Daily Papers

While perspective is a well-studied topic in art, it is generally taken for granted in images. However, for the recent wave of high-quality image synthesis methods such as latent diffusion models, perspective accuracy is not an explicit requirement. Since these methods are capable of outputting a wide gamut of possible images, it is difficult for these synthesized images to adhere to the principles of linear perspective. We introduce a novel geometric constraint in the training process of generative models to enforce perspective accuracy. We show that outputs of models trained with this constraint both appear more realistic and improve performance of downstream models trained on generated images. Subjective human trials show that images generated with latent diffusion models trained with our constraint are preferred over images from the Stable Diffusion V2 model 70% of the time. SOTA monocular depth estimation models such as DPT and PixelFormer, fine-tuned on our images, outperform the original models trained on real images by up to 7.03% in RMSE and 19.3% in SqRel on the KITTI test set for zero-shot transfer.

Zero-shot novel view synthesis (NVS) from a single image is an essential problem in 3D object understanding. While recent approaches that leverage pre-trained generative models can synthesize high-quality novel views from in-the-wild inputs, they still struggle to maintain 3D consistency across different views. In this paper, we present Consistent-1-to-3, which is a generative framework that significantly mitigate this issue. Specifically, we decompose the NVS task into two stages: (i) transforming observed regions to a novel view, and (ii) hallucinating unseen regions. We design a scene representation transformer and view-conditioned diffusion model for performing these two stages respectively. Inside the models, to enforce 3D consistency, we propose to employ epipolor-guided attention to incorporate geometry constraints, and multi-view attention to better aggregate multi-view information. Finally, we design a hierarchy generation paradigm to generate long sequences of consistent views, allowing a full 360 observation of the provided object image. Qualitative and quantitative evaluation over multiple datasets demonstrate the effectiveness of the proposed mechanisms against state-of-the-art approaches. Our project page is at https://jianglongye.com/consistent123/

Most self-supervised 6D object pose estimation methods can only work with additional depth information or rely on the accurate annotation of 2D segmentation masks, limiting their application range. In this paper, we propose a 6D object pose estimation method that can be trained with pure RGB images without any auxiliary information. We first obtain a rough pose initialization from networks trained on synthetic images rendered from the target's 3D mesh. Then, we introduce a refinement strategy leveraging the geometry constraint in synthetic-to-real image pairs from multiple different views. We formulate this geometry constraint as pixel-level flow consistency between the training images with dynamically generated pseudo labels. We evaluate our method on three challenging datasets and demonstrate that it outperforms state-of-the-art self-supervised methods significantly, with neither 2D annotations nor additional depth images.

Point management is a critical component in optimizing 3D Gaussian Splatting (3DGS) models, as the point initiation (e.g., via structure from motion) is distributionally inappropriate. Typically, the Adaptive Density Control (ADC) algorithm is applied, leveraging view-averaged gradient magnitude thresholding for point densification, opacity thresholding for pruning, and regular all-points opacity reset. However, we reveal that this strategy is limited in tackling intricate/special image regions (e.g., transparent) as it is unable to identify all the 3D zones that require point densification, and lacking an appropriate mechanism to handle the ill-conditioned points with negative impacts (occlusion due to false high opacity). To address these limitations, we propose a Localized Point Management (LPM) strategy, capable of identifying those error-contributing zones in the highest demand for both point addition and geometry calibration. Zone identification is achieved by leveraging the underlying multiview geometry constraints, with the guidance of image rendering errors. We apply point densification in the identified zone, whilst resetting the opacity of those points residing in front of these regions so that a new opportunity is created to correct ill-conditioned points. Serving as a versatile plugin, LPM can be seamlessly integrated into existing 3D Gaussian Splatting models. Experimental evaluation across both static 3D and dynamic 4D scenes validate the efficacy of our LPM strategy in boosting a variety of existing 3DGS models both quantitatively and qualitatively. Notably, LPM improves both vanilla 3DGS and SpaceTimeGS to achieve state-of-the-art rendering quality while retaining real-time speeds, outperforming on challenging datasets such as Tanks & Temples and the Neural 3D Video Dataset.

The recent advancements in 3D Gaussian Splatting (3DGS) have demonstrated remarkable potential in novel view synthesis tasks. The divide-and-conquer paradigm has enabled large-scale scene reconstruction, but significant challenges remain in scene partitioning, optimization, and merging processes. This paper introduces BlockGaussian, a novel framework incorporating a content-aware scene partition strategy and visibility-aware block optimization to achieve efficient and high-quality large-scale scene reconstruction. Specifically, our approach considers the content-complexity variation across different regions and balances computational load during scene partitioning, enabling efficient scene reconstruction. To tackle the supervision mismatch issue during independent block optimization, we introduce auxiliary points during individual block optimization to align the ground-truth supervision, which enhances the reconstruction quality. Furthermore, we propose a pseudo-view geometry constraint that effectively mitigates rendering degradation caused by airspace floaters during block merging. Extensive experiments on large-scale scenes demonstrate that our approach achieves state-of-the-art performance in both reconstruction efficiency and rendering quality, with a 5x speedup in optimization and an average PSNR improvement of 1.21 dB on multiple benchmarks. Notably, BlockGaussian significantly reduces computational requirements, enabling large-scale scene reconstruction on a single 24GB VRAM device. The project page is available at https://github.com/SunshineWYC/BlockGaussian

Recent advances in single-view 3D scene reconstruction have highlighted the challenges in capturing fine geometric details and ensuring structural consistency, particularly in high-fidelity outdoor scene modeling. This paper presents Niagara, a new single-view 3D scene reconstruction framework that can faithfully reconstruct challenging outdoor scenes from a single input image for the first time. Our approach integrates monocular depth and normal estimation as input, which substantially improves its ability to capture fine details, mitigating common issues like geometric detail loss and deformation. Additionally, we introduce a geometric affine field (GAF) and 3D self-attention as geometry-constraint, which combines the structural properties of explicit geometry with the adaptability of implicit feature fields, striking a balance between efficient rendering and high-fidelity reconstruction. Our framework finally proposes a specialized encoder-decoder architecture, where a depth-based 3D Gaussian decoder is proposed to predict 3D Gaussian parameters, which can be used for novel view synthesis. Extensive results and analyses suggest that our Niagara surpasses prior SoTA approaches such as Flash3D in both single-view and dual-view settings, significantly enhancing the geometric accuracy and visual fidelity, especially in outdoor scenes.

Recent advancements in text-to-image generation have enabled significant progress in zero-shot 3D shape generation. This is achieved by score distillation, a methodology that uses pre-trained text-to-image diffusion models to optimize the parameters of a 3D neural presentation, e.g. Neural Radiance Field (NeRF). While showing promising results, existing methods are often not able to preserve the geometry of complex shapes, such as human bodies. To address this challenge, we present ZeroAvatar, a method that introduces the explicit 3D human body prior to the optimization process. Specifically, we first estimate and refine the parameters of a parametric human body from a single image. Then during optimization, we use the posed parametric body as additional geometry constraint to regularize the diffusion model as well as the underlying density field. Lastly, we propose a UV-guided texture regularization term to further guide the completion of texture on invisible body parts. We show that ZeroAvatar significantly enhances the robustness and 3D consistency of optimization-based image-to-3D avatar generation, outperforming existing zero-shot image-to-3D methods.

3D Gaussian splatting (3DGS) has shown its detailed expressive ability and highly efficient rendering speed in the novel view synthesis (NVS) task. The application to inverse rendering still faces several challenges, as the discrete nature of Gaussian primitives makes it difficult to apply geometry constraints. Recent works introduce the signed distance field (SDF) as an extra continuous representation to regularize the geometry defined by Gaussian primitives. It improves the decomposition quality, at the cost of increasing memory usage and complicating training. Unlike these works, we introduce a discretized SDF to represent the continuous SDF in a discrete manner by encoding it within each Gaussian using a sampled value. This approach allows us to link the SDF with the Gaussian opacity through an SDF-to-opacity transformation, enabling rendering the SDF via splatting and avoiding the computational cost of ray marching.The key challenge is to regularize the discrete samples to be consistent with the underlying SDF, as the discrete representation can hardly apply the gradient-based constraints (\eg Eikonal loss). For this, we project Gaussians onto the zero-level set of SDF and enforce alignment with the surface from splatting, namely a projection-based consistency loss. Thanks to the discretized SDF, our method achieves higher relighting quality, while requiring no extra memory beyond GS and avoiding complex manually designed optimization. The experiments reveal that our method outperforms existing Gaussian-based inverse rendering methods. Our code is available at https://github.com/NK-CS-ZZL/DiscretizedSDF.

We present SPAD, a novel approach for creating consistent multi-view images from text prompts or single images. To enable multi-view generation, we repurpose a pretrained 2D diffusion model by extending its self-attention layers with cross-view interactions, and fine-tune it on a high quality subset of Objaverse. We find that a naive extension of the self-attention proposed in prior work (e.g. MVDream) leads to content copying between views. Therefore, we explicitly constrain the cross-view attention based on epipolar geometry. To further enhance 3D consistency, we utilize Plucker coordinates derived from camera rays and inject them as positional encoding. This enables SPAD to reason over spatial proximity in 3D well. In contrast to recent works that can only generate views at fixed azimuth and elevation, SPAD offers full camera control and achieves state-of-the-art results in novel view synthesis on unseen objects from the Objaverse and Google Scanned Objects datasets. Finally, we demonstrate that text-to-3D generation using SPAD prevents the multi-face Janus issue. See more details at our webpage: https://yashkant.github.io/spad

Generalizing metric monocular depth estimation presents a significant challenge due to its ill-posed nature, while the entanglement between camera parameters and depth amplifies issues further, hindering multi-dataset training and zero-shot accuracy. This challenge is particularly evident in autonomous vehicles and mobile robotics, where data is collected with fixed camera setups, limiting the geometric diversity. Yet, this context also presents an opportunity: the fixed relationship between the camera and the ground plane imposes additional perspective geometry constraints, enabling depth regression via vertical image positions of objects. However, this cue is highly susceptible to overfitting, thus we propose a novel canonical representation that maintains consistency across varied camera setups, effectively disentangling depth from specific parameters and enhancing generalization across datasets. We also propose a novel architecture that adaptively and probabilistically fuses depths estimated via object size and vertical image position cues. A comprehensive evaluation demonstrates the effectiveness of the proposed approach on five autonomous driving datasets, achieving accurate metric depth estimation for varying resolutions, aspect ratios and camera setups. Notably, we achieve comparable accuracy to existing zero-shot methods, despite training on a single dataset with a single-camera setup.

