △  1  口  I  I  V  △△  口  V  口  口  口  I  口  口  I  1  口  1  口口  口  口口  口  口  V  V  V  V  V  V  V  V  VV  V口  0VX  W  V  V  口  口  口  口  口  ■  口  口  口Preprint  Table 2: Number of updates required to estimate the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Method  Number of updates  Standard deviation  Our method  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='53  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='83  AND search 1  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='72  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='43  AND search 2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='69  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='71  Table 3: Comparison with the use of probabilistic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Method  Nuber of updates  Standard deviation  Our method (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='93  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='38  Our method (λ = 0)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='11  Probabilistic value (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='13  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='79  The most common information given directly is “AND search2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' By contrast, the proposed method can reduce the  number of updates by vectorizing queries and utilizing prior knowledge embedded in the representation space as  additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='4  Experiment 4: Comparison with the use of probabilistic evaluation  We compare the proposed method with the case where the probability value of each environment satisfying the query is  used as the evaluation function instead of CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The proposed method uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (3) as the evaluation function, while the  comparison method uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  S(vi, vobs, Vq) =  �  j  P(aquery|vi, squery) − λ · distance(vi, vobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  (9)  The procedure for this experiment is the same as in experiment 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' however, the initial world model is not completely  random, and the coordinates of either the key or the door are assumed to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' This condition replicates the  assumption that the agent’s world model and the user’s world model are similar6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Experimental results showed that the proposed method, which takes into account the distance in the representation  space (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05), was able to estimate the user’s world model with the fewest number of updates (Table3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The results  of the corresponding two-tailed t test showed that the proposed method with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05 compared to λ = 0 showed  t(99) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='59 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='005, while the proposed method with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05 compared to the method using probability values  showed t(99) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='44 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='005, both of which are significantly different from each other7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The proposed method, which takes into account the distance in the representation space, was able to estimate the  environment with a significantly smaller number of updates than the method using the same coordinates for either the  key or the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Compared to the method using probability value as the evaluation function, the proposed method  was able to absorb small errors in probability values, resulting in a significantly smaller number of updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' When the  distance in the representation space corresponds to the similarity of the state transition structure between environments,  as in the present verification, it is effective to use CAV to obtain environments that are perpendicular to the virtual  separation boundary as the user’s world models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='5  Experiment 5: Number of samples and accuracy of CAV  We evaluate the number of queries required to correctly estimate the world model when the number of samples (number  of environments) used to compute the CAV is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The evaluation procedure is the same as in Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this  experiment, the maximum number of samples is 300 because 300 environments are embedded in the representation  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' We also set λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  6Without this assumption, the world model with minimal modification to satisfy the query (the optimal environment) is not  necessarily the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' However, theoretically, when the evaluation value is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (3), other environments  that satisfy the query will have a lower evaluation value compared to the optimal environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Therefore, if the assumption that the  world models of the agent and user are similar cannot be made, it is desirable to use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' However, in a real environment, the  world models of the agent and user are not completely independent, and similarity can be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  7Because the t test was applied twice in this verification, a significant difference was found at the significance level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='01  based on the Bonferroni method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  9  Preprint  Table 4: Relationship between the number of CAV samples and the order of appearance of the optimal environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Each value represents the cumulative frequency, and the number of trials is 100 for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Nuber of samples  Order of appearance  1  2  3  300  40  62  69  250  44  61  64  200  42  51  62  150  40  51  55  100  35  46  54  50  9  18  26  Prior distribution 1  Prior distribution 2  P(door_y = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='4  P(door_y = 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='3  P(door_y = 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='2  P(door_y = 4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  P(door_y = 5) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='0  P(door_y = 6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='0  P(door_y = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='0  P(door_y = 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  P(door_y = 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='4  P(door_y = 4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='4  P(door_y = 5) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='1  P(door_y = 6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='0  Figure 9: The prior distribution used in experiment 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The prior distribution for the y-coordinate of the door (in the  vertical direction) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The experimental results show that the accuracy deteriorates much more slowly up to 100 samples than when the CAV  is generated with 300 samples (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' This suggests that a specific level of estimation accuracy can be maintained  even when the number of samples is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The fact that the user’s world model is estimated taking into account  the distance in the representation space may also contribute to maintaining accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' On the other hand, the accuracy  dropped drastically when the number of samples was 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' This may be because of an increase in the number of trials in  which the number of positive data (the number of data satisfying the query) in the sample is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='6  Experiment 6: Use of the user vector  We test whether the estimation accuracy of the other-world model can be improved by using the user vector defined  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (5), as opposed to using only CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this verification, two types of prior distributions are defined for the  y-coordinate of the door (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 9), and the user vector is calculated for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The user vector is treated as one  of the query vectors in the evaluation value calculation for each environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Note that λ = 0 is assumed in this  verification because there is no assumption that the world models held by agents A and B are similar8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The estimation results of the other-world model for the same reference environment and query are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  User vectors 1 and 2 correspond to prior distributions 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' It can be seen that while the results of the  inference of the door location are unstable when only the query vector is applied, the results of the inference using the  other vectors show that the door coordinates are concentrated in locations that have high probability values in the prior  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The same validation as in experiment 3 was conducted by applying the prior distribution shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 9, and the  number of queries required for both distributions 1 and 2 was lower when using the user vector (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The results  8If this experiment is conducted under the assumption that the world models of agents A and B are similar, it is necessary to set  the number of objects that determine the state transition structure of the environment to three or more and that they are placed at the  same coordinates as the current environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Under these conditions, it is desirable to set λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  10  Fo  Preprint  Why don’t you  pick up the key  at the state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Base environment  Query vector  Query vector  +User vector 1  Query vector  +User vector 2  Highest  Score  Lower  Figure 10: User vector and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Table 5: Change in estimation accuracy when using user vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The number of trials is 100 for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Distribution 1  Distribution 2  Method  Number of updates  Standard deviation  Number of updates  Standard deviation  Query + User  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='59  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='34  Query  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='39  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='46  of the corresponding two-tailed t test showed that t(99) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='34 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='05 for distribution 1 and t(99) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='45 and  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='15 for distribution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' These results suggest that the number of queries required for estimation can be reduced by  using the user vector, although the size of the effect depends on the shape of the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='7  Experiment 7: Explanation by language  We test whether the language vectors learned in advance can be used to correctly output language that describes the  relationship between the agent’s world model and user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The language vectors used in this study are eight  different explanations, such as “In the world model assumed by the user, {key, door} is located at {upper, lower, right,  left} than in the world model maintained by the agent.” In the experiment, language explanations were first given to n  pairs of world models with different coordinates of keys or doors, and the language vectors were obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  We then generated linguistic explanations for randomly selected pairs of world models using the same conditions as  those used to generate the language vectors and evaluated the percentage of the explanations that correctly explained  the relationships between the world models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=', the percentage of correct responses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  The percentage of correct responses after 100 trials is shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Note that if more than one linguistic explanation  correctly represented the relationship between the world models, both were considered as correct answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' For example,  if the key is located on the upper right, “the key is on the right” and “the key is on the top” are treated as correct  answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The “1st” in the table indicates the percentage of correct explanations generated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The “1st and 2nd”  represents the percentage of correctness of the two languages that were the first and second most similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' Note that “1st  and 2nd” was evaluated only in cases where more than one language explanation was considered to be a correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Although the number of world model pairs considered in this experiment is approximately 9000, the experimental  results show that language explanation generation is possible with high accuracy even when the number of data used  for language vector acquisition is n = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The accuracy was also maintained even when the number of data was  extremely reduced to n = 100 and n = 50, suggesting that the learning in the representation space is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In this  experiment, only eight language vectors were set, but it is expected that more accurate explanation generation will  become possible by setting more detailed language vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' However, it should be noted that in these cases, sufficient  teacher data is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  11  -------Fo  Preprint  Table 6: Accuracy of language description generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  n  1st  1st and 2nd  5000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='67  3000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='73  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='68  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='63  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='57  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='54  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='37  4  Conclusion  In this study, a novel method was proposed for estimating the user’s world model from the robot’s world model and the  query given by the user to obtain XAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' The proposed method can estimate others’ world models more efficiently than  using the “AND” search of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content=' In the future, user vectors should be introduced, and the methods for generating  explanations using differences in world models should be devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  Acknowledgments  This study was supported by the New Energy and Industrial Technology Development Organization (NEDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
+page_content='  References  [1] Sakai, Tatsuya and Nagai, Takayuki, "Explainable Autonomous Robots: A Survey and Perspective," Advanced  Robotics, 36(5-6), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'}
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+A Theory of Human-Like Few-Shot Learning
+Zhiying Jiang1, Rui Wang2, Dongbo Bu2, Ming Li1∗
+1David Cheriton School of Computer Science, University of Waterloo,
+200 University Ave W, Waterloo, ON N2L 3G1, Canada
+2Institute of Computing Technology, Chinese Academy of Science, Beijing, China
+∗To whom correspondence should be addressed; E-mail: [email protected]
+We aim to bridge the gap between our common-sense few-sample human learn-
+ing and large-data machine learning. We derive a theory of human-like few-
+shot learning from von-Neuman-Landauer’s principle. Modelling human learn-
+ing is difficult as how people learn varies from one to another. Under com-
+monly accepted definitions, we prove that all human or animal few-shot learn-
+ing, and major models including Free Energy Principle and Bayesian Program
+Learning that model such learning, approximate our theory, under Church-
+Turing thesis. We find that deep generative model like variational autoencoder
+(VAE) can be used to approximate our theory and perform significantly better
+than baseline models including deep neural networks, for image recognition,
+low resource language processing, and character recognition.