Vector graphics are widely used in digital art and highly favored by designers due to their scalability and layer-wise properties. However, the process of creating and editing vector graphics requires creativity and design expertise, making it a time-consuming task. Recent advancements in text-to-vector (T2V) generation have aimed to make this process more accessible. However, existing T2V methods directly optimize control points of vector graphics paths, often resulting in intersecting or jagged paths due to the lack of geometry constraints. To overcome these limitations, we propose a novel neural path representation by designing a dual-branch Variational Autoencoder (VAE) that learns the path latent space from both sequence and image modalities. By optimizing the combination of neural paths, we can incorporate geometric constraints while preserving expressivity in generated SVGs. Furthermore, we introduce a two-stage path optimization method to improve the visual and topological quality of generated SVGs. In the first stage, a pre-trained text-to-image diffusion model guides the initial generation of complex vector graphics through the Variational Score Distillation (VSD) process. In the second stage, we refine the graphics using a layer-wise image vectorization strategy to achieve clearer elements and structure. We demonstrate the effectiveness of our method through extensive experiments and showcase various applications. The project page is https://intchous.github.io/T2V-NPR.

Absolute Pose Regression (APR) methods use deep neural networks to directly regress camera poses from RGB images. However, the predominant APR architectures only rely on 2D operations during inference, resulting in limited accuracy of pose estimation due to the lack of 3D geometry constraints or priors. In this work, we propose a test-time refinement pipeline that leverages implicit geometric constraints using a robust feature field to enhance the ability of APR methods to use 3D information during inference. We also introduce a novel Neural Feature Synthesizer (NeFeS) model, which encodes 3D geometric features during training and directly renders dense novel view features at test time to refine APR methods. To enhance the robustness of our model, we introduce a feature fusion module and a progressive training strategy. Our proposed method achieves state-of-the-art single-image APR accuracy on indoor and outdoor datasets.

We present a novel system for sketch-based face image editing, enabling users to edit images intuitively by sketching a few strokes on a region of interest. Our interface features tools to express a desired image manipulation by providing both geometry and color constraints as user-drawn strokes. As an alternative to the direct user input, our proposed system naturally supports a copy-paste mode, which allows users to edit a given image region by using parts of another exemplar image without the need of hand-drawn sketching at all. The proposed interface runs in real-time and facilitates an interactive and iterative workflow to quickly express the intended edits. Our system is based on a novel sketch domain and a convolutional neural network trained end-to-end to automatically learn to render image regions corresponding to the input strokes. To achieve high quality and semantically consistent results we train our neural network on two simultaneous tasks, namely image completion and image translation. To the best of our knowledge, we are the first to combine these two tasks in a unified framework for interactive image editing. Our results show that the proposed sketch domain, network architecture, and training procedure generalize well to real user input and enable high quality synthesis results without additional post-processing.

Text-to-3D generation from a single-view image is a popular but challenging task in 3D vision. Although numerous methods have been proposed, existing works still suffer from the inconsistency issues, including 1) semantic inconsistency, 2) geometric inconsistency, and 3) saturation inconsistency, resulting in distorted, overfitted, and over-saturated generations. In light of the above issues, we present Consist3D, a three-stage framework Chasing for semantic-, geometric-, and saturation-Consistent Text-to-3D generation from a single image, in which the first two stages aim to learn parameterized consistency tokens, and the last stage is for optimization. Specifically, the semantic encoding stage learns a token independent of views and estimations, promoting semantic consistency and robustness. Meanwhile, the geometric encoding stage learns another token with comprehensive geometry and reconstruction constraints under novel-view estimations, reducing overfitting and encouraging geometric consistency. Finally, the optimization stage benefits from the semantic and geometric tokens, allowing a low classifier-free guidance scale and therefore preventing oversaturation. Experimental results demonstrate that Consist3D produces more consistent, faithful, and photo-realistic 3D assets compared to previous state-of-the-art methods. Furthermore, Consist3D also allows background and object editing through text prompts.

Maps are a key component in image-based camera localization and visual SLAM systems: they are used to establish geometric constraints between images, correct drift in relative pose estimation, and relocalize cameras after lost tracking. The exact definitions of maps, however, are often application-specific and hand-crafted for different scenarios (e.g. 3D landmarks, lines, planes, bags of visual words). We propose to represent maps as a deep neural net called MapNet, which enables learning a data-driven map representation. Unlike prior work on learning maps, MapNet exploits cheap and ubiquitous sensory inputs like visual odometry and GPS in addition to images and fuses them together for camera localization. Geometric constraints expressed by these inputs, which have traditionally been used in bundle adjustment or pose-graph optimization, are formulated as loss terms in MapNet training and also used during inference. In addition to directly improving localization accuracy, this allows us to update the MapNet (i.e., maps) in a self-supervised manner using additional unlabeled video sequences from the scene. We also propose a novel parameterization for camera rotation which is better suited for deep-learning based camera pose regression. Experimental results on both the indoor 7-Scenes dataset and the outdoor Oxford RobotCar dataset show significant performance improvement over prior work. The MapNet project webpage is https://goo.gl/mRB3Au.

Powered by large-scale text-to-image generation models, text-to-3D avatar generation has made promising progress. However, most methods fail to produce photorealistic results, limited by imprecise geometry and low-quality appearance. Towards more practical avatar generation, we present SEEAvatar, a method for generating photorealistic 3D avatars from text with SElf-Evolving constraints for decoupled geometry and appearance. For geometry, we propose to constrain the optimized avatar in a decent global shape with a template avatar. The template avatar is initialized with human prior and can be updated by the optimized avatar periodically as an evolving template, which enables more flexible shape generation. Besides, the geometry is also constrained by the static human prior in local parts like face and hands to maintain the delicate structures. For appearance generation, we use diffusion model enhanced by prompt engineering to guide a physically based rendering pipeline to generate realistic textures. The lightness constraint is applied on the albedo texture to suppress incorrect lighting effect. Experiments show that our method outperforms previous methods on both global and local geometry and appearance quality by a large margin. Since our method can produce high-quality meshes and textures, such assets can be directly applied in classic graphics pipeline for realistic rendering under any lighting condition. Project page at: https://seeavatar3d.github.io.

Accurate 6D object pose estimation from images is a key problem in object-centric scene understanding, enabling applications in robotics, augmented reality, and scene reconstruction. Despite recent advances, existing methods often produce physically inconsistent pose estimates, hindering their deployment in real-world scenarios. We introduce PhysPose, a novel approach that integrates physical reasoning into pose estimation through a postprocessing optimization enforcing non-penetration and gravitational constraints. By leveraging scene geometry, PhysPose refines pose estimates to ensure physical plausibility. Our approach achieves state-of-the-art accuracy on the YCB-Video dataset from the BOP benchmark and improves over the state-of-the-art pose estimation methods on the HOPE-Video dataset. Furthermore, we demonstrate its impact in robotics by significantly improving success rates in a challenging pick-and-place task, highlighting the importance of physical consistency in real-world applications.

In this study, we address the challenge of 3D scene structure recovery from monocular depth estimation. While traditional depth estimation methods leverage labeled datasets to directly predict absolute depth, recent advancements advocate for mix-dataset training, enhancing generalization across diverse scenes. However, such mixed dataset training yields depth predictions only up to an unknown scale and shift, hindering accurate 3D reconstructions. Existing solutions necessitate extra 3D datasets or geometry-complete depth annotations, constraints that limit their versatility. In this paper, we propose a learning framework that trains models to predict geometry-preserving depth without requiring extra data or annotations. To produce realistic 3D structures, we render novel views of the reconstructed scenes and design loss functions to promote depth estimation consistency across different views. Comprehensive experiments underscore our framework's superior generalization capabilities, surpassing existing state-of-the-art methods on several benchmark datasets without leveraging extra training information. Moreover, our innovative loss functions empower the model to autonomously recover domain-specific scale-and-shift coefficients using solely unlabeled images.

How can Transformers model and learn enumerative geometry? What is a robust procedure for using Transformers in abductive knowledge discovery within a mathematician-machine collaboration? In this work, we introduce a Transformer-based approach to computational enumerative geometry, specifically targeting the computation of psi-class intersection numbers on the moduli space of curves. By reformulating the problem as a continuous optimization task, we compute intersection numbers across a wide value range from 10^{-45} to 10^{45}. To capture the recursive nature inherent in these intersection numbers, we propose the Dynamic Range Activator (DRA), a new activation function that enhances the Transformer's ability to model recursive patterns and handle severe heteroscedasticity. Given precision requirements for computing the intersections, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window adaptive to the partitions of equivalent number of marked points. To the best of our knowledge, there has been no prior work on modeling recursive functions with such a high-variance and factorial growth. Beyond simply computing intersection numbers, we explore the enumerative "world-model" of Transformers. Our interpretability analysis reveals that the network is implicitly modeling the Virasoro constraints in a purely data-driven manner. Moreover, through abductive hypothesis testing, probing, and causal inference, we uncover evidence of an emergent internal representation of the the large-genus asymptotic of psi-class intersection numbers. These findings suggest that the network internalizes the parameters of the asymptotic closed-form and the polynomiality phenomenon of psi-class intersection numbers in a non-linear manner.

Unsupervised Domain Adaptation has been an efficient approach to transferring the semantic segmentation model across data distributions. Meanwhile, the recent Open-vocabulary Semantic Scene understanding based on large-scale vision language models is effective in open-set settings because it can learn diverse concepts and categories. However, these prior methods fail to generalize across different camera views due to the lack of cross-view geometric modeling. At present, there are limited studies analyzing cross-view learning. To address this problem, we introduce a novel Unsupervised Cross-view Adaptation Learning approach to modeling the geometric structural change across views in Semantic Scene Understanding. First, we introduce a novel Cross-view Geometric Constraint on Unpaired Data to model structural changes in images and segmentation masks across cameras. Second, we present a new Geodesic Flow-based Correlation Metric to efficiently measure the geometric structural changes across camera views. Third, we introduce a novel view-condition prompting mechanism to enhance the view-information modeling of the open-vocabulary segmentation network in cross-view adaptation learning. The experiments on different cross-view adaptation benchmarks have shown the effectiveness of our approach in cross-view modeling, demonstrating that we achieve State-of-the-Art (SOTA) performance compared to prior unsupervised domain adaptation and open-vocabulary semantic segmentation methods.

Novel-view synthesis (NVS) can be tackled through different approaches, depending on the general setting: a single source image to a short video sequence, exact or noisy camera pose information, 3D-based information such as point clouds etc. The most challenging scenario, the one where we stand in this work, only considers a unique source image to generate a novel one from another viewpoint. However, in such a tricky situation, the latest learning-based solutions often struggle to integrate the camera viewpoint transformation. Indeed, the extrinsic information is often passed as-is, through a low-dimensional vector. It might even occur that such a camera pose, when parametrized as Euler angles, is quantized through a one-hot representation. This vanilla encoding choice prevents the learnt architecture from inferring novel views on a continuous basis (from a camera pose perspective). We claim it exists an elegant way to better encode relative camera pose, by leveraging 3D-related concepts such as the epipolar constraint. We, therefore, introduce an innovative method that encodes the viewpoint transformation as a 2D feature image. Such a camera encoding strategy gives meaningful insights to the network regarding how the camera has moved in space between the two views. By encoding the camera pose information as a finite number of coloured epipolar lines, we demonstrate through our experiments that our strategy outperforms vanilla encoding.

We present GeoManip, a framework to enable generalist robots to leverage essential conditions derived from object and part relationships, as geometric constraints, for robot manipulation. For example, cutting the carrot requires adhering to a geometric constraint: the blade of the knife should be perpendicular to the carrot's direction. By interpreting these constraints through symbolic language representations and translating them into low-level actions, GeoManip bridges the gap between natural language and robotic execution, enabling greater generalizability across diverse even unseen tasks, objects, and scenarios. Unlike vision-language-action models that require extensive training, operates training-free by utilizing large foundational models: a constraint generation module that predicts stage-specific geometric constraints and a geometry parser that identifies object parts involved in these constraints. A solver then optimizes trajectories to satisfy inferred constraints from task descriptions and the scene. Furthermore, GeoManip learns in-context and provides five appealing human-robot interaction features: on-the-fly policy adaptation, learning from human demonstrations, learning from failure cases, long-horizon action planning, and efficient data collection for imitation learning. Extensive evaluations on both simulations and real-world scenarios demonstrate GeoManip's state-of-the-art performance, with superior out-of-distribution generalization while avoiding costly model training.

In representation learning, uniformity refers to the uniform feature distribution in the latent space (i.e., unit hypersphere). Previous work has shown that improving uniformity contributes to the learning of under-represented classes. However, most of the previous work focused on classification; the representation space of imbalanced regression remains unexplored. Classification-based methods are not suitable for regression tasks because they cluster features into distinct groups without considering the continuous and ordered nature essential for regression. In a geometric aspect, we uniquely focus on ensuring uniformity in the latent space for imbalanced regression through two key losses: enveloping and homogeneity. The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness, with representations evenly spaced at consistent intervals. Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning (SRL) framework. Experiments with real-world regression and operator learning tasks highlight the importance of uniformity in imbalanced regression and validate the efficacy of our geometry-based loss functions.

Text-driven 3D indoor scene generation could be useful for gaming, film industry, and AR/VR applications. However, existing methods cannot faithfully capture the room layout, nor do they allow flexible editing of individual objects in the room. To address these problems, we present Ctrl-Room, which is able to generate convincing 3D rooms with designer-style layouts and high-fidelity textures from just a text prompt. Moreover, Ctrl-Room enables versatile interactive editing operations such as resizing or moving individual furniture items. Our key insight is to separate the modeling of layouts and appearance. %how to model the room that takes into account both scene texture and geometry at the same time. To this end, Our proposed method consists of two stages, a `Layout Generation Stage' and an `Appearance Generation Stage'. The `Layout Generation Stage' trains a text-conditional diffusion model to learn the layout distribution with our holistic scene code parameterization. Next, the `Appearance Generation Stage' employs a fine-tuned ControlNet to produce a vivid panoramic image of the room guided by the 3D scene layout and text prompt. In this way, we achieve a high-quality 3D room with convincing layouts and lively textures. Benefiting from the scene code parameterization, we can easily edit the generated room model through our mask-guided editing module, without expensive editing-specific training. Extensive experiments on the Structured3D dataset demonstrate that our method outperforms existing methods in producing more reasonable, view-consistent, and editable 3D rooms from natural language prompts.

We introduce TetSphere Splatting, a Lagrangian geometry representation designed for high-quality 3D shape modeling. TetSphere splatting leverages an underused yet powerful geometric primitive -- volumetric tetrahedral meshes. It represents 3D shapes by deforming a collection of tetrahedral spheres, with geometric regularizations and constraints that effectively resolve common mesh issues such as irregular triangles, non-manifoldness, and floating artifacts. Experimental results on multi-view and single-view reconstruction highlight TetSphere splatting's superior mesh quality while maintaining competitive reconstruction accuracy compared to state-of-the-art methods. Additionally, TetSphere splatting demonstrates versatility by seamlessly integrating into generative modeling tasks, such as image-to-3D and text-to-3D generation.

Recent advances in 3D Gaussian Splatting (3DGS) have enabled RGB-only SLAM systems to achieve high-fidelity scene representation. However, the heavy reliance of existing systems on photometric rendering loss for camera tracking undermines their robustness, especially in unbounded outdoor environments with severe viewpoint and illumination changes. To address these challenges, we propose TVG-SLAM, a robust RGB-only 3DGS SLAM system that leverages a novel tri-view geometry paradigm to ensure consistent tracking and high-quality mapping. We introduce a dense tri-view matching module that aggregates reliable pairwise correspondences into consistent tri-view matches, forming robust geometric constraints across frames. For tracking, we propose Hybrid Geometric Constraints, which leverage tri-view matches to construct complementary geometric cues alongside photometric loss, ensuring accurate and stable pose estimation even under drastic viewpoint shifts and lighting variations. For mapping, we propose a new probabilistic initialization strategy that encodes geometric uncertainty from tri-view correspondences into newly initialized Gaussians. Additionally, we design a Dynamic Attenuation of Rendering Trust mechanism to mitigate tracking drift caused by mapping latency. Experiments on multiple public outdoor datasets show that our TVG-SLAM outperforms prior RGB-only 3DGS-based SLAM systems. Notably, in the most challenging dataset, our method improves tracking robustness, reducing the average Absolute Trajectory Error (ATE) by 69.0\% while achieving state-of-the-art rendering quality. The implementation of our method will be released as open-source.

We present a method for 3D object detection and pose estimation from a single image. In contrast to current techniques that only regress the 3D orientation of an object, our method first regresses relatively stable 3D object properties using a deep convolutional neural network and then combines these estimates with geometric constraints provided by a 2D object bounding box to produce a complete 3D bounding box. The first network output estimates the 3D object orientation using a novel hybrid discrete-continuous loss, which significantly outperforms the L2 loss. The second output regresses the 3D object dimensions, which have relatively little variance compared to alternatives and can often be predicted for many object types. These estimates, combined with the geometric constraints on translation imposed by the 2D bounding box, enable us to recover a stable and accurate 3D object pose. We evaluate our method on the challenging KITTI object detection benchmark both on the official metric of 3D orientation estimation and also on the accuracy of the obtained 3D bounding boxes. Although conceptually simple, our method outperforms more complex and computationally expensive approaches that leverage semantic segmentation, instance level segmentation and flat ground priors and sub-category detection. Our discrete-continuous loss also produces state of the art results for 3D viewpoint estimation on the Pascal 3D+ dataset.

2D diffusion model, which often contains unwanted baked-in shading effects and results in unrealistic rendering effects in the downstream applications. Generating Physically Based Rendering (PBR) materials instead of just RGB textures would be a promising solution. However, directly distilling the PBR material parameters from 2D diffusion models still suffers from incorrect material decomposition, such as baked-in shading effects in albedo. We introduce DreamMat, an innovative approach to resolve the aforementioned problem, to generate high-quality PBR materials from text descriptions. We find out that the main reason for the incorrect material distillation is that large-scale 2D diffusion models are only trained to generate final shading colors, resulting in insufficient constraints on material decomposition during distillation. To tackle this problem, we first finetune a new light-aware 2D diffusion model to condition on a given lighting environment and generate the shading results on this specific lighting condition. Then, by applying the same environment lights in the material distillation, DreamMat can generate high-quality PBR materials that are not only consistent with the given geometry but also free from any baked-in shading effects in albedo. Extensive experiments demonstrate that the materials produced through our methods exhibit greater visual appeal to users and achieve significantly superior rendering quality compared to baseline methods, which are preferable for downstream tasks such as game and film production.

Alignment is no longer a luxury, it is a necessity. As large language models (LLMs) enter high-stakes domains like education, healthcare, governance, and law, their behavior must reliably reflect human-aligned values and safety constraints. Yet current evaluations rely heavily on behavioral proxies such as refusal rates, G-Eval scores, and toxicity classifiers, all of which have critical blind spots. Aligned models are often vulnerable to jailbreaking, stochasticity of generation, and alignment faking. To address this issue, we introduce the Alignment Quality Index (AQI). This novel geometric and prompt-invariant metric empirically assesses LLM alignment by analyzing the separation of safe and unsafe activations in latent space. By combining measures such as the Davies-Bouldin Score (DBS), Dunn Index (DI), Xie-Beni Index (XBI), and Calinski-Harabasz Index (CHI) across various formulations, AQI captures clustering quality to detect hidden misalignments and jailbreak risks, even when outputs appear compliant. AQI also serves as an early warning signal for alignment faking, offering a robust, decoding invariant tool for behavior agnostic safety auditing. Additionally, we propose the LITMUS dataset to facilitate robust evaluation under these challenging conditions. Empirical tests on LITMUS across different models trained under DPO, GRPO, and RLHF conditions demonstrate AQI's correlation with external judges and ability to reveal vulnerabilities missed by refusal metrics. We make our implementation publicly available to foster future research in this area.

Cosmological data probe massive neutrinos via their effects on the geometry of the Universe and the growth of structure, both of which are degenerate with the late-time expansion history. We clarify the nature of these degeneracies and the individual roles of both probes in neutrino mass inference. Geometry is strongly sensitive to neutrino masses: within LambdaCDM, the primary cosmic microwave background anisotropies alone impose that the matter fraction Omega_m must increase fivefold with increasing neutrino mass. Moreover, large-scale structure observables, like weak lensing of the CMB, are dimensionless and thus depend not on the matter density (as often quoted) but in fact the matter fraction. We explore the consequential impact of this distinction on the interplay between probes of structure, low-redshift distances, and CMB anisotropies. We derive constraints on the neutrino's masses independently from their suppression of structure and impact on geometry, showing that the latter is at least as important as the former. While the Dark Energy Spectroscopic Instrument's recent baryon acoustic oscillation data place stringent bounds largely deriving from their geometric incompatibility with massive neutrinos, all recent type Ia supernova datasets drive marginal preferences for nonzero neutrino masses because they prefer substantially larger matter fractions. Recent CMB lensing data, however, neither exclude neutrinos' suppression of structure nor constrain it strongly enough to discriminate between mass hierarchies. Current data thus evince not a need for modified dynamics of neutrino perturbations or structure growth but rather an inconsistent compatibility with massive neutrinos' impact on the expansion history. We identify two of DESI's measurements that strongly influence its constraints, and we also discuss neutrino mass measurements in models that alter the sound horizon.