+Introduction
+During the past decade, fast progress in deep learning (1) has empowered computer speech
+recognition, image processing, natural language processing, protein folding, game playing and
+many other applications. However, these great progresses fell short when we try to understand
+our own learning mechanism: How to model human learning (2), (3), (4)?
+Species in nature learn quickly to survive. When a dragonfly is hatched, within hours it
+firms up its wings and then flies to catch mosquitoes; a newborn does not need tons of repeated
+examples or transfer learning to identify an apple. Most human or animal learning exhibits a
+mixture of inherited intelligence, few-shot learning without prior knowledge, as well as long
+term many-shot learning. It is interesting to note that these learning programs are encoded in
+our genomes but they are not all the same, even for individuals within the same species. The
+diversity of these learning algorithms is vividly expressed by Spearman’s "g" factor (2).
+Work in progress.
+1
+arXiv:2301.01047v1  [cs.LG]  3 Jan 2023
+
+Unlike data-laden, model-heavy, and energy-hungry deep learning approaches, most human
+learning appear to be simple and easy. Merely scaling up current deep learning approaches may
+not be sufficient for achieving human level intelligence. We miss certain major components
+when modelling human or animal learning.
+Diversity is one of the missing part when modelling human or animal few-shot learning.
+There are eight billion people on earth, each with a unique few-shot learning model (5). Even if
+we just want to model one person, a single person often uses different parameters, features, and
+perhaps different algorithms to deal with different learning tasks. Ideally we want a framework
+that can cover the diversity in human and animal few-shot learning. Facing such a seemingly
+formidable task, traditional thinking in machine learning will only lead us to various traps. To
+avoid such traps we need to go back to the very first principles of physics.
+Specifically, we start from an agreed-upon law in thermodynamics, to formally derive our
+model for few-shot learning, and prove this is the optimal model within our framework in the
+sense that all other models including human ones may be viewed as approximations to our
+framework. We show a deep connection between our framework and the free energy principle (3)
+and the Bayesian Program Learning model (4). By the end of this process, a component of data
+compression during the inference phase of learning emerges as a key component of all few-shot
+learning models.
+First, we formalize our intuitive and commonly accepted concept of human-like few-shot
+learning. For example, our definition below is consistent with what is used in (4), and in the
+same spirit of (3).
+Definition 1. Consider a universe Ω, partitioned into H disjoint concept classes: Ch, h =
+1, 2, . . . , H. Few-shot (k-shot) learning is described as follows:
+1. n elements in or outside Ω are given as unlabelled samples y1, . . . , yn;
+2. There are k labelled examples for each class Ch, for small k;
+3. The learning program, using a computable metric M, few-shot learns Ch, h = 1, 2, ...H,
+if it uses the n unlabelled samples and k labelled samples and minimizes the objective
+function:
+H
+�
+h=1
+|Ch|
+�
+i=1
+M(xi, coreh) | y1, . . . , yn, xi ∈ Ch,
+where coreh = ψ(k samples of Ch) representing a transformed representation of the k
+labelled samples from Ch.
+This definition covers most of our common sense few-shot learning scenarios and other
+studies. In particular, this is used in one-shot learning by (4). As each independent individual,
+we do not all use a same metric, or even similar metric, to few-shot learning. For example,
+MN Hebart et al (6) identified 49 highly reproducible dimensions to 1854 objects to measure
+2
+
+their similarity. Different people can be equipped to better observe some of these dimensional
+features.
+We explain the intuition behind Definition 1 via a simple example. A human toddler may
+have already seen many unlabelled samples of fruits which, for example, contains two classes:
+apples and pears. Then given a new labelled sample from each class, the toddler learns how to
+differentiate between these two fruits. The number of labelled data required for one to classify
+may vary as people have different learning algorithms.
+Current deep learning based approaches for few-shot learning generally depend on 1) many
+auxiliary labelled training samples or task-specific data augmentation for transfer learning or
+meta learning (7); or 2) very large scale self-supervised pre-training (8). These approaches thus
+fall short to model few-shot learning in nature by humans and animals as they can hardly account
+for the diversity in learning algorithms and they either neglect the unsupervised scenario that
+humans are mostly exposed to or use the scale of unlabelled data and training parameters that
+are far beyond creatures need.
+Many attempts have been made to understand human learning through cognitive, biological,
+and behavior sciences. Some studies have established basic principles a human learning model
+should obey. One theory is the two-factor theory of intelligence by Charles Spearman in 1904 (2),
+where the “g” factor is an indicator of the overall cognitive ability, and the “s” factor stands for
+the aptitude that a person possesses in specific areas. As “g” factor is genetically-related (9), it
+indicates the necessity of a learning theory that can account for the diversity in creatures’ learning
+ability. Another theory is the Free Energy Principle by Karl Friston (3) that human (and all
+biological systems) learning tends to minimize the free energy between internal understanding in
+the sense of Bayesian (under internal perceived distribution p) and that of the environmental event
+(under distribution q), measured by KL-divergence (10). In a similar spirit, Lake, Salakhutdinov
+and Tenenbaum (4) proposed a Bayesian program learning (BPL) model, learning a probabilistic
+model for each concept and achieve human-level performance. Two articles by Schmidhuber (11)
+and by Chater and Vitanyi (12) linked simplicity to human cognition and appreciation of arts.
+Instead of exploring a biological basis for few-shot learning, we think it is possible to
+mathematically derive an optimal framework that can unify the above theories. We further
+demonstrate by experiments that our new model indeed works significantly better than other
+classical deep learning neural networks for few-shot learning. As a byproduct of our new model,
+a new concept class of "interestingness" is learned; this class implies where our appreciation
+of art, music, science and games comes from. Extending this observation, some aspects of
+consciousness may be modelled as a set of few-shot learned concepts. Consequently, we
+hypothesize the ability of labelling input data becomes a key step to acquiring some aspects of
+consciousness.
+3
+
+A theory of few-shot learning
+We mathematically derive an optimal few-shot learning model for Definition 1 that is effective
+and is able to cover enormous diversities existed in different species. The task may appear to be
+formidable because of conflicting and seemingly very general goals: each individual is allowed
+to have a different learning model, yet our model has just one program to model everybody;
+we do not yet exactly know the complete underlying biological mechanisms, yet we need to
+implement the right functionality; there are infinite number of models, yet we need to choose
+one that is optimal; we are not really interested in "proposing models" out of blue, yet we wish
+our model to be a mathematical consequence of some basic laws of physics; the model needs to
+be theoretically sound, yet practically useful.
+For simplicity and readability, we begin with one-shot learning, k = 1 in Definition 1. Thus,
+coreh in Definition 1 is just the single labelled sample xh. For larger k, coreh can be some form
+of average of the k samples. As Definition 1 defined, some unlabelled objects are assumed and
+it’s also possible to extend the definition by adding distribution, learnt from either unlabelled or
+labelled data, to Ω. Using metric M that is responsible for k-shot learning of an individual, the
+learning system seeks to minimize the energy function
+H
+�
+h=1
+|Ch|
+�
+i=1
+M(xi, xh|y1, . . . , yn),
+or, assuming H(y1, . . . , yn) is a pre-trained model of y1, . . . , yn, or other labelled samples,
+capturing the distribution.
+H
+�
+h=1
+|Ch|
+�
+i=1
+M(xi, xh|H(y1, . . . , yn)),
+Now the question is, what sort of M should we use? Indeed, this varies from person to person.
+Can we unify all such measures, algorithms and inferences? Let’s go back to the fundamentals.
+Principle 1 (von-Neuman-Landauer Principle). Irreversibly processing 1 bit of information costs
+1kT; reversible computation is free.
+Then for two objects x, y, the minimum energy needed to convert between x and y in our
+brain is:
+EU(x, y) = min{|p| : U(x, p) = y, U(y, p) = x},
+where U is a universal Turing machine or our brain, assuming Church-Turing thesis. Since we
+can prove a theorem showing all Universal Turing machines are equivalent modulo a constant and
+efficiency, we will drop the index U (see (13)). To interpret, E(x, y) is the length of the shortest
+program that reversibly converts between x and y. These bits used in the shortest program p
+when they are erased will cost |p|kT of energy, according to the John von Neuman and Rolf
+Landuaer’s law. This leads us to a fundamental theorem (14):
+4
+
+...
+...
+degree
+degree
+Figure 1: Bipartite Graph
+Theorem 1. E(x, y) = max{K(x|y), K(y|x)} + O(1).
+K(x|y) is the Kolmogorov complexity of x given y, or informally, the length of the shortest
+program that outputs x given input y (details are shown in (13)). As this theorem was proved
+thirty years ago and it is vital in our theory, to help our readers, we will provide an intuitive but
+less formal proof here.
+Proof. By the definition of E(x, y), it follows E(x, y) ≥ K(x|y) and E(x, y) ≥ K(y|x), thus
+we have E(x, y) ≥ max{K(x|y), K(y|x)}.
+To prove the other direction E(x, y) ≤ max{K(x|y), K(y|x)}, we need to construct a
+program p such that p outputs y on input x and p outputs x on input y, and length of p is
+bounded by max{K(x|y), K(y|x)} + O(1).
+Let k1 = K(x|y), and k2 = K(y|x). Without loss of generality, assume k1 ≤ k2. We first
+define a bipartite graph {X, Y, E}, where X, Y = {0, 1}∗, as shown in Figure 1 and E is a finite
+set of edges defined between X and Y as follows:
+E = {{u, v}, u ∈ X, v ∈ Y, K(u|v) ≤ k1, K(v|u) ≤ k2}
+Note that a particular edge (x, y) is in E. If we find edge (x, y), then given x, p can output y,
+and vice versa. So the idea of the proof is to partition E properly so that we can identify (x, y)
+easily. Two edges are disjoint if they do not share nodes on either end. A matching in graph
+theory is a set of disjoint edges in E.