3D-aware GANs aim to synthesize realistic 3D scenes such that they can be rendered in arbitrary perspectives to produce images. Although previous methods produce realistic images, they suffer from unstable training or degenerate solutions where the 3D geometry is unnatural. We hypothesize that the 3D geometry is underdetermined due to the insufficient constraint, i.e., being classified as real image to the discriminator is not enough. To solve this problem, we propose to approximate the background as a spherical surface and represent a scene as a union of the foreground placed in the sphere and the thin spherical background. It reduces the degree of freedom in the background field. Accordingly, we modify the volume rendering equation and incorporate dedicated constraints to design a novel 3D-aware GAN framework named BallGAN. BallGAN has multiple advantages as follows. 1) It produces more reasonable 3D geometry; the images of a scene across different viewpoints have better photometric consistency and fidelity than the state-of-the-art methods. 2) The training becomes much more stable. 3) The foreground can be separately rendered on top of different arbitrary backgrounds.

From a single image, visual cues can help deduce intrinsic and extrinsic camera parameters like the focal length and the gravity direction. This single-image calibration can benefit various downstream applications like image editing and 3D mapping. Current approaches to this problem are based on either classical geometry with lines and vanishing points or on deep neural networks trained end-to-end. The learned approaches are more robust but struggle to generalize to new environments and are less accurate than their classical counterparts. We hypothesize that they lack the constraints that 3D geometry provides. In this work, we introduce GeoCalib, a deep neural network that leverages universal rules of 3D geometry through an optimization process. GeoCalib is trained end-to-end to estimate camera parameters and learns to find useful visual cues from the data. Experiments on various benchmarks show that GeoCalib is more robust and more accurate than existing classical and learned approaches. Its internal optimization estimates uncertainties, which help flag failure cases and benefit downstream applications like visual localization. The code and trained models are publicly available at https://github.com/cvg/GeoCalib.

Non-line-of-sight (NLOS) imaging is conducted to infer invisible scenes from indirect light on visible objects. The neural transient field (NeTF) was proposed for representing scenes as neural radiance fields in NLOS scenes. We propose NLOS neural implicit surface (NLOS-NeuS), which extends the NeTF to neural implicit surfaces with a signed distance function (SDF) for reconstructing three-dimensional surfaces in NLOS scenes. We introduce two constraints as loss functions for correctly learning an SDF to avoid non-zero level-set surfaces. We also introduce a lower bound constraint of an SDF based on the geometry of the first-returning photons. The experimental results indicate that these constraints are essential for learning a correct SDF in NLOS scenes. Compared with previous methods with discretized representation, NLOS-NeuS with the neural continuous representation enables us to reconstruct smooth surfaces while preserving fine details in NLOS scenes. To the best of our knowledge, this is the first study on neural implicit surfaces with volume rendering in NLOS scenes.

Unsupervised learning of depth from indoor monocular videos is challenging as the artificial environment contains many textureless regions. Fortunately, the indoor scenes are full of specific structures, such as planes and lines, which should help guide unsupervised depth learning. This paper proposes PLNet that leverages the plane and line priors to enhance the depth estimation. We first represent the scene geometry using local planar coefficients and impose the smoothness constraint on the representation. Moreover, we enforce the planar and linear consistency by randomly selecting some sets of points that are probably coplanar or collinear to construct simple and effective consistency losses. To verify the proposed method's effectiveness, we further propose to evaluate the flatness and straightness of the predicted point cloud on the reliable planar and linear regions. The regularity of these regions indicates quality indoor reconstruction. Experiments on NYU Depth V2 and ScanNet show that PLNet outperforms existing methods. The code is available at https://github.com/HalleyJiang/PLNet.

Data heterogeneity in federated learning, characterized by a significant misalignment between local and global distributions, leads to divergent local optimization directions and hinders global model training. Existing studies mainly focus on optimizing local updates or global aggregation, but these indirect approaches demonstrate instability when handling highly heterogeneous data distributions, especially in scenarios where label skew and domain skew coexist. To address this, we propose a geometry-guided data generation method that centers on simulating the global embedding distribution locally. We first introduce the concept of the geometric shape of an embedding distribution and then address the challenge of obtaining global geometric shapes under privacy constraints. Subsequently, we propose GGEUR, which leverages global geometric shapes to guide the generation of new samples, enabling a closer approximation to the ideal global distribution. In single-domain scenarios, we augment samples based on global geometric shapes to enhance model generalization; in multi-domain scenarios, we further employ class prototypes to simulate the global distribution across domains. Extensive experimental results demonstrate that our method significantly enhances the performance of existing approaches in handling highly heterogeneous data, including scenarios with label skew, domain skew, and their coexistence. Code published at: https://github.com/WeiDai-David/2025CVPR_GGEUR

Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an encoder, which maps a high-dimensional dataset and its low-dimensional embedding, and a decoder, which takes the embedded data back to the high-dimensional space. Stacking the encoder and decoder together constructs an autoencoder, which we term a diffusion net, that performs out-of-sample-extension as well as outlier detection. We introduce new neural net constraints for the encoder, which preserves the local geometry of the points, and we prove rates of convergence for the encoder. Also, our approach is efficient in both computational complexity and memory requirements, as opposed to previous methods that require storage of all training points in both the high-dimensional and the low-dimensional spaces to calculate the out-of-sample-extension and the pre-image.

Emerging from low-level vision theory, steerable filters found their counterpart in prior work on steerable convolutional neural networks equivariant to rigid transformations. In our work, we propose a steerable feed-forward learning-based approach that consists of neurons with spherical decision surfaces and operates on point clouds. Such spherical neurons are obtained by conformal embedding of Euclidean space and have recently been revisited in the context of learning representations of point sets. Focusing on 3D geometry, we exploit the isometry property of spherical neurons and derive a 3D steerability constraint. After training spherical neurons to classify point clouds in a canonical orientation, we use a tetrahedron basis to quadruplicate the neurons and construct rotation-equivariant spherical filter banks. We then apply the derived constraint to interpolate the filter bank outputs and, thus, obtain a rotation-invariant network. Finally, we use a synthetic point set and real-world 3D skeleton data to verify our theoretical findings. The code is available at https://github.com/pavlo-melnyk/steerable-3d-neurons.

A commonly observed failure mode of Neural Radiance Field (NeRF) is fitting incorrect geometries when given an insufficient number of input views. One potential reason is that standard volumetric rendering does not enforce the constraint that most of a scene's geometry consist of empty space and opaque surfaces. We formalize the above assumption through DS-NeRF (Depth-supervised Neural Radiance Fields), a loss for learning radiance fields that takes advantage of readily-available depth supervision. We leverage the fact that current NeRF pipelines require images with known camera poses that are typically estimated by running structure-from-motion (SFM). Crucially, SFM also produces sparse 3D points that can be used as "free" depth supervision during training: we add a loss to encourage the distribution of a ray's terminating depth matches a given 3D keypoint, incorporating depth uncertainty. DS-NeRF can render better images given fewer training views while training 2-3x faster. Further, we show that our loss is compatible with other recently proposed NeRF methods, demonstrating that depth is a cheap and easily digestible supervisory signal. And finally, we find that DS-NeRF can support other types of depth supervision such as scanned depth sensors and RGB-D reconstruction outputs.

Inspired by cognitive theories, we introduce AnyHome, a framework that translates any text into well-structured and textured indoor scenes at a house-scale. By prompting Large Language Models (LLMs) with designed templates, our approach converts provided textual narratives into amodal structured representations. These representations guarantee consistent and realistic spatial layouts by directing the synthesis of a geometry mesh within defined constraints. A Score Distillation Sampling process is then employed to refine the geometry, followed by an egocentric inpainting process that adds lifelike textures to it. AnyHome stands out with its editability, customizability, diversity, and realism. The structured representations for scenes allow for extensive editing at varying levels of granularity. Capable of interpreting texts ranging from simple labels to detailed narratives, AnyHome generates detailed geometries and textures that outperform existing methods in both quantitative and qualitative measures.

Holistic understanding of urban scenes based on RGB images is a challenging yet important problem. It encompasses understanding both the geometry and appearance to enable novel view synthesis, parsing semantic labels, and tracking moving objects. Despite considerable progress, existing approaches often focus on specific aspects of this task and require additional inputs such as LiDAR scans or manually annotated 3D bounding boxes. In this paper, we introduce a novel pipeline that utilizes 3D Gaussian Splatting for holistic urban scene understanding. Our main idea involves the joint optimization of geometry, appearance, semantics, and motion using a combination of static and dynamic 3D Gaussians, where moving object poses are regularized via physical constraints. Our approach offers the ability to render new viewpoints in real-time, yielding 2D and 3D semantic information with high accuracy, and reconstruct dynamic scenes, even in scenarios where 3D bounding box detection are highly noisy. Experimental results on KITTI, KITTI-360, and Virtual KITTI 2 demonstrate the effectiveness of our approach.

This paper addresses the challenge of reconstructing 3D indoor scenes from multi-view images. Many previous works have shown impressive reconstruction results on textured objects, but they still have difficulty in handling low-textured planar regions, which are common in indoor scenes. An approach to solving this issue is to incorporate planer constraints into the depth map estimation in multi-view stereo-based methods, but the per-view plane estimation and depth optimization lack both efficiency and multi-view consistency. In this work, we show that the planar constraints can be conveniently integrated into the recent implicit neural representation-based reconstruction methods. Specifically, we use an MLP network to represent the signed distance function as the scene geometry. Based on the Manhattan-world assumption, planar constraints are employed to regularize the geometry in floor and wall regions predicted by a 2D semantic segmentation network. To resolve the inaccurate segmentation, we encode the semantics of 3D points with another MLP and design a novel loss that jointly optimizes the scene geometry and semantics in 3D space. Experiments on ScanNet and 7-Scenes datasets show that the proposed method outperforms previous methods by a large margin on 3D reconstruction quality. The code is available at https://zju3dv.github.io/manhattan_sdf.

Video depth estimation aims to infer temporally consistent depth. Some methods achieve temporal consistency by finetuning a single-image depth model during test time using geometry and re-projection constraints, which is inefficient and not robust. An alternative approach is to learn how to enforce temporal consistency from data, but this requires well-designed models and sufficient video depth data. To address these challenges, we propose a plug-and-play framework called Neural Video Depth Stabilizer (NVDS) that stabilizes inconsistent depth estimations and can be applied to different single-image depth models without extra effort. We also introduce a large-scale dataset, Video Depth in the Wild (VDW), which consists of 14,203 videos with over two million frames, making it the largest natural-scene video depth dataset to our knowledge. We evaluate our method on the VDW dataset as well as two public benchmarks and demonstrate significant improvements in consistency, accuracy, and efficiency compared to previous approaches. Our work serves as a solid baseline and provides a data foundation for learning-based video depth models. We will release our dataset and code for future research.

We present a novel meta-learning approach for 6D pose estimation on unknown objects. In contrast to ``instance-level" and ``category-level" pose estimation methods, our algorithm learns object representation in a category-agnostic way, which endows it with strong generalization capabilities across object categories. Specifically, we employ a neural process-based meta-learning approach to train an encoder to capture texture and geometry of an object in a latent representation, based on very few RGB-D images and ground-truth keypoints. The latent representation is then used by a simultaneously meta-trained decoder to predict the 6D pose of the object in new images. Furthermore, we propose a novel geometry-aware decoder for the keypoint prediction using a Graph Neural Network (GNN), which explicitly takes geometric constraints specific to each object into consideration. To evaluate our algorithm, extensive experiments are conducted on the \linemod dataset, and on our new fully-annotated synthetic datasets generated from Multiple Categories in Multiple Scenes (MCMS). Experimental results demonstrate that our model performs well on unseen objects with very different shapes and appearances. Remarkably, our model also shows robust performance on occluded scenes although trained fully on data without occlusion. To our knowledge, this is the first work exploring cross-category level 6D pose estimation.

3D Gaussian Splatting (3DGS) has shown its impressive power in novel view synthesis. However, creating relightable 3D assets, especially for objects with ill-defined shapes (e.g., fur), is still a challenging task. For these scenes, the decomposition between the light, geometry, and material is more ambiguous, as neither the surface constraints nor the analytical shading model hold. To address this issue, we propose RNG, a novel representation of relightable neural Gaussians, enabling the relighting of objects with both hard surfaces or fluffy boundaries. We avoid any assumptions in the shading model but maintain feature vectors, which can be further decoded by an MLP into colors, in each Gaussian point. Following prior work, we utilize a point light to reduce the ambiguity and introduce a shadow-aware condition to the network. We additionally propose a depth refinement network to help the shadow computation under the 3DGS framework, leading to better shadow effects under point lights. Furthermore, to avoid the blurriness brought by the alpha-blending in 3DGS, we design a hybrid forward-deferred optimization strategy. As a result, we achieve about 20times faster in training and about 600times faster in rendering than prior work based on neural radiance fields, with 60 frames per second on an RTX4090.

Recovering 3D scenes from sparse views is a challenging task due to its inherent ill-posed problem. Conventional methods have developed specialized solutions (e.g., geometry regularization or feed-forward deterministic model) to mitigate the issue. However, they still suffer from performance degradation by minimal overlap across input views with insufficient visual information. Fortunately, recent video generative models show promise in addressing this challenge as they are capable of generating video clips with plausible 3D structures. Powered by large pretrained video diffusion models, some pioneering research start to explore the potential of video generative prior and create 3D scenes from sparse views. Despite impressive improvements, they are limited by slow inference time and the lack of 3D constraint, leading to inefficiencies and reconstruction artifacts that do not align with real-world geometry structure. In this paper, we propose VideoScene to distill the video diffusion model to generate 3D scenes in one step, aiming to build an efficient and effective tool to bridge the gap from video to 3D. Specifically, we design a 3D-aware leap flow distillation strategy to leap over time-consuming redundant information and train a dynamic denoising policy network to adaptively determine the optimal leap timestep during inference. Extensive experiments demonstrate that our VideoScene achieves faster and superior 3D scene generation results than previous video diffusion models, highlighting its potential as an efficient tool for future video to 3D applications. Project Page: https://hanyang-21.github.io/VideoScene

Since its inception in "Attention Is All You Need", transformer architecture has led to revolutionary advancements in NLP. The attention layer within the transformer admits a sequence of input tokens X and makes them interact through pairwise similarities computed as softmax(XQK^top X^top), where (K,Q) are the trainable key-query parameters. In this work, we establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem that separates optimal input tokens from non-optimal tokens using linear constraints on the outer-products of token pairs. This formalism allows us to characterize the implicit bias of 1-layer transformers optimized with gradient descent: (1) Optimizing the attention layer with vanishing regularization, parameterized by (K,Q), converges in direction to an SVM solution minimizing the nuclear norm of the combined parameter W=KQ^top. Instead, directly parameterizing by W minimizes a Frobenius norm objective. We characterize this convergence, highlighting that it can occur toward locally-optimal directions rather than global ones. (2) Complementing this, we prove the local/global directional convergence of gradient descent under suitable geometric conditions. Importantly, we show that over-parameterization catalyzes global convergence by ensuring the feasibility of the SVM problem and by guaranteeing a benign optimization landscape devoid of stationary points. (3) While our theory applies primarily to linear prediction heads, we propose a more general SVM equivalence that predicts the implicit bias with nonlinear heads. Our findings are applicable to arbitrary datasets and their validity is verified via experiments. We also introduce several open problems and research directions. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.

Multi-view image generation attracts particular attention these days due to its promising 3D-related applications, e.g., image viewpoint editing. Most existing methods follow a paradigm where a 3D representation is first synthesized, and then rendered into 2D images to ensure photo-consistency across viewpoints. However, such explicit bias for photo-consistency sacrifices photo-realism, causing geometry artifacts and loss of fine-scale details when these methods are applied to edit real images. To address this issue, we propose ray conditioning, a geometry-free alternative that relaxes the photo-consistency constraint. Our method generates multi-view images by conditioning a 2D GAN on a light field prior. With explicit viewpoint control, state-of-the-art photo-realism and identity consistency, our method is particularly suited for the viewpoint editing task.

3D reconstruction from in-the-wild images remains a challenging task due to inconsistent lighting conditions and transient distractors. Existing methods typically rely on heuristic strategies to handle the low-quality training data, which often struggle to produce stable and consistent reconstructions, frequently resulting in visual artifacts. In this work, we propose Asymmetric Dual 3DGS, a novel framework that leverages the stochastic nature of these artifacts: they tend to vary across different training runs due to minor randomness. Specifically, our method trains two 3D Gaussian Splatting (3DGS) models in parallel, enforcing a consistency constraint that encourages convergence on reliable scene geometry while suppressing inconsistent artifacts. To prevent the two models from collapsing into similar failure modes due to confirmation bias, we introduce a divergent masking strategy that applies two complementary masks: a multi-cue adaptive mask and a self-supervised soft mask, which leads to an asymmetric training process of the two models, reducing shared error modes. In addition, to improve the efficiency of model training, we introduce a lightweight variant called Dynamic EMA Proxy, which replaces one of the two models with a dynamically updated Exponential Moving Average (EMA) proxy, and employs an alternating masking strategy to preserve divergence. Extensive experiments on challenging real-world datasets demonstrate that our method consistently outperforms existing approaches while achieving high efficiency. Codes and trained models will be released.

Crystals are the foundation of numerous scientific and industrial applications. While various learning-based approaches have been proposed for crystal generation, existing methods seldom consider the space group constraint which is crucial in describing the geometry of crystals and closely relevant to many desirable properties. However, considering space group constraint is challenging owing to its diverse and nontrivial forms. In this paper, we reduce the space group constraint into an equivalent formulation that is more tractable to be handcrafted into the generation process. In particular, we translate the space group constraint into two parts: the basis constraint of the invariant logarithmic space of the lattice matrix and the Wyckoff position constraint of the fractional coordinates. Upon the derived constraints, we then propose DiffCSP++, a novel diffusion model that has enhanced a previous work DiffCSP by further taking space group constraint into account. Experiments on several popular datasets verify the benefit of the involvement of the space group constraint, and show that our DiffCSP++ achieves promising performance on crystal structure prediction, ab initio crystal generation and controllable generation with customized space groups.

Synthesizing semantic-aware, long-horizon, human-object interaction is critical to simulate realistic human behaviors. In this work, we address the challenging problem of generating synchronized object motion and human motion guided by language descriptions in 3D scenes. We propose Controllable Human-Object Interaction Synthesis (CHOIS), an approach that generates object motion and human motion simultaneously using a conditional diffusion model given a language description, initial object and human states, and sparse object waypoints. While language descriptions inform style and intent, waypoints ground the motion in the scene and can be effectively extracted using high-level planning methods. Naively applying a diffusion model fails to predict object motion aligned with the input waypoints and cannot ensure the realism of interactions that require precise hand-object contact and appropriate contact grounded by the floor. To overcome these problems, we introduce an object geometry loss as additional supervision to improve the matching between generated object motion and input object waypoints. In addition, we design guidance terms to enforce contact constraints during the sampling process of the trained diffusion model.

We present En3D, an enhanced generative scheme for sculpting high-quality 3D human avatars. Unlike previous works that rely on scarce 3D datasets or limited 2D collections with imbalanced viewing angles and imprecise pose priors, our approach aims to develop a zero-shot 3D generative scheme capable of producing visually realistic, geometrically accurate and content-wise diverse 3D humans without relying on pre-existing 3D or 2D assets. To address this challenge, we introduce a meticulously crafted workflow that implements accurate physical modeling to learn the enhanced 3D generative model from synthetic 2D data. During inference, we integrate optimization modules to bridge the gap between realistic appearances and coarse 3D shapes. Specifically, En3D comprises three modules: a 3D generator that accurately models generalizable 3D humans with realistic appearance from synthesized balanced, diverse, and structured human images; a geometry sculptor that enhances shape quality using multi-view normal constraints for intricate human anatomy; and a texturing module that disentangles explicit texture maps with fidelity and editability, leveraging semantical UV partitioning and a differentiable rasterizer. Experimental results show that our approach significantly outperforms prior works in terms of image quality, geometry accuracy and content diversity. We also showcase the applicability of our generated avatars for animation and editing, as well as the scalability of our approach for content-style free adaptation.