+Claim. E can be partitioned into at most 2k2+2 matchings.
+Proof of Claim. Consider edge (u, v) ∈ E. The degree of a node u ∈ X is bounded by 2k2+1
+because there are at most 2k2+1 different strings v such that K(v|u) ≤ k2, accumulating possible
+strings from i = 1 to i = k2 gives us �i=k2
+i=1 = 2k2+1 − 2. Hence u belongs to at most 2k2+1
+matchings. Similarly, node v ∈ Y belongs to at most 2k1+1 matchings. We just need to put edge
+(u, v) in an unused matching. (End of Proof of Claim)
+Let Mi be the matching that contains edge (x, y) We now construct our program p. p operates
+as follows:
+• Generate Mi following the proof of Claim, i.e. enumerating the matchings. This uses
+information k1, k2, and i. K(i) ≤ k2 + O(1)
+5
+
+• Given x, p uses Mi to output y, and given y, p uses Mi to output x.
+A conditional version of Theorem 1, using information in Definition 1, can be obtained
+E(x, y|y1, . . . , yn) = max{K(x|y, y1, . . . , yn), K(y|x, y1, . . . , yn)}, conditioning on unlabelled
+samples y1, . . . , yn. According to (14), this distance is universal, in the sense that E(x, y) is the
+minimum among any other computable distances:
+Theorem 2. For any computable metric D, there is a constant c, such that for all x, y, E(x, y) ≤
+D(x, y) + c.
+This theorem implies: if D metric finds some similarity between x and y, so will E. Thus,
+the above theorem implies, up to some constant O(H)
+H
+�
+h=1
+|Ch|
+�
+i=1
+E(xi ∈ Ch, coreh|y1, . . . , yn) ≤
+H
+�
+h=1
+|Ch|
+�
+i=1
+M(xi ∈ Ch, coreh|y1, . . . , yn).
+When unlabelled samples y1, . . . , yn plus other irrelevant historical labelled samples are modeled
+by some model H such as a generative model (e.g., VAE), then the above inequality can be
+rewritten as:
+H
+�
+h=1
+|Ch|
+�
+i=1
+E(xi ∈ Ch, coreh|H) ≤
+H
+�
+h=1
+|Ch|
+�
+i=1
+M(xi ∈ Ch, coreh|H).
+(1)
+Thus, E gives optimal metric for few-shot learning algorithm. Other algorithms satisfied
+Definition 1 are the approximation to this optimal solution. 1
+In addition, we show that our theory’s deep connection to two well-established principles
+of learning in neuroscience and psychology. Friston’s Free Energy Principle (FEP) (3), derived
+from Bayesian brain hypothesis (15), states that brain seeks to minimize surprises. Specifically,
+it assumes the brain has its internal state (a.k.a. generative model) that implicitly models the
+environment according to the sensory data. Hidden (latent) variables need to be defined for
+the internal state, which are drawn from prior beliefs. Ideally, these prior knowledge is also
+modelled, which is made possible by hierarchical generative models. The free energy principle
+(FEP) is often interpreted as Bayesian optimization, using the Evidence Lower Bound (ELBO)
+as ELBO = log p(x; θ) − D(q(z)∥p(z|x; θ) optimization function. Here the evidence log p(x; θ)
+is the encoding length of x under probability p, and the Kullback-Leibler divergence term is the
+p-expected encoding length difference. This is half of Theorem 1 and FEP is asymmetric if we
+view it as a distance. However, the symmetry is important to few-shot learning. For example, a
+scarlet king snake may look like a coral snake, but the latter certainly has more deadly features
+the former lacks, one way compression K(ScarletKingSnake|CoralSnake) is not sufficient to
+1Note that E is a metric: it is symmetric, and satisfies triangle inequality
+6
+
+Compressor
+Unlabeled Data
+Distribution
+Test Instance
+Figure 2: Illustration of our framework, dashed line indicates optional component when learning.
+distinguish the two. Despite of the fact H. influnza with genome size 1.8 million and E. coli with
+genome size 5 million they are sister species but E. coli would be much closer to a species with
+zero genome G0 or just a covid-19 genome with this asymmetric measure (K(G0|E.coli) than
+with H. influnza (K(H. influnza|E. coli)). A symmetric interpretation of Friston’s FEP can be
+derived by requiring minimum conversion energy as we show in Theorem 1.
+Different individuals may use different compression algorithms to do data abstraction and
+inference. It can be viewed that these algorithms all approximate E(x, y). Some are more efficient
+than others in different situations. The individuals with better compression algorithms have
+bigger “g” factor. Diversified compression algorithms also guarantee better survival chances of a
+community when facing a pandemic. As compression neural networks are genetically encoded,
+the “g” factor is thus inheritable. This can be seen via Figure 2, compression algorithms vary
+from one to another. The distribution of the data to be learnt is either implicitly or explicitly
+captured by creatures. Those who can better utilize unlabelled data to capture distribution may
+have a more efficient compression algorithm.
+Experimental Results
+Image Experiments
+To approximate our universal few-shot learning model, we use a hierarchical VAE as our
+underlying model H in Inequality 1 to model the unlabelled samples y1, . . . , yn. This hierarchical
+structure coincides with our visual cortex and brain structure (16). According to integrated
+information theory (17), an input y may come from all sensing terminals: vision, hearing,
+smell, taste, sensation. Often, creatures are exposed to an unsupervised environment where
+objects are unknown and unlabelled. Revisiting the negative ELBO, we can see it can be
+interpreted as changing perceptions to minimize discrepancy (minimize KL divergence) or
+changing observations to maximize evidence, in the context of FEP. When the creatures are
+7
+
+exposed to a “tree” and they do not fully realize what it is, the sensory information of the
+objects are internalized with hidden states (inner belief) that can describes how it believes the
+generation process of a “tree”. This process of generation, helps the creatures to identify the
+latent similarities among objects that belong to the same category, without the full awareness.
+This process of "unconsciously" training to generate helps the creatures to better categorize in
+future. When the identity of a “tree” is finally revealed, they can generalize quickly. This explains
+our rationale of using a VAE to process unlabelled samples. Consequently, the Kolmogorov
+complexity terms in Inequality 1 are naturally approximated by a VAE based compressor (18).
+To test the hypothesis, we carry out the experiment on five datasets, MNIST, KMNIST,
+FashionMNIST, STL-10 and CIFAR-10. We first train a hierarchical VAE on unlabelled data
+to learn to generate ˆx that’s as close to x as possible. This corresponds to the time when
+creatures exposed to a environment without knowing the object, implicitly learning the latent
+representation among objects. When the identity of objects are revealed, a VAE based universal
+compressor can be used to identify the new objects. Specifically, after training a hierarchical
+VAE unsupervisedly, we compare the E energy function between a labelled image and a test
+image, as in Definition 1. In our experiment, we use 5 labelled samples per class to test the
+accuracy of classification. The energy function E relies on a compressor to approximate. We thus
+use the bits-back argument to directly use our trained VAE for the compressor in (18). Our result
+shows that using only 5 samples, our method outperforms traditional supervised models like
+SVM, CNN, VGG and Vision Transformer (ViT) on all five datasets. These supervised methods
+are chosen to represent different model complexity with wide range of number of parameters.
+As we can see, when labelled data are scarce, supervised methods are not effective: complex
+models like VGG cannot perform better than SVM and this tendency is more obvious on ViT
+without pre-training. The improvement that our method brings is more obvious on more complex
+datasets like STL-10 and CIFAR-10. Similar result is also obtained in the recent work, across
+different shot settings (19).
+We also compare with using latent representation directly with k-Nearest-Neighbor classifier,
+labelled as “Latent” in the table. The architecture and training procedure for “Latent” method
+is exactly the same to our method — we train on unlabelled data to generate the sample and
+then take the latent representation for classification. We can see using latent representation
+outperforms all supervised methods on four out of five datasets. But the accuracy is still way
+lower than our method, indicating our method can better utilize the generative models.
+Text Experiments
+Our theory is generally applicable, even without pre-training on unlabelled data. Here, we
+demonstrate significant advantages of our approach with a simple compressor gzip over lower
+resource languages.
+Languages with Abundant Resources
+We first test our method on datasets with abundant
+resources. Specifically, we compare with three datasets — AG News, SogouNews and DBpedia.
+8
+
+MNIST
+KMNIST
+FashionMNIST
+STL-10
+CIFAR-10
+SVM
+69.4±2.2
+40.3±3.6
+67.1±2.1
+21.3±2.8
+21.1±1.9
+CNN
+72.4±3.5
+41.2±1.9
+67.4±1.9
+24.8±1.5
+23.4±2.9
+VGG
+69.4±5.7
+36.4±4.7
+62.8±4.1
+20.6±2.0
+22.2±1.6
+ViT (disc)
+58.8±4.6
+35.8±4.1
+61.5±2.2
+24.2±2.5
+22.3±1.8
+Latent
+73.6±3.1
+48.1±3.3
+69.5±3.5
+31.5±3.7
+22.2±1.6
+Ours
+77.6±0.4
+55.4±4.3
+74.1±3.2
+39.6±3.1
+35.3±2.9
+Table 1: 5-shot image classification accuracy on five datasets.
+AG News
+SogouNews
+DBpedia
+fasttext
+27.3±2.1
+54.5±5.3
+47.5±4.1
+Bi-LSTM+Attn
+26.9±2.2
+53.4±4.2
+50.6±4.1
+HAN
+27.4±2.4
+42.5±7.2
+35.0± 1.2
+W2V
+38.8±18.6
+14.4±0.5
+32.5±11.3
+BERT
+80.3±2.6
+22.1±4.1
+96.4±4.1
+Ours
+58.7±4.8
+64.9±6.1
+62.2±2.2
+Table 2: 5-shot text classification accuracy on three datasets.
+Similar to image classification, we compare with both supervised methods, including fasttext (20),
+BiLSTM (21) with attention mechanism (22) and Hierarchical Attention Network (HAN) (23),
+and non-parametric methods that use Word2Vec (W2V) (24) as representation. We also compare
+with pre-trained language models like BERT (25) We use five labelled data for each class (5-shot)
+for all the methods.
+Surprisingly, even without any pre-training and with a simple compressor like gzip, our
+method outperforms all non-pretrained supervised methods and non-parametric methods in
+low data regime. This indicates that compressor serves as an efficient method to capture the
+regularity and our information distance is effective in comparing the similarity based on the
+essential information. When comparing with pre-trained models like BERT, we can see our
+method is significantly higher on SogouNews, a special dataset that includes Pinyin — a phonetic
+romanization of Chinese, which can be viewed as an Out-Of-Distributed (OOD) dataset as it
+uses the same alphabet as english corpus.
+Low-Resource Languages
+Sufficiently pre-trained language models are exceptional few-shot
+learners (8). However, when faced with low resource data or distributions that are significantly
+different from any pre-trained data, those pre-trained language models lose their advantages
+to our method. We compare our method with BERT on four different low-resource language
+datasets - Kinyarwanda, Kirundi, Swahili and Filipino. These datasets are curated
+to have the Latin alphabets, same as english corpus. BERT has performed extremely well as
+9
+
+Kinnews
+Kirnews
+Swahili
+Filipino
+BERT
+24.0±6.0
+38.6±10.0
+39.6±9.6
+40.9±5.8
+mBERT
+22.9±6.6
+32.4±7.1
+55.8±16.9
+46.5±4.8
+Ours
+45.8±6.5
+54.1±5.6
+62.7±7.2
+65.2±4.8
+Table 3: 5-shot text classification accuracy on low-resource datasets
+shown in Table 2 due to pre-training on billions of tokens. However, when facing low-resource
+datasets, BERT perform significantly worse than our method only using gzip as we can see
+in Table 3, no matter using multilingual pre-trained version or the original one. Note that mBERT
+is pre-trained on 104 languages including Swahili and Tagalog (on which Filipino is based
+on). As we can see on Swahili and Filipino, mBERT performs better than BERT, but still
+significantly lower than our method.
+Omniglot one-shot-classification dataset
+Figure 3: Distance between two Bezier curves
+In (4),
+a one-shot learning framework
+Bayesian program learning (BPL) was pro-
+posed. It learns a simple probabilistic model
+for each concept. Taking a negative logarithm
+converts a Bayesian formula to a description
+length paradigm, hence BPL can be viewed
+as one particular approximation to our theory.
+Here we provide another simple approxima-
+tion of our theory for the Omniglot one-shot-
+classification dataset of (4).
+Our system first decompose a given char-
+acter into strokes, then compute E(a, b) be-
+tween characters a and b, using all their possi-
+ble stroke decomposition. We provide how to
+calculate E(a, b) here and details of decompo-
+sition program is given in Appendix A.
+1. Fit a stroke by a Bezier curve;
+2. Ensure the number of points on two curves are same. This algorithm utilize equally split
+method to select certain same number of points on each curve Figure 3;
+3. Ensure the area of the convex hull and the barycenter of the compared characters are the
+same;
+10
+
+0
+-20
+-40
+-60
+-80
+-100-
+0
+20
+40
+60
+80
+1004. Use max Cartesian distance between parallel points on two Bezier curves to approximate
+the minimum encoding distance between two Bezier curves, as shown in Figure 3;
+5. Choose the character with minimum distance.
+This simple implementation achieves 92.25% accuracy 20-way-1-shot on this dataset. The
+point here is to demonstrate various approximations of our theory that work rather than com-
+paring accuracy. At 96.75% (4) or at 92.25% might be two different individuals with different
+compression algorithms.
+Unification
+Our framework can unify other popular deep neural networks for few-shot learning.
+Siamese Network: Siamese network uses twin subnetwork to rank the similarity between
+two inputs in order to learn useful features. M here is often a contrastive loss. This framework
+shows strong performance in one-shot image recognition (26).
+Prototypical Network: Prototypical networks (27) propose to optimize the distance metric
+M directly by learning coreh in representation space. coreh are represented as the mean of
+embedded support samples.
+Bi-Encoder: In the context of natural language processing, one of the dominant structure
+is the Bi-Encoder design with each encoder being a pre-trained language model. For example,
+in information retrieval, Dense Passage Retrieval (DPR), with two encoders encoding query
+and document respectively, has become the new state of the art. To capture semantic similarity,
+sentenceBERT (28) also adopts the bi-encoder design and becoming one of the most prevalent
+methods for semantic textual similarity. M in both cases can either be cosine similarity or
+Euclidean distance between the representation learned through pre-trained models.
+Information Distance based Methods: Hundreds of algorithms were published, before the
+deep learning era, on parameter-free data mining, clustering, anomaly detection, classification
+using information distance E (29–34), with a comprehensive list in (13). Recently (19) have
+discovered using information distance with deep neural networks and leverage the generalizability
+of few-shot image classification. This work shows that with the help of deep generative models,
+unlabelled data can be better utilized for few-shot learning under our framework.
+Conclusion and a discussion on consciousness
+We have defined human-like few-shot learning and derived an optimal form of such few-shot
+learning. Note there is an interesting difference between our theory and classical learning
+theory. In classical learning theory, it is well-known that if we compress training data to a
+smaller consistent description, whether it is a classical Bayesian network or a deep neural
+networks (13,35), we would achieve learning. In this paper, we demonstrate that in the inference
+11
+
+stage, compression is also important, especially when there are not enough labelled data to
+train a small model. On the biological side, compression circuits using predictive coding in
+human cortex has been studied by (36). Experiments have also strongly supported our theory.
+We expect to see more practical systems approximating our theory can be implemented to
+solve commonplace few-shot learning problems when large amounts of labelled data for deep
+learning is lacking. We now wish to explore two consequences of our few-shot learning model,
+to consciousness.
+A binary classifier of interestingness
+Our few-shot learning model has a by-product. We have proved compression is a universal goal
+that few-shot learning algorithms approximate. Thus this implies immediately a (subconscious)
+binary classifier: if something is compressed, then something interesting happens, and attention
+is given. It turns out that this "Interestingness" has been theoretically studied as logical depth first
+proposed by Charles Bennett (13). According to Bennett, a structure is deep if it is superficially
+random but subtly redundant. When few-shot learning happens, significant compression happens,
+and these deep objects gain attention. Such a binary classifier might explain our appreciation
+of arts, music, games, and science, since these all share a common feature of dealing with
+non-trivially compressible objects: whether it is a shorter description of the data that gives rise
+of Newton’s laws (13), or a piece of art or music that itself is compressible or that reminds us of
+something we have experienced before, hence very compressible, we feel we understand it and
+hence appreciate it. Science is nothing but compressing data into simpler descriptions of nature.
+Consciousness and the ability of labelling data
+Do other species have consciousness? It is difficult to answer this question as consciousness is
+not testable. Thomas Nagel (37) made a comment: We will never know if a bat is conscious
+because we are not bats.
+Consider an alternative data-driven approach by asking what a species can do instead of how
+they feel. That is, if we treat some aspects of consciousness as a collection of learned concepts,
+then given a compression network, the ability of acquiring the relevant concepts becomes a matter
+of labelling relevant data. We know learning and consciousness are both located at posterior
+cortex region (38). This is in agreement with some injured patients when they lost consciousness.
+This is also in agreement with “bistable perception” training results with monkeys (39).
+Varieties of consciousness are being pragmatically studied (40). These include: 1) the ability
+of consciously perceive the environment; 2) the ability of evaluating conscious emotions; 3) the
+ability of having a unified conscious experience; 4) the ability of integrating across time as a
+continuous stream, one moment flowing into the next; 5) the conscious awareness of oneself
+as distinct from the world outside. Many of these abilities may be seen as a few-shot learnable
+concepts, given properly labelled data.
+12
+
+Different animals have various levels of some of such consciousness by passing certain tests.
+For example, chimpanzees, dolphins, Asian elephants, and magpies can recognize themselves
+by passing some mirror-mark tests. The corvids display some emotions, and are able to plan
+ahead. Octopus have powerful perceptual facilities obtaining and processing data independently
+with each tentacle. Experimentally, awareness emerges when information travels back and forth
+between brain areas (41) instead of a linear chain of command.
+According to our theory, the brain really only needs to use a universal compressor to compress
+information, regardless of one processor in the head or a few processors in the tentacle (in case
+of Cephalopods). Thus we can conjecture that "consciousness” then is a matter of ability of
+labelling the data from sensory terminals. Food or enemy in the environment are easy to label.
+Emotional labelling requires some level of abstraction. Self-awareness of “me” and “others”
+thus is just another binary classifier trainable depending on if the species is able to do “displaced
+reference” mental labelling. Other than the human beings, only orangutans are known to have
+limited displaced reference ability (42).
+Thus we have just reduced the non-testable question of whether an animal has consciousness
+in some aspects to if it is able to label the corresponding data properly.
+Acknowledgement
+We thank Dr. Hang Li for suggestions and bringing (43) to our attention and Dr. Amy Sun for
+bringing (44) to our attention. The work is supported in part by Canada’s NSERC operating grant
+OGP0046506, Canada Research Chair Program, and the Leading Innovative and Entrepreneur
+teams program of Zhejiang, number 2019R02002, and NSFC grant 61832019.