Scene generation with 3D assets presents a complex challenge, requiring both high-level semantic understanding and low-level geometric reasoning. While Multimodal Large Language Models (MLLMs) excel at semantic tasks, their application to 3D scene generation is hindered by their limited grounding on 3D geometry. In this paper, we investigate how to best work with MLLMs in an object placement task. Towards this goal, we introduce a novel framework, FirePlace, that applies existing MLLMs in (1) 3D geometric reasoning and the extraction of relevant geometric details from the 3D scene, (2) constructing and solving geometric constraints on the extracted low-level geometry, and (3) pruning for final placements that conform to common sense. By combining geometric reasoning with real-world understanding of MLLMs, our method can propose object placements that satisfy both geometric constraints as well as high-level semantic common-sense considerations. Our experiments show that these capabilities allow our method to place objects more effectively in complex scenes with intricate geometry, surpassing the quality of prior work.

Long-term forecasting of chaotic systems from short-term observations remains a fundamental and underexplored challenge due to the intrinsic sensitivity to initial conditions and the complex geometry of strange attractors. Existing approaches often rely on long-term training data or focus on short-term sequence correlations, struggling to maintain predictive stability and dynamical coherence over extended horizons. We propose PhyxMamba, a novel framework that integrates a Mamba-based state-space model with physics-informed principles to capture the underlying dynamics of chaotic systems. By reconstructing the attractor manifold from brief observations using time-delay embeddings, PhyxMamba extracts global dynamical features essential for accurate forecasting. Our generative training scheme enables Mamba to replicate the physical process, augmented by multi-token prediction and attractor geometry regularization for physical constraints, enhancing prediction accuracy and preserving key statistical invariants. Extensive evaluations on diverse simulated and real-world chaotic systems demonstrate that PhyxMamba delivers superior long-term forecasting and faithfully captures essential dynamical invariants from short-term data. This framework opens new avenues for reliably predicting chaotic systems under observation-scarce conditions, with broad implications across climate science, neuroscience, epidemiology, and beyond. Our code is open-source at https://github.com/tsinghua-fib-lab/PhyxMamba.

Neural implicit fields are powerful for representing 3D scenes and generating high-quality novel views, but it remains challenging to use such implicit representations for creating a 3D human avatar with a specific identity and artistic style that can be easily animated. Our proposed method, AvatarCraft, addresses this challenge by using diffusion models to guide the learning of geometry and texture for a neural avatar based on a single text prompt. We carefully design the optimization framework of neural implicit fields, including a coarse-to-fine multi-bounding box training strategy, shape regularization, and diffusion-based constraints, to produce high-quality geometry and texture. Additionally, we make the human avatar animatable by deforming the neural implicit field with an explicit warping field that maps the target human mesh to a template human mesh, both represented using parametric human models. This simplifies animation and reshaping of the generated avatar by controlling pose and shape parameters. Extensive experiments on various text descriptions show that AvatarCraft is effective and robust in creating human avatars and rendering novel views, poses, and shapes. Our project page is: https://avatar-craft.github.io/.

This paper introduces DCT-Net, a novel image translation architecture for few-shot portrait stylization. Given limited style exemplars (sim100), the new architecture can produce high-quality style transfer results with advanced ability to synthesize high-fidelity contents and strong generality to handle complicated scenes (e.g., occlusions and accessories). Moreover, it enables full-body image translation via one elegant evaluation network trained by partial observations (i.e., stylized heads). Few-shot learning based style transfer is challenging since the learned model can easily become overfitted in the target domain, due to the biased distribution formed by only a few training examples. This paper aims to handle the challenge by adopting the key idea of "calibration first, translation later" and exploring the augmented global structure with locally-focused translation. Specifically, the proposed DCT-Net consists of three modules: a content adapter borrowing the powerful prior from source photos to calibrate the content distribution of target samples; a geometry expansion module using affine transformations to release spatially semantic constraints; and a texture translation module leveraging samples produced by the calibrated distribution to learn a fine-grained conversion. Experimental results demonstrate the proposed method's superiority over the state of the art in head stylization and its effectiveness on full image translation with adaptive deformations.

Despite recent advances in multimodal content generation enabled by vision-language models (VLMs), their ability to reason about and generate structured 3D scenes remains largely underexplored. This limitation constrains their utility in spatially grounded tasks such as embodied AI, immersive simulations, and interactive 3D applications. We introduce a new paradigm that enables VLMs to generate, understand, and edit complex 3D environments by injecting a continually evolving spatial context. Constructed from multimodal input, this context consists of three components: a scene portrait that provides a high-level semantic blueprint, a semantically labeled point cloud capturing object-level geometry, and a scene hypergraph that encodes rich spatial relationships, including unary, binary, and higher-order constraints. Together, these components provide the VLM with a structured, geometry-aware working memory that integrates its inherent multimodal reasoning capabilities with structured 3D understanding for effective spatial reasoning. Building on this foundation, we develop an agentic 3D scene generation pipeline in which the VLM iteratively reads from and updates the spatial context. The pipeline features high-quality asset generation with geometric restoration, environment setup with automatic verification, and ergonomic adjustment guided by the scene hypergraph. Experiments show that our framework can handle diverse and challenging inputs, achieving a level of generalization not observed in prior work. Further results demonstrate that injecting spatial context enables VLMs to perform downstream tasks such as interactive scene editing and path planning, suggesting strong potential for spatially intelligent systems in computer graphics, 3D vision, and embodied applications.

Geometry problem solving (GPS) represents a critical frontier in artificial intelligence, with profound applications in education, computer-aided design, and computational graphics. Despite its significance, automating GPS remains challenging due to the dual demands of spatial understanding and rigorous logical reasoning. Recent advances in large models have enabled notable breakthroughs, particularly for SAT-level problems, yet the field remains fragmented across methodologies, benchmarks, and evaluation frameworks. This survey systematically synthesizes GPS advancements through three core dimensions: (1) benchmark construction, (2) textual and diagrammatic parsing, and (3) reasoning paradigms. We further propose a unified analytical paradigm, assess current limitations, and identify emerging opportunities to guide future research toward human-level geometric reasoning, including automated benchmark generation and interpretable neuro-symbolic integration.

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such constraints can be softly imposed via loss function penalties, recent advancements in differentiable physics and optimization improve performance by incorporating PDE-constrained optimization as individual layers in neural networks. This enables a stricter adherence to physical constraints. However, imposing hard constraints significantly increases computational and memory costs, especially for complex dynamical systems. This is because it requires solving an optimization problem over a large number of points in a mesh, representing spatial and temporal discretizations, which greatly increases the complexity of the constraint. To address this challenge, we develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE), which can be used with any neural network architecture. Our approach imposes the constraint over smaller decomposed domains, each of which is solved by an "expert" through differentiable optimization. During training, each expert independently performs a localized backpropagation step by leveraging the implicit function theorem; the independence of each expert allows for parallelization across multiple GPUs. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting for predicting the dynamics of challenging non-linear systems. We also improve training stability and require significantly less computation time during both training and inference stages.

Learning spatial-temporal correspondences in cardiac motion from images is important for understanding the underlying dynamics of cardiac anatomical structures. Many methods explicitly impose smoothness constraints such as the L_2 norm on the displacement vector field (DVF), while usually ignoring biomechanical feasibility in the transformation. Other geometric constraints either regularize specific regions of interest such as imposing incompressibility on the myocardium or introduce additional steps such as training a separate network-based regularizer on physically simulated datasets. In this work, we propose an explicit biomechanics-informed prior as regularization on the predicted DVF in modeling a more generic biomechanically plausible transformation within all cardiac structures without introducing additional training complexity. We validate our methods on two publicly available datasets in the context of 2D MRI data and perform extensive experiments to illustrate the effectiveness and robustness of our proposed methods compared to other competing regularization schemes. Our proposed methods better preserve biomechanical properties by visual assessment and show advantages in segmentation performance using quantitative evaluation metrics. The code is publicly available at https://github.com/Voldemort108X/bioinformed_reg.

This is the first paper in a series of work we have accomplished over the past three years. In this paper, we have constructed a consistent formal plane geometry system. This will serve as a crucial bridge between IMO-level plane geometry challenges and readable AI automated reasoning. Within this formal framework, we have been able to seamlessly integrate modern AI models with our formal system. AI is now capable of providing deductive reasoning solutions to IMO-level plane geometry problems, just like handling other natural languages, and these proofs are readable, traceable, and verifiable. We propose the geometry formalization theory (GFT) to guide the development of the geometry formal system. Based on the GFT, we have established the FormalGeo, which consists of 88 geometric predicates and 196 theorems. It can represent, validate, and solve IMO-level geometry problems. we also have crafted the FGPS (formal geometry problem solver) in Python. It serves as both an interactive assistant for verifying problem-solving processes and an automated problem solver. We've annotated the formalgeo7k and formalgeo-imo datasets. The former contains 6,981 (expand to 133,818 through data augmentation) geometry problems, while the latter includes 18 (expand to 2,627 and continuously increasing) IMO-level challenging geometry problems. All annotated problems include detailed formal language descriptions and solutions. Implementation of the formal system and experiments validate the correctness and utility of the GFT. The backward depth-first search method only yields a 2.42% problem-solving failure rate, and we can incorporate deep learning techniques to achieve lower one. The source code of FGPS and datasets are available at https://github.com/BitSecret/FGPS.

We present a computational framework that transforms single images into 3D physical objects. The visual geometry of a physical object in an image is determined by three orthogonal attributes: mechanical properties, external forces, and rest-shape geometry. Existing single-view 3D reconstruction methods often overlook this underlying composition, presuming rigidity or neglecting external forces. Consequently, the reconstructed objects fail to withstand real-world physical forces, resulting in instability or undesirable deformation -- diverging from their intended designs as depicted in the image. Our optimization framework addresses this by embedding physical compatibility into the reconstruction process. We explicitly decompose the three physical attributes and link them through static equilibrium, which serves as a hard constraint, ensuring that the optimized physical shapes exhibit desired physical behaviors. Evaluations on a dataset collected from Objaverse demonstrate that our framework consistently enhances the physical realism of 3D models over existing methods. The utility of our framework extends to practical applications in dynamic simulations and 3D printing, where adherence to physical compatibility is paramount.

Geometry problem solving is a key area of mathematical reasoning, which is widely involved in many important fields such as education, mathematical ability assessment of artificial intelligence, and multimodal ability assessment. In recent years, the rapid development of deep learning technology, especially the rise of multimodal large language models, has triggered a widespread research boom. This paper provides a survey of the applications of deep learning in geometry problem solving, including (i) a comprehensive summary of the relevant tasks in geometry problem solving; (ii) a thorough review of related deep learning methods; (iii) a detailed analysis of evaluation metrics and methods; and (iv) a critical discussion of the current challenges and future directions that can be explored. Our goal is to provide a comprehensive and practical reference of deep learning for geometry problem solving to promote further developments in this field. We create a continuously updated list of papers on GitHub: https://github.com/majianz/dl4gps.

Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and proving, especially in geometry for its combination of numerical and spatial elements. We introduce TongGeometry, a Euclidean geometry system supporting tree-search-based guided problem proposing and solving. The efficient geometry system establishes the most extensive repository of geometry theorems to date: within the same computational budget as the existing state-of-the-art, TongGeometry discovers 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion exhibiting geometric symmetry. Among them, 10 theorems were proposed to regional mathematical olympiads with 3 of TongGeometry's proposals selected in real competitions, earning spots in a national team qualifying exam or a top civil olympiad in China and the US. Guided by fine-tuned large language models, TongGeometry solved all International Mathematical Olympiad geometry in IMO-AG-30, outperforming gold medalists for the first time. It also surpasses the existing state-of-the-art across a broader spectrum of olympiad-level problems. The full capabilities of the system can be utilized on a consumer-grade machine, making the model more accessible and fostering widespread democratization of its use. By analogy, unlike existing systems that merely solve problems like students, TongGeometry acts like a geometry coach, discovering, presenting, and proving theorems.

In this paper, we present a deep learning-based framework for solving geometric construction problems through visual reasoning, which is useful for automated geometry theorem proving. Constructible problems in geometry often ask for the sequence of straightedge-and-compass constructions to construct a given goal given some initial setup. Our EuclidNet framework leverages the neural network architecture Mask R-CNN to extract the visual features from the initial setup and goal configuration with extra points of intersection, and then generate possible construction steps as intermediary data models that are used as feedback in the training process for further refinement of the construction step sequence. This process is repeated recursively until either a solution is found, in which case we backtrack the path for a step-by-step construction guide, or the problem is identified as unsolvable. Our EuclidNet framework is validated on complex Japanese Sangaku geometry problems, demonstrating its capacity to leverage backtracking for deep visual reasoning of challenging problems.

Geometry problem solving is a well-recognized testbed for evaluating the high-level multi-modal reasoning capability of deep models. In most existing works, two main geometry problems: calculation and proving, are usually treated as two specific tasks, hindering a deep model to unify its reasoning capability on multiple math tasks. However, in essence, these two tasks have similar problem representations and overlapped math knowledge which can improve the understanding and reasoning ability of a deep model on both two tasks. Therefore, we construct a large-scale Unified Geometry problem benchmark, UniGeo, which contains 4,998 calculation problems and 9,543 proving problems. Each proving problem is annotated with a multi-step proof with reasons and mathematical expressions. The proof can be easily reformulated as a proving sequence that shares the same formats with the annotated program sequence for calculation problems. Naturally, we also present a unified multi-task Geometric Transformer framework, Geoformer, to tackle calculation and proving problems simultaneously in the form of sequence generation, which finally shows the reasoning ability can be improved on both two tasks by unifying formulation. Furthermore, we propose a Mathematical Expression Pretraining (MEP) method that aims to predict the mathematical expressions in the problem solution, thus improving the Geoformer model. Experiments on the UniGeo demonstrate that our proposed Geoformer obtains state-of-the-art performance by outperforming task-specific model NGS with over 5.6% and 3.2% accuracies on calculation and proving problems, respectively.

Recent advances in large language models (LLMs) have demonstrated remarkable capabilities across diverse domains, particularly in mathematical reasoning, amid which geometry problem solving remains a challenging area where auxiliary construction plays a enssential role. Existing approaches either achieve suboptimal performance or rely on massive LLMs (e.g., GPT-4o), incurring massive computational costs. We posit that reinforcement learning with verifiable reward (e.g., GRPO) offers a promising direction for training smaller models that effectively combine auxiliary construction with robust geometric reasoning. However, directly applying GRPO to geometric reasoning presents fundamental limitations due to its dependence on unconditional rewards, which leads to indiscriminate and counterproductive auxiliary constructions. To address these challenges, we propose Group Contrastive Policy Optimization (GCPO), a novel reinforcement learning framework featuring two key innovations: (1) Group Contrastive Masking, which adaptively provides positive or negative reward signals for auxiliary construction based on contextual utility, and a (2) length reward that promotes longer reasoning chains. Building on GCPO, we develop GeometryZero, a family of affordable-size geometric reasoning models that judiciously determine when to employ auxiliary construction. Our extensive empirical evaluation across popular geometric benchmarks (Geometry3K, MathVista) demonstrates that GeometryZero models consistently outperform baselines (e.g. GRPO), achieving an average improvement of 4.29% across all benchmarks.

Using Quadrics as the object representation has the benefits of both generality and closed-form projection derivation between image and world spaces. Although numerous constraints have been proposed for dual quadric reconstruction, we found that many of them are imprecise and provide minimal improvements to localization.After scrutinizing the existing constraints, we introduce a concise yet more precise convex hull-based algebraic constraint for object landmarks, which is applied to object reconstruction, frontend pose estimation, and backend bundle adjustment.This constraint is designed to fully leverage precise semantic segmentation, effectively mitigating mismatches between complex-shaped object contours and dual quadrics.Experiments on public datasets demonstrate that our approach is applicable to both monocular and RGB-D SLAM and achieves improved object mapping and localization than existing quadric SLAM methods. The implementation of our method is available at https://github.com/tiev-tongji/convexhull-based-algebraic-constraint.

In document image rectification, there exist rich geometric constraints between the distorted image and the ground truth one. However, such geometric constraints are largely ignored in existing advanced solutions, which limits the rectification performance. To this end, we present DocGeoNet for document image rectification by introducing explicit geometric representation. Technically, two typical attributes of the document image are involved in the proposed geometric representation learning, i.e., 3D shape and textlines. Our motivation arises from the insight that 3D shape provides global unwarping cues for rectifying a distorted document image while overlooking the local structure. On the other hand, textlines complementarily provide explicit geometric constraints for local patterns. The learned geometric representation effectively bridges the distorted image and the ground truth one. Extensive experiments show the effectiveness of our framework and demonstrate the superiority of our DocGeoNet over state-of-the-art methods on both the DocUNet Benchmark dataset and our proposed DIR300 test set. The code is available at https://github.com/fh2019ustc/DocGeoNet.

We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with other additions, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that combines multiple search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry2 to 84% for all geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.

Geometric methods for solving open-world off-road navigation tasks, by learning occupancy and metric maps, provide good generalization but can be brittle in outdoor environments that violate their assumptions (e.g., tall grass). Learning-based methods can directly learn collision-free behavior from raw observations, but are difficult to integrate with standard geometry-based pipelines. This creates an unfortunate conflict -- either use learning and lose out on well-understood geometric navigational components, or do not use it, in favor of extensively hand-tuned geometry-based cost maps. In this work, we reject this dichotomy by designing the learning and non-learning-based components in a way such that they can be effectively combined in a self-supervised manner. Both components contribute to a planning criterion: the learned component contributes predicted traversability as rewards, while the geometric component contributes obstacle cost information. We instantiate and comparatively evaluate our system in both in-distribution and out-of-distribution environments, showing that this approach inherits complementary gains from the learned and geometric components and significantly outperforms either of them. Videos of our results are hosted at https://sites.google.com/view/hybrid-imitative-planning

The aim of this article is to introduce a new methodology for constructing morphings between shapes that have identical topology. This morphing is obtained by deforming a reference shape, through the resolution of a sequence of linear elasticity equations, onto the target shape. In particular, our approach does not assume any knowledge of a boundary parametrization. Furthermore, we demonstrate how constraints can be imposed on specific points, lines and surfaces in the reference domain to ensure alignment with their counterparts in the target domain after morphing. Additionally, we show how the proposed methodology can be integrated in an offline and online paradigm, which is useful in reduced-order modeling scenarii involving variable shapes. This framework facilitates the efficient computation of the morphings in various geometric configurations, thus improving the versatility and applicability of the approach. The methodology is illustrated on the regression problem of the drag and lift coefficients of airfoils of non-parameterized variable shapes.

Creating CAD digital twins from the physical world is crucial for manufacturing, design, and simulation. However, current methods typically rely on costly 3D scanning with labor-intensive post-processing. To provide a user-friendly design process, we explore the problem of reverse engineering from unconstrained real-world CAD images that can be easily captured by users of all experiences. However, the scarcity of real-world CAD data poses challenges in directly training such models. To tackle these challenges, we propose CADCrafter, an image-to-parametric CAD model generation framework that trains solely on synthetic textureless CAD data while testing on real-world images. To bridge the significant representation disparity between images and parametric CAD models, we introduce a geometry encoder to accurately capture diverse geometric features. Moreover, the texture-invariant properties of the geometric features can also facilitate the generalization to real-world scenarios. Since compiling CAD parameter sequences into explicit CAD models is a non-differentiable process, the network training inherently lacks explicit geometric supervision. To impose geometric validity constraints, we employ direct preference optimization (DPO) to fine-tune our model with the automatic code checker feedback on CAD sequence quality. Furthermore, we collected a real-world dataset, comprised of multi-view images and corresponding CAD command sequence pairs, to evaluate our method. Experimental results demonstrate that our approach can robustly handle real unconstrained CAD images, and even generalize to unseen general objects.

We propose GeoUni, the first unified geometry expert model capable of generating problem solutions and diagrams within a single framework in a way that enables the creation of unique and individualized geometry problems. Traditionally, solving geometry problems and generating diagrams have been treated as separate tasks in machine learning, with no models successfully integrating both to support problem creation. However, we believe that mastery in geometry requires frictionless integration of all of these skills, from solving problems to visualizing geometric relationships, and finally, crafting tailored problems. Our extensive experiments demonstrate that GeoUni, with only 1.5B parameters, achieves performance comparable to larger models such as DeepSeek-R1 with 671B parameters in geometric reasoning tasks. GeoUni also excels in generating precise geometric diagrams, surpassing both text-to-image models and unified models, including the GPT-4o image generation. Most importantly, GeoUni is the only model capable of successfully generating textual problems with matching diagrams based on specific knowledge points, thus offering a wider range of capabilities that extend beyond current models.