+References and Notes
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+Algorithm for extracting strokes from a character
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+(1) Extract its skeleton so that the stroke width is 1 pixel point. Then convert the image to
+a graph and shrink adjacent cross points. (2) Randomly select an endpoint as starting point,
+endpoint at top left has a greater chance of being selected. Walk until a cross point or endpoint.
+If there is a circle then select a cross point of a top left point if there is no cross point. Record
+this stroke and mark it on the character. Allow small number of marked pixel points to make
+the decomposition more natural. (3) When meeting a cross point, then enumerate two situations
+of pen-up and turning, randomly. Pen-up means end of a stroke, go to step (2) with the marked
+graph. Turning means continuation hence repeat step (2). If walking to an endpoint, then attempt
+to turn by going back to find a new unmarked pixels within some small number of pixels or
+directly end the stroke and repeat step (2) with marked graph.
+16
+

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+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023
+1
+Coarse-to-fine Hybrid 3D Mapping System with
+Co-calibrated Omnidirectional Camera and
+Non-repetitive LiDAR
+Ziliang Miao1, Buwei He1, Wenya Xie1, Wenquan Zhao1, Xiao Huang1, Jian Bai2, and Xiaoping Hong1
+Abstract—This paper presents a novel 3D mapping robot with
+an omnidirectional field-of-view (FoV) sensor suite composed of a
+non-repetitive LiDAR and an omnidirectional camera. Thanks to
+the non-repetitive scanning nature of the LiDAR, an automatic
+targetless co-calibration method is proposed to simultaneously
+calibrate the intrinsic parameters for the omnidirectional camera
+and the extrinsic parameters for the camera and LiDAR, which
+is crucial for the required step in bringing color and texture
+information to the point clouds in surveying and mapping
+tasks. Comparisons and analyses are made to target-based
+intrinsic calibration and mutual information (MI)-based extrinsic
+calibration, respectively. With this co-calibrated sensor suite,
+the hybrid mapping robot integrates both the odometry-based
+mapping mode and stationary mapping mode. Meanwhile, we
+proposed a new workflow to achieve coarse-to-fine mapping,
+including efficient and coarse mapping in a global environment
+with odometry-based mapping mode; planning for viewpoints in
+the region-of-interest (ROI) based on the coarse map (relies on the
+previous work [1]); navigating to each viewpoint and performing
+finer and more precise stationary scanning and mapping of the
+ROI. The fine map is stitched with the global coarse map,
+which provides a more efficient and precise result than the
+conventional stationary approaches and the emerging odometry-
+based approaches, respectively.
+Index Terms—Mapping, Robotic Systems, Omnidirectional
+Vision, Calibration and Identification, SLAM.
+I. INTRODUCTION
+T
+HREE-DIMENSIONAL scanning (obtain the raw points)
+and mapping (register or stitch the points into a
+point cloud map) are becoming increasingly important in
+robotics [2], digital construction [3], and virtual reality [4],
+where digitization of the physical 3D space could provide
+tremendous insights in modeling, planning, management,
+optimization, and quality assurance. Photogrammetry has been
+developed to capture the 3D world. However, its application
+has been limited in aviation settings where accurate GPS
+Manuscript received: Nov. 21, 2022; Revised Jan. 19, 2023; Accepted Jan.
+28, 2023.
+This paper was recommended for publication by Editor Javier Civera
+upon
+evaluation
+of
+the
+Associate
+Editor
+and
+Reviewers’
+comments.
+This work was supported by Shenzhen Science and Technology Project
+(JSGG20211029095803004,
+JSGG20201103100401004)
+and
+SUSTech
+startup fund. (Ziliang Miao and Buwei He contributed equally to this work;
+Corresponding author: Xiaoping Hong)
+1These authors are with School of System Design and Intelligent
+Manufacturing (SDIM), Southern University of Science and Technology
+(SUSTech),
+China
+[email protected],
+[email protected], [email protected]
+2Jian Bai is with State Key Laboratory of Modern Optical Instrumentation,
+Zhejiang University, China
+Digital Object Identifier (DOI): see top of this page.
+RTK signals are required. Recently, the need for large-scale
+mapping of building environments has been rising, mainly
+due to the requirements from Building Information Modeling
+(BIM) systems. Thanks to the availability of emerging 3D
+robotic LiDAR sensors [5], [6], Mobile Laser Scanner (MLS)
+systems are increasingly adopted [7] (Fig. 1a, #3 and #4),
+where point clouds from these sensors could be registered
+to the global frame through sensor motion estimation (i.e.,
+odometry) at each instance. However, due to the movement
+nature, such approaches largely depend on estimations of
+temporal characteristics such as translation and rotation, or
+spatial characteristics such as sensor FoV and landmark
+coverages. The results vary from scan to scan with no
+guarantee of precision. Hence, a more robust and precise
+method is desired.
+On
+the
+other
+hand,
+the
+traditional
+Terrestrial
+Laser
+Scanner (TLS) has been employed in many precision-stringent
+applications (Fig. 1a, #1 and #2). The TLS-based stationary
+mapping is usually inefficient (due to the accurate but slow
+laser rotation) but could provide precise results. Viewpoints
+(also known as stationary scanning locations) need to be
+carefully planned to ensure the spatial coverage and enough
+overlapping regions of adjacent viewpoints to make accurate
+point cloud stitching [8], but on the other hand, as fewer as
+possible to reduce scanning time and cost. The planning for
+viewpoints largely relies on the overall layout of the scene,
+which has been done by human experience so far [9].
+#1
+#2
+#3
+#4
+(a)
+Omnidirectional
+camera
+Livox Mid-360 LiDAR
+(with integrated IMU)
+Gimbal mount
+Mobile platform
+(synchronized)
+(b)
+Fig. 1. 3D mapping systems: (a) the current TLS (#1 FARO Focus Premium,
+#2 LEICA BLK360) and MLS (#3 LEICA BLK2GO, #4 NavVis VLX)
+systems; (b) the proposed hybrid mapping robotic system.
+Combining the strength from both worlds would be ideal in
+large-scale 3D mapping applications. As shown in Fig. 1b,
+arXiv:2301.12934v1  [cs.RO]  30 Jan 2023
+
+OISEE
+SCOUT2
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023
+the proposed hybrid mapping robot is developed carrying
+a gimbal mount and a novel sensor suite consisting of an
+omnidirectional non-repetitive Livox Mid-360 LiDAR1 and
+an omnidirectional camera. The sensors’ FoV and the non-
+repetitive scanning nature are shown in Fig. 2a. In the
+odometry-based mapping mode, the sensor suite is kept
+horizontal by fixing the gimbal mount to coarsely and
+efficiently map the entire space with the mobile platform.
+Based on the coarse map, a few viewpoints are planned for the
+stationary mapping of targeted ROIs. In the stationary mapping
+mode, the robot will navigate and stay still at each viewpoint,
+performing 360°×300° scanning by traversing the vertical FoV
+through the gimbal mount. These precise scans are registered
+with each other and then stitched with the pre-generated coarse
+map forming a global map with fine ROIs.
+The main contributions of this work are as follows:
+1) The
+first
+hybrid
+3D
+mapping
+robot
+system
+that
+integrates odometry-based and stationary mapping modes
+is proposed. The consistency of point clouds in two
+modes can be guaranteed with the single omnidirectional
+non-repetitive Livox Mid-360 LiDAR.
+2) An omnidirectional camera is introduced in the proposed
+system to complement the omnidirectional LiDAR.
+A
+novel
+automatic
+targetless
+co-calibration
+method
+is proposed to simultaneously calibrate the intrinsic
+parameters and the extrinsic parameters.
+3) An automated coarse-to-fine hybrid mapping workflow is
+demonstrated, including odometry-based coarse mapping
+in the global environment, planning for the viewpoints
+in the ROIs, and finer stationary mapping at viewpoints.
+The entire project is open-sourced on GitHub2 to aid the
+development of this emerging field.
+II. RELATED WORKS
+A. Mapping Solutions
+3D mapping solutions are of great interest in many
+emerging fields [3]. TLS-based and MLS-based approaches
+are commonly adopted.
+The traditional TLS-based approach uses a heavy-duty
+single-laser scanner and traverses the entire FoV through
+step-wise rotations about the horizontal and vertical axes.
+It provides sufficiently dense points with good precision.
+However, this method is slow and laborious. It has to be
+repeated on many viewpoints, which need to be chosen wisely
+because a lack of viewpoints will cause missing information
+in the desired ROI, while the excess of viewpoints will lead
+to longer scanning hours and poorer efficiency. Currently,
+viewpoints planning relies on human intuition or experiences,
+making it challenging to plan effectively in large and complex
+working environments like the construction scenes [9].
+On the contrary, the MLS-based approach provides real-
+time scanning and mapping results as the LiDAR moves.
+The current MLS devices are classified by their usage
+configurations, such as handheld (Fig. 1a, #3), backpack
+1The authors gratefully acknowledge Livox Technology for the equipment
+support.
+2https://github.com/ZiliangMiao/Hybrid Mapping Cocalibration.git
+(Fig. 1a, #4), and trolley. Most of these mobile systems rely on
+conventional LiDARs (16, 32, or 64 lines) and construct the
+3D map by registering the point cloud with LiDAR odometry
+or LiDAR-IMU odometry. Such mobile systems greatly speed
+up the mapping process without planning for viewpoints.
+However, it cannot replace the TLS-based approaches due to
+insufficient mapping precision and sparse point clouds [3]. The
+repetitive scanning nature of mechanical LiDAR is unsuitable
+for stationary scanning due to limited FoV coverage (20%
+coverage for 32-line LiDAR). Therefore, the indispensable
+motion for more coverage will cause errors in pose estimation,
+which are accumulated throughout the process, limiting the
+usage in high-precision applications.