Automatic math problem solving has recently attracted increasing attention as a long-standing AI benchmark. In this paper, we focus on solving geometric problems, which requires a comprehensive understanding of textual descriptions, visual diagrams, and theorem knowledge. However, the existing methods were highly dependent on handcraft rules and were merely evaluated on small-scale datasets. Therefore, we propose a Geometric Question Answering dataset GeoQA, containing 4,998 geometric problems with corresponding annotated programs, which illustrate the solving process of the given problems. Compared with another publicly available dataset GeoS, GeoQA is 25 times larger, in which the program annotations can provide a practical testbed for future research on explicit and explainable numerical reasoning. Moreover, we introduce a Neural Geometric Solver (NGS) to address geometric problems by comprehensively parsing multimodal information and generating interpretable programs. We further add multiple self-supervised auxiliary tasks on NGS to enhance cross-modal semantic representation. Extensive experiments on GeoQA validate the effectiveness of our proposed NGS and auxiliary tasks. However, the results are still significantly lower than human performance, which leaves large room for future research. Our benchmark and code are released at https://github.com/chen-judge/GeoQA .

3D Gaussian Splatting (3D-GS) has recently attracted great attention with real-time and photo-realistic renderings. This technique typically takes perspective images as input and optimizes a set of 3D elliptical Gaussians by splatting them onto the image planes, resulting in 2D Gaussians. However, applying 3D-GS to panoramic inputs presents challenges in effectively modeling the projection onto the spherical surface of {360^circ} images using 2D Gaussians. In practical applications, input panoramas are often sparse, leading to unreliable initialization of 3D Gaussians and subsequent degradation of 3D-GS quality. In addition, due to the under-constrained geometry of texture-less planes (e.g., walls and floors), 3D-GS struggles to model these flat regions with elliptical Gaussians, resulting in significant floaters in novel views. To address these issues, we propose 360-GS, a novel 360^{circ} Gaussian splatting for a limited set of panoramic inputs. Instead of splatting 3D Gaussians directly onto the spherical surface, 360-GS projects them onto the tangent plane of the unit sphere and then maps them to the spherical projections. This adaptation enables the representation of the projection using Gaussians. We guide the optimization of 360-GS by exploiting layout priors within panoramas, which are simple to obtain and contain strong structural information about the indoor scene. Our experimental results demonstrate that 360-GS allows panoramic rendering and outperforms state-of-the-art methods with fewer artifacts in novel view synthesis, thus providing immersive roaming in indoor scenarios.

Geometry problem solving has attracted much attention in the NLP community recently. The task is challenging as it requires abstract problem understanding and symbolic reasoning with axiomatic knowledge. However, current datasets are either small in scale or not publicly available. Thus, we construct a new large-scale benchmark, Geometry3K, consisting of 3,002 geometry problems with dense annotation in formal language. We further propose a novel geometry solving approach with formal language and symbolic reasoning, called Interpretable Geometry Problem Solver (Inter-GPS). Inter-GPS first parses the problem text and diagram into formal language automatically via rule-based text parsing and neural object detecting, respectively. Unlike implicit learning in existing methods, Inter-GPS incorporates theorem knowledge as conditional rules and performs symbolic reasoning step by step. Also, a theorem predictor is designed to infer the theorem application sequence fed to the symbolic solver for the more efficient and reasonable searching path. Extensive experiments on the Geometry3K and GEOS datasets demonstrate that Inter-GPS achieves significant improvements over existing methods. The project with code and data is available at https://lupantech.github.io/inter-gps.

Modern machine learning algorithms, especially deep learning based techniques, typically involve careful hyperparameter tuning to achieve the best performance. Despite the surge of intense interest in practical techniques like Bayesian optimization and random search based approaches to automating this laborious and compute intensive task, the fundamental learning theoretic complexity of tuning hyperparameters for deep neural networks is poorly understood. Inspired by this glaring gap, we initiate the formal study of hyperparameter tuning complexity in deep learning through a recently introduced data driven setting. We assume that we have a series of deep learning tasks, and we have to tune hyperparameters to do well on average over the distribution of tasks. A major difficulty is that the utility function as a function of the hyperparameter is very volatile and furthermore, it is given implicitly by an optimization problem over the model parameters. To tackle this challenge, we introduce a new technique to characterize the discontinuities and oscillations of the utility function on any fixed problem instance as we vary the hyperparameter; our analysis relies on subtle concepts including tools from differential/algebraic geometry and constrained optimization. This can be used to show that the learning theoretic complexity of the corresponding family of utility functions is bounded. We instantiate our results and provide sample complexity bounds for concrete applications tuning a hyperparameter that interpolates neural activation functions and setting the kernel parameter in graph neural networks.

The process of camera calibration involves estimating the intrinsic and extrinsic parameters, which are essential for accurately performing tasks such as 3D reconstruction, object tracking and augmented reality. In this work, we propose a novel constraints-based loss for measuring the intrinsic (focal length: (f_x, f_y) and principal point: (p_x, p_y)) and extrinsic (baseline: (b), disparity: (d), translation: (t_x, t_y, t_z), and rotation specifically pitch: (theta_p)) camera parameters. Our novel constraints are based on geometric properties inherent in the camera model, including the anatomy of the projection matrix (vanishing points, image of world origin, axis planes) and the orthonormality of the rotation matrix. Thus we proposed a novel Unsupervised Geometric Constraint Loss (UGCL) via a multitask learning framework. Our methodology is a hybrid approach that employs the learning power of a neural network to estimate the desired parameters along with the underlying mathematical properties inherent in the camera projection matrix. This distinctive approach not only enhances the interpretability of the model but also facilitates a more informed learning process. Additionally, we introduce a new CVGL Camera Calibration dataset, featuring over 900 configurations of camera parameters, incorporating 63,600 image pairs that closely mirror real-world conditions. By training and testing on both synthetic and real-world datasets, our proposed approach demonstrates improvements across all parameters when compared to the state-of-the-art (SOTA) benchmarks. The code and the updated dataset can be found here: https://github.com/CVLABLUMS/CVGL-Camera-Calibration

Given a set of dissimilarity measurements amongst data points, determining what metric representation is most "consistent" with the input measurements or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. Existing methods are restricted to specific kinds of metrics or small problem sizes because of the large number of metric constraints in such problems. In this paper, we provide an active set algorithm, Project and Forget, that uses Bregman projections, to solve metric constrained problems with many (possibly exponentially) inequality constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithm converges to the global optimal solution and that the L_2 distance of the current iterate to the optimal solution decays asymptotically at an exponential rate. We demonstrate that using our method we can solve large problem instances of three types of metric constrained problems: general weight correlation clustering, metric nearness, and metric learning; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes.

Although polygon meshes have been a standard representation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning-based applications. In this work, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neural shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the boundary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target surface alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.

A modified LAB algorithm is introduced in this paper. It builds upon the original LAB algorithm (Reddy et al. 2023), which is a socio-inspired algorithm that models competitive and learning behaviours within a group, establishing hierarchical roles. The proposed algorithm incorporates the roulette wheel approach and a reduction factor introducing inter-group competition and iteratively narrowing down the sample space. The algorithm is validated by solving the benchmark test problems from CEC 2005 and CEC 2017. The solutions are validated using standard statistical tests such as two-sided and pairwise signed rank Wilcoxon test and Friedman rank test. The algorithm exhibited improved and superior robustness as well as search space exploration capabilities. Furthermore, a Clustering-Based Search Space Reduction (C-SSR) method is proposed, making the algorithm capable to solve constrained problems. The C-SSR method enables the algorithm to identify clusters of feasible regions, satisfying the constraints and contributing to achieve the optimal solution. This method demonstrates its effectiveness as a potential alternative to traditional constraint handling techniques. The results obtained using the Modified LAB algorithm are then compared with those achieved by other recent metaheuristic algorithms.

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane's compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified F\"{o}ppl's model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain-displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. In the primal problem a linear functional is maximized with respect to displacement functions while enforcing that point-wisely the strain lies in an unbounded closed convex set. The dual problem consists in finding equilibrated stresses that are to minimize a convex integral functional of linear growth defined on the space of Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

LLMs have demonstrated strong mathematical reasoning abilities by leveraging reinforcement learning with long chain-of-thought, yet they continue to struggle with theorem proving due to the lack of clear supervision signals when solely using natural language. Dedicated domain-specific languages like Lean provide clear supervision via formal verification of proofs, enabling effective training through reinforcement learning. In this work, we propose Seed-Prover, a lemma-style whole-proof reasoning model. Seed-Prover can iteratively refine its proof based on Lean feedback, proved lemmas, and self-summarization. To solve IMO-level contest problems, we design three test-time inference strategies that enable both deep and broad reasoning. Seed-Prover proves 78.1% of formalized past IMO problems, saturates MiniF2F, and achieves over 50\% on PutnamBench, outperforming the previous state-of-the-art by a large margin. To address the lack of geometry support in Lean, we introduce a geometry reasoning engine Seed-Geometry, which outperforms previous formal geometry engines. We use these two systems to participate in IMO 2025 and fully prove 5 out of 6 problems. This work represents a significant advancement in automated mathematical reasoning, demonstrating the effectiveness of formal verification with long chain-of-thought reasoning.

Proving geometric theorems constitutes a hallmark of visual reasoning combining both intuitive and logical skills. Therefore, automated theorem proving of Olympiad-level geometry problems is considered a notable milestone in human-level automated reasoning. The introduction of AlphaGeometry, a neuro-symbolic model trained with 100 million synthetic samples, marked a major breakthrough. It solved 25 of 30 International Mathematical Olympiad (IMO) problems whereas the reported baseline based on Wu's method solved only ten. In this note, we revisit the IMO-AG-30 Challenge introduced with AlphaGeometry, and find that Wu's method is surprisingly strong. Wu's method alone can solve 15 problems, and some of them are not solved by any of the other methods. This leads to two key findings: (i) Combining Wu's method with the classic synthetic methods of deductive databases and angle, ratio, and distance chasing solves 21 out of 30 methods by just using a CPU-only laptop with a time limit of 5 minutes per problem. Essentially, this classic method solves just 4 problems less than AlphaGeometry and establishes the first fully symbolic baseline strong enough to rival the performance of an IMO silver medalist. (ii) Wu's method even solves 2 of the 5 problems that AlphaGeometry failed to solve. Thus, by combining AlphaGeometry with Wu's method we set a new state-of-the-art for automated theorem proving on IMO-AG-30, solving 27 out of 30 problems, the first AI method which outperforms an IMO gold medalist.

The topic of stitching images with globally natural structures holds paramount significance. Current methodologies exhibit the ability to preserve local geometric structures, yet fall short in maintaining relationships between these geometric structures. In this paper, we endeavor to safeguard the overall, OBJect-level structures within images based on Global Similarity Prior, while concurrently mitigating distortion and ghosting artifacts with OBJ-GSP