+Both TLS-based and MLS-based approaches have their
+unique advantages and drawbacks. It is desired to devise
+a mechanism to combine both modes. For example, a
+combination of TLS and MLS is used to solve the registration
+problem between non-overlapping spaces [8] or use TLS scans
+as references to MLS mapping registration to achieve low
+mapping errors [10]. Moreover, MLS is also used to provide a
+3D map to solve the viewpoints planning problem of TLS [9].
+However, all these methods are based on heterogeneous
+sensors for different modes, with different synchronization,
+data structure, and protocols, which are difficult to construct
+a one-stop mapping robot with a streamlined and automated
+workflow.
+The unique non-repetitive scanning nature of the Livox
+LiDAR provides a combination of an instantaneous high
+density at a short time interval for odometry (with effective
+point density as 32-line LiDAR within 0.1 seconds) and an
+image-level resolution at relatively long time intervals for
+scanning (within 3 seconds, as shown in Fig. 2b), which makes
+it surprisingly suitable for such hybrid working mechanism.
+The
+feature
+provides
+sufficiently
+good
+performance
+in
+odometry scenarios [11] and a dense FoV coverage for image-
+like feature processing [6], [12], [13]. In this paper, the two
+working modes are integrated into the same robot, ensuring
+overall mapping efficiency and precision with an automated
+coarse-to-fine hybrid mapping workflow.
+B. Calibration Methods
+In addition to LiDAR, Cameras are usually required
+in
+3D
+mapping
+systems
+to
+give
+an
+overview
+of
+the
+mapped environment [14]. Cameras could provide high-quality
+geometric, color, and texture information [15], which enables
+further modeling and rendering [16] of the point clouds
+and permits tasks in object detection, segmentation, and
+classification [17]. Meanwhile, for autonomous navigation, the
+camera is also vital to visual-LiDAR odometry through sensor
+fusion [4]. All these functions would rely on the accurate
+calibration of the intrinsic parameters of the camera and
+extrinsic parameters between the cameras and LiDAR [15].
+Traditionally, multiple cameras are usually required to
+be complementary to the omnidirectional FoV of LiDAR.
+This work employs an omnidirectional camera over the
+traditional multi-camera vision to avoid bulky construction,
+high cost, shutter synchronization, and cascaded extrinsic
+
+MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR
+3
+calibrations. The intrinsic and extrinsic parameters of this
+novel omnidirectional sensor suite are essentially needed.
+The intrinsic parameters of the omnidirectional camera
+must be well calibrated since these types usually possess
+much larger and more complex distortions than pin-hole
+cameras [18]. In [18]–[20], higher-order polynomial-based
+intrinsic
+models
+are
+introduced
+with
+many
+degrees
+of
+freedom to obtain satisfactory results. A popular OcamCalib
+toolbox based on the checkerboard is provided [19]. These
+methods could be susceptible to over-fitting with high-order
+polynomials and often require evenly distributed artificial
+targets and dense features across the entire space. Typically,
+these calibration processes are manual and could lead to
+tedious procedures with a large margin of error. Additionally,
+the omnidirectional camera in our work is constructed with
+a refractive-reflective geometry to capture a ring-like FoV
+beyond 180°. This construction makes intrinsic calibration
+even more difficult. An accurate, automatic, and targetless
+calibration method is desired.
+The
+extrinsic
+calibration
+method
+between
+the
+omnidirectional camera and LiDAR has only been explored
+in [21] using edge correspondence to match point clouds
+and images. The bearing angle images highlight the edge
+features, which are manually positioned. Targetless extrinsic
+calibration methods for monocular cameras and LiDAR have
+been developed recently. With the non-repetitive LiDARs,
+CamVox [12] could project the image-like LiDAR point
+clouds onto the camera image plane and extract edge pixels
+using the grayscale images based on reflectivity and depth.
+The method proposed in [13] uses voxels to extract the edge
+points in 3D space and classifies the edges based on depth
+continuity. Both methods work well with conventional pin-hole
+cameras and need to be extended toward the omnidirectional
+cameras with significantly larger distortions. An additional
+targetless extrinsic calibration method employing mutual
+information (MI) is also developed [22], which maximizes
+the intensity correlations of LiDAR and camera. However,
+the misrepresented information caused by lighting conditions,
+surface
+reflection
+properties,
+and
+spectral
+reflectance
+disagreement could result in worse calibration than the
+edge-based methods.
+In the proposed targetless co-calibration method, the high-
+resolution dense point cloud of the non-repetitive scanning
+LiDAR gives abundant and ground-truth-level features, which
+eliminates the artificial targets and manual involvement and
+reduces the error caused by insufficient coverage and sparse
+features of the targets. With the co-calibration method, the
+intrinsic and extrinsic parameters are obtained simultaneously
+and can be re-calibrated fast and reliably in work scenes.
+III. PROPOSED SYSTEM
+A. Co-calibrated Omnidirectional Sensor Suite
+The Livox Mid-360 LiDAR has a 360° × 55° FoV and
+features a non-repetitive scanning pattern, with increasingly
+denser points over time (the coverage of FoV approaches
+100%), as shown in Fig. 2b. The unique feature specifically
+benefits both odometry-based and stationary mapping modes.
+The omnidirectional camera provides color information of
+the surroundings and has a corresponding 360° × 70° FoV
+(Fig. 2a). Both sensors are synchronized and are mounted
+on a two-axis gimbal (Fig. 1) to extend the scanning FoV
+to 360° × 300°.
+-7°
+-10°
++60°
++52°
+-10°
++60°
+-7°
++52°
+Omnidirectional Camera
+Livox Mid-360 LiDAR
+(a)
+T = 0.1s
+T = 0.5s
+T = 3.0s
+(b)
+Fig. 2. Configuration of the sensors: (a) omnidirectional camera and Livox
+Mid-360 LiDAR, both on the gimbal mount; (b) point cloud accumulation
+over time due to the non-repetitive scanning nature of the Livox LiDAR.
+（Color represents reflectivity of LiDAR points)
+#1
+#2
+#3
+...
+0
+5.1E-3
+Probability density
+0
+5.1E-3
+Probability density
+Fig. 3. Proposed co-calibration process. * The grayscale value indicates the
+average reflectivity of the projected LiDAR points within a pixel.
+The co-calibration simultaneously obtains the intrinsic
+(camera) and extrinsic (camera-LiDAR) parameters, defined
+respectively as Θ ≜ [u0, v0, c, d, e, a0, . . . , an]T and ∆ ≜
+[α, β, γ, tx, ty, tz]T, which will be introduced later. With
+
+HO
+DJP2002NOOEORXDHZDZDIHZp)120,△=argma
+0,1
+ax
+nn
+Cf(
+-14
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023
+the unique benefit of the non-repetitive scanning LiDAR,
+an extremely dense point cloud is always available, which
+provides a 3D ground truth of the environment. This high-
+resolution point cloud could be projected onto the 2D image
+plane with pixel values from LiDAR reflectivity, from which
+clear edge features could be extracted. To align the edges
+from LiDAR and the camera, the co-calibration iteratively
+maximizes the correspondence of projected LiDAR edge
+points with the omnidirectional camera edge pixels. Kernel
+Density Estimation (KDE) is employed to estimate the camera
+edge distribution with different distribution smoothness (by
+varying bandwidth coefficient) to obtain global optimum.
+The entire process of co-calibration can be divided into the
+following two steps (Fig. 3):
+1) Edge Extraction: Edge extractions are performed for
+both camera and LiDAR. For the camera, exposure fusion [23]
+is adopted to enhance the dynamic range of images to capture
+more details for low and high-brightness objects. Canny edge
+extraction [24] is performed on the enhanced image, with
+edge points Q = [q1, q2 . . . , qn]. For LiDAR, since the
+FoV is smaller, point clouds scanned from different pitch
+angles are stitched together. The stitching is performed by the
+generalized iterative closest point (GICP) algorithm [25] with
+the initial transformation given by the state of the gimbal.
+The stitched point cloud with reflectivity is then projected
+to an image plane with the azimuthal angle and elevation
+angle as the coordinates, generating a grayscale image by
+taking the average reflectivity of the projected LiDAR points
+within each pixel. The Canny edge extraction is performed
+on this grayscale image. Uniform sampling is performed in
+each stage to remove the non-uniform point distribution. The
+edge pixels are then identified in the original 3D point cloud
+P = [LP1, LP2 . . . , LPm].
+2) Iterative Optimization:
+The iterative optimization is
+performed in the omnidirectional image space. The LiDAR
+edge points are projected to the image coordinates through
+the following equations:
+CP = C
+LT(LP; ∆) = C
+LR · LP + C
+Lt, LP ∈ P,
+(1)
+p = Π(CP; Θ) =
+�c
+d
+e
+1
+� �r cos φ − u0
+r sin φ − v0
+�
+,
+(2)
+r = F(θ; a0, . . . , an) = a0 + a1θ1 + . . . + anθn,
+(3)
+θ = arccos(
+z
+�
+x2 + y2 + z2 ),
+(4)
+φ = arccos(
+x
+�
+x2 + y2 ),
+(5)
+where CP and LP denote the 3D point coordinates in camera
+and LiDAR coordinate systems, respectively, and they are
+related through the extrinsic transformation C
+LT(LP; ∆), i.e.,
+rotation C
+LR and translation C
+Lt with the extrinsic parameters
+∆. The symbol p denotes the location of the point in the
+camera image space, and Π(CP; Θ) expresses the intrinsic
+transformation from
+CP
+=
+[x, y, z]T (3D point) to p
+(2D point), with the distortion correction matrix
+�
+c
+d
+e
+1
+�
+.
+The pixel radius r from the image center [u0, v0]T is
+transformed from the elevation angles θ by a polynomial
+function F(θ; a0, . . . , an) in the camera model; θ and φ are the
+elevation and azimuth angle of CP (Note the omnidirectional
+camera features a ring image).
+To facilitate the alignment between the camera edges
+and the LiDAR edges, the camera edge distribution with
+nonparametric probability density function is constructed
+with the Gaussian Kernel by Kernel Density Estimation
+(KDE) [26]. The optimization is based on maximizing the
+probabilities of the projected LiDAR edge points onto the
+camera edge distribution:
+ˆΘ, ˆ∆ = arg max
+Θ, ∆
+1
+n
+m
+�
+i=1
+|| ˆf(pi; h, Q)||2,
+(6)
+ˆf(pi; h, Q) = 1
+nh
+n
+�
+j=1
+K
+�pi − qj
+h
+�
+,
+(7)
+K(x) =
+1
+√
+2π det(Σ)e− 1
+2 (x−µ)TΣ−1(x−µ),
+(8)
+µ = [0, 0]T, Σ = I2×2,
+(9)
+where h denotes the bandwidth of the KDE.
+Several rounds of iterative optimization with reducing
+bandwidth are carried out to approach the correct calibration
+values smoothly. At the start of the process, the bandwidth
+is set at a large number to get a continuous and smooth
+cost function, which allows the optimization to approach
+the optimal region quickly without many local optima.
+Then the bandwidth is reduced gradually to increase the
+gradient, ensuring a sensitive optimization around the optimum
+(optimization of the x-axis translation is shown in Fig. 4).
+0.20        0.25 
+   0.30 
+0.35     
+Bandwidth=16
+Bandwidth=4
+Bandwidth=1
+1
+0
+Normalized cost
+Translation in the x-axis (m)
+(a)
+0.265         0.275         0.285         0.295            
+Translation in the x-axis (m)
+1
+2
+4
+3
+(b)
+Fig. 4.
+Iterative optimization with the reducing KDE bandwidth: (a) the
+normalized cost w.r.t. the translation in the x-axis under the different values
+of bandwidth; (b) zoom in to a sub-region of (a) to demonstrate the iterative
+process.
+The
+optimization
+uses
+the
+Levenberg-Marquardt
+method
+implemented
+in
+Ceres-solver
+[27].
+For
+computational
+efficiency,
+the
+parabolic
+Epanechnikov
+kernel K(x) =
+3
+4(1 − xTx) can be substituted for the
+Gaussian kernel.
+B. Coarse-to-fine Hybrid Mapping
+The coarse-to-fine hybrid mapping workflow is outlined in
+Fig. 5. With the co-calibration and synchronization, all the
+obtained LiDAR points are represented in both coordinates
+and color. Odometry/SLAM methods are used as a backbone
+to provide localization in both coarse and fine mapping. We
+used FAST-LIO (LiDAR-Inertial odometry [11]) in our current
+
+MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR
+5
+Fig.
+5.
+Proposed
+coarse-to-fine
+hybrid
+mapping
+workflow.
+The
+odometry/SLAM serves as a backbone to provide localization results.
+system, but the choice is not limited; other odometry/SLAM
+methods could be utilized as well. At the coarse mapping
+stage, the robot obtains the localization and motion results
+from the odometry, from which the scanned points are
+converted and registered to the global map. Based on the
+coarse map, a few viewpoints for stationary mapping are
+planned for the targeted ROIs, which is well developed in
+previous work by considering the constraints such as range,
+grazing angle, FoV, and overlap [1]. The robot then navigates
+to the generated viewpoints one-by-one through the backbone
+odometry/SLAM and performs the fine mapping, respectively.
+At each viewpoint, stationary scans are performed at several
+gimbal states, with overlapping FoV regions between the
+adjacent two states, and cover a large overall FoV (360° ×
+300°). These point clouds will be pre-registered based on the
+gimbal angles (as initial angles) at each viewpoint. The scans
+from all the viewpoints are then combined with the global
+coarse map based on robot localization (again provided by the
+LiDAR-Inertial odometry) as the initial state for optimization.
+Finally, the GICP [25] algorithm is used to optimize all the
+localization results and gimbal states and refine all stationary
+scans and the coarse map to form the fine map. Notably,
+we could choose either odometry or SLAM methods in
+the localization backbone. Although SLAM has more loop-
+closure functions than odometry, the final GICP optimization
+is accurate enough to yield a much better localization result.
+IV. EXPERIMENTS AND RESULTS
+A. Co-calibration Results
+The effectiveness of the proposed co-calibration method is
+demonstrated in three natural scenes, as shown in Fig. 6. The
+projection error (in pixels) is defined as:
+e = 1
+n
+n
+�
+i=1
+d(pi; Q),
+(10)
+where d is to calculate the distance from the LiDAR projected
+point pi to the nearest point in target set Q. Note that the
+largest 10% of the distances are considered outliers with no
+correspondences and are eliminated. Overall, the co-calibration
+works well in all scenes with projection errors on the order of
+3 pixels or less. The colorized point clouds after co-calibration
+also show much better consistency, as seen in Fig. 6b.
+(a)
+(b)
+Fig. 6. Co-calibration results in three scenes: (a) aligned LiDAR edge points
+(red) on camera images; (b) comparison of colorized point clouds before and
+after co-calibration with the average projection errors in pixels.
+We
+further
+compare
+our
+co-calibration
+results
+with
+the classical target-based intrinsic calibration [19], [28],
+and the state-of-the-art MI-based extrinsic calibration [22],
+respectively, as shown below.
+1) Analysis of the Intrinsic Results: As a comparison, the
+target-based intrinsic calibration for omnidirectional cameras
+is performed [19]. Thirty checkerboards are manually selected
+as a reference set (Fig. 7a). As the number and position
+of the targets affect the calibration profoundly, we evaluate
+the calibration result as a function of the targets’ number
+and randomly select a specific number of checkerboards
+from the reference set for calibration (repeated 100 times
+independently). The mean reprojection error is used to
+represent the calibration accuracy. The results in Fig. 7b
+show that as the number of checkerboards increases, the
+calibration is more accurate and converged. It is likely that
+more checkerboards would increase the FoV coverage and
+feature points density and improve the effectiveness of the
+target-based method. However, it is labor-intensive to place
+many checkerboards uniformly and densely around the sensor
+and manually select the appropriate ones, which may be
+impossible in the field. The co-calibration method, on the
+contrary, employs dense LiDAR points as abundant, well-
+covered, and accurate features; and the elimination of artificial
+targets and human involvement enables an accurate, efficient,
+and field-friendly approach. Our co-calibration result yields
+a significantly improved performance on the same reference
+set, compared with the conventional method (orange and blue
+boxplot in Fig. 7b, respectively).
+2) Analysis
+of
+the
+Extrinsic
+Results:
+The
+mutual
+information
+(MI)-based
+extrinsic
+calibration
+method
+utilizes the fact that the reflectivity of LiDAR points
+and corresponding grayscale intensity values of camera
+pixels are correlated since both of them capture the spectral
+response of the object at light frequencies (LiDAR 905 nm,
+camera 400-800 nm), which are usually similar. These values
+
+Projection Error: 7.15 →3.17
+Projection Error: 7.36 →2.85
+皖A·35
+天国310
+Proiection Error: 7.08 → 2.63Projection Error: 7.15 →3.17
+Projection Error: 7.36 →2.85
+皖A·35
+天国310
+Proiection Error: 7.08 → 2.636
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023
+-0.4
+-0.3
+-0.2
+-0.1
+0.6
+0.5
+0
+X
+0.4
+Z
+0.2
+0
+0 -0.2 -0.4
+-0.5
+Y
+x-axis (m)
+y-axis (m)
+z-axis (m)
+(a)
+Proposed
+0
+10
+20
+30
+Avg.
+14
+15
+16
+13
+12
+11
+10
+9
+8
+7
+6
+5
+Number of checkerboards
+Projection error (px)
+3.19
+5.89
+5.74
+5.93
+6.64
+6.97
+7.86
+7.71
+8.43
+9.28
+9.64
+10.3
+11.9
+40
+(b)
+Fig. 7. Comparison with the target-based intrinsic calibration: (a) the poses
+of the thirty checkerboards; (b) boxplots of projection errors of target-based
+calibration (blue) and the proposed co-calibration (orange).
+are then used to calibrate the extrinsic parameters between
+the camera and LiDAR by maximizing the MI of the two
+distributions [22]. Fig. 8 shows the comparisons of the two
+optimization methods demonstrating the normalized costs on
+different extrinsic parameters. The proposed co-calibration
+method shows a much more sensitive and reliable gradient
+in the cost function near the optimum than the MI-based
+method.
+0
+1
+Normalized cost
+MI-based
+Proposed
+Rotation in the x-axis (rad)
+-0.4 -0.2
+0
+0.2
+0.4
+-0.4
+-0.2
+0
+0.2
+0.4
+Rotation in the y-axis (rad)
+0
+1
+Normalized cost
+MI-based
+Proposed
+MI-based
+Proposed
+Rotation in the z-axis (rad)
+1.2
+1.4
+1.6
+1.8
+2
+0
+1
+Normalized cost
+Translation in the x-axis (m)
+-0.5
+0
+0.5
+1
+0
+1
+Normalized cost
+MI-based
+Proposed
+Translation in the y-axis (m)
+0
+1
+Normalized cost
+-1
+-0.5
+0
+0.5
+1
+MI-based
+Proposed
+0
+1
+Normalized cost
+Translation in the z-axis (m)
+-0.5
+0
+0.5
+1
+MI-based
+Proposed
+Fig. 8. Comparisons of the normalized cost function between the proposed
+method and the MI-based method. The optimal values should lie in the gray
+areas estimated based on manufacturing.
+The inaccurate calibration result of the MI-based method
+could be attributed mainly to three reasons: the lighting
+conditions, the surface reflection properties, and the spectral
+reflectance disagreement. The camera’s light source Ii is the
+external ambient lighting which does not change with the
+camera pose. On the contrary, LiDAR uses an active laser from
+the sensor and therefore differs significantly from the camera,
+as shown in Fig. 10a. Besides the lighting, the surfaces of the
+objects are important. The detected intensity could be modeled
+as follows:
+Ir = Kd · Ii · f(θ),
+(11)
+where Ir and Ii indicate the reflection intensity and incident
+intensity, respectively, Kd
+is the reflectance, and f(θ)
+describes the surface properties of the object with respect to
+incident angle θ. For most objects, the surface is Lambertian
+(diffusive), and in that case, f(θ) = cos θ. However, many
+surfaces do not follow this property, and it could be a specular
+reflection that the LiDAR does not collect any signal; or the
+retroreflection that the majority of the energy will be directed
+back toward the LiDAR itself and gives a strong intensity, such
+as those on traffic signs and warning stickers, which show a
+contrast difference in the LiDAR intensities from the camera
+intensities shown in the red boxes in Fig. 9b. Additionally,
+the spectral reflectance of objects at various light wavelengths
+could be different. For instance, materials composed of plant
+fibers show a large reflectance at around 905 nm, even those
+dyed in black colors. As a result, no contrast could be seen in
+LiDAR intensities of materials with different colors, as shown
+in green boxes in Fig. 9b. All three factors mentioned above
+could cause significant differences in intensity response from
+the LiDAR and the camera and reduce the applicability of the
+MI-based method.
+LiDAR
+Camera
+Ambient Light
+Retro
+A
+B
+A
+B
+Spread
+Lamber�an
+(Lamber�an+Retro)
+(a)
+0
+255
+Intensity and reflectivity
+(in grayscale)
+Camera
+LiDAR
+(b)
+Fig. 9.
+Analysis of the MI-based extrinsic calibration: (a) the types of
+reflection of the LiDAR and camera w.r.t. the rough surface and the
+retroreflective surface; (b) the inconsistent intensity cases between LiDAR and
+camera, including retroreflection cases (red boxes), and the special spectral
+reflectance cases (green boxes).
+B. Coarse-to-fine Hybrid Mapping Results
+The proposed coarse-to-fine hybrid mapping method is
+demonstrated in an academic building on the SUSTech
+campus. The global coarse map is generated by Fast-LIO in
+ten minutes, and the ROI is selected based on this global
+coarse map (Fig. 10a). In this case, five viewpoints are
+properly planned in this ROI (Fig. 10b), and perform stationary
+scanning for three minutes in each (Fig. 10c).
+Plane thickness could be used as a quantitative metric for
+precision evaluation and comparison between coarse and fine
+mapping. Local planes with a small third eigenvalue λ3 are
+selected by diagonalizing the covariance matrix. Assuming
+the points along the plane’s normal direction follow the
+Gaussian distribution (corresponding to the third eigenvalue
+λ3 with the normal direction of the plane defined by its
+eigenvector), we could set the thickness of the plane as 4√λ3.
+
+MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR
+7
+(a)
+(b)
+(c)
+Fig. 10.
+Coarse-to-fine hybrid mapping: (a) odometry-based global coarse
+mapping; (b) coarse map of the selected ROI, with markers indicating the
+planned viewpoints; (c) fine map of the ROI, the color illustrates the scans
+from respective viewpoints.
+TABLE I
+SPECS COMPARISON OF CURRENT MAPPING SYSTEMS
+Proposed
+#1 FARO Focus
+Premium 150
+Type
+Hybrid Mapping
+TLS
+FoV
+360° × 300°
+360° × 300°
+Range
+0.1-40 m
+0.5-150 m
+PPS
+200,000 pts/s
+2,000,000 pts/s
+Precision
+∼ 40 mm (coarse)
+∼ 20 mm (fine)
+∼ 1mm [29]
+Accuracy
+∼ 10 mm (coarse)
+∼ 2 mm (fine)
+∼ 1mm [29]
+Registration
+Odometry+Optimization
+Optimization
+Work Manner
+Mobile Robot
+Manual (tripod)
+Viewpoints Planning
+Coarse map-based
+Intuition-based
+Vision
+1-omni camera
+1-camera
+#2 LEICA
+BLK360
+#3 LEICA
+BLK2GO
+#4 NavVis
+VLX
+TLS
+MLS
+MLS
+360° × 300°
+360° × 270°
+360° × 30°(×2)
+0.5-45 m
+0.5-25 m
+0.9-100 m
+680,000 pts/s
+420,000 pts/s
+300,000 pts/s (×2)
+∼ 20 mm [30]
+∼ 20 mm [30]
+15-50 mm(walls, 80.5%) [31]
+∼ 1 mm [30]
+∼ 30 mm [30]
+15-50 mm(beams, 98.2%) [31]
+Optimization
+Odometry/SLAM
+Odometry/SLAM
+Manual (tripod)
+Manual (handheld)
+Manual (backpack)
+Intuition-based
+No need
+No need
+3-camera
+3-camera
+4-camera
+The coarse and fine maps of the three different scenes are
+shown in Fig. 11a, whereas the zoomed views show the point
+cloud quality with the top view of the selected planes to
+demonstrate the mapping quality. The quantitative evaluations
+of the plane thickness (the mapping precision) in these scenes
+are summarized in Fig. 11b. Besides precision (spread of data),
+accuracy (correctness) is also important to examine. Fig. 11c
+illustrates the measurement accuracy (compared to results
+from a TLS system, which we regard as ground truth). It is
+evident that both the precision and accuracy of fine mapping
+outperform coarse mapping. Although odometry-based coarse
+mapping has good performances in best-case scenarios, it
+could be significantly improved by fine mapping in the average
+values and worse-case scenarios, which are the main concerns
+of the surveying and mapping industry.
+With the accurate co-calibration results, LiDAR points
+(a)
+#3
+#2
+#1
+Scenes
+0
+10
+20
+30
+40
+50
+Mapping precision (mm)
+Coarse Mapping
+Fine Mapping
+60
+(b)
+#3
+#2
+#1
+Scenes
+-20
+0
+20
+40
+60
+80
+Mapping accuracy (mm)
+Coarse Mapping
+Fine Mapping
+100
+(c)
+(d)
+(e)
+Fig. 11. Comparison of coarse and fine mapping: (a) coarse and fine maps
+in three scenes (scene #1 is from Fig. 10b, scene #2 and #3 are new). The
+left column shows the large-scale coarse map, and the right column shows
+the zoomed-in coarse and fine map in top view (to visualize wall thickness)
+and third person view (to visualize scene); (b) mapping precision from the
+three scenes; (c) mapping accuracy from the three scenes; (d) top view of the
+colorized fine map; (e) third-person view of the colorized ROI.
+can be colorized from the image information through the
+transformation in Eqn. 1 and Eqn. 2. Fig. 11d shows the
+colorized hybrid mapping, and Fig. 11e illustrates the fine
+mapping of the zoomed-in ROI. The coarse-to-fine map with
+great precision and accurate colorization pave the way for
+higher precision with a single unified setup and workflow. It
+benefits industries requiring both efficiency and accuracy, such
+as construction automation and building inspection.
+Lastly, a detailed comparison of the proposed system
+with the current widely used TLS and MLS systems
+(shown in Fig. 1a) is made in Table I, where several key
+
+0
+2m
+ROI
+10m
+2m0
+2m
+ROI
+10m
+2m0
+2m
+ROI
+10m
+2mCoarse Mapping
+Fine
+Scene #1
+Top view
+Third-person
+view
+Scene #2
+Top view
+Third-person
+view
+Scene #3MappingTop vie
+Third-person
+vlewX
+取消ROI8
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023
+parameters are listed. The most crucial difference is that the
+proposed system integrates two working modes in a single
+streamlined workflow, ensuring overall mapping efficiency
+and precision/accuracy. All other systems are either TLS
+which only works in stationary mode, or MLS in mobile
+mode. Due to this capability, it is the first robotic system
+that allows automatic viewpoint planning instead of human
+intuition-based viewpoints selection. In addition, the mobile
+robot could navigate itself with overall good localization and
+provide good initial states for fine map optimization. The
+mapping precision and accuracy of the proposed system are
+also compared with these systems [29]–[31]. The proposed
+system achieves performance close to the LEICA TLS but
+allows mobility as MLS, agreeing with the purpose of the
+system.
+V. CONCLUSION
+This paper proposed a coarse-to-fine hybrid 3D mapping
+robotic system based on an omnidirectional camera and a non-
+repetitive Livox LiDAR. A hybrid mapping approach with both
+odometry-based and stationary mapping modes is integrated
+into one mobile mapping robot, achieving a streamlined
+and automated mapping workflow with the assurance of
+efficiency and mapping precision and accuracy. Meanwhile,
+the proposed automatic and targetless co-calibration method
+provides accurate parameters to generate colorized mapping.
+Specifically, the calibration is based on edges extracted
+from camera images and LiDAR reflectivity, and the result
+is compared with the mutual-information-based calibration
+method, which was under-performing possibly due to varied
+reflection nature in light sources, surface reflection properties,
+and the spectral reflectance disagreement in the MI-based
+method. In future work, more complicated planning strategies
+could be developed to further optimize both the objectives
+of scanning time and spatial coverage. We believe this new
+automated mapping robot will open up a new horizon for
+surveying and inspection robotics.
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+

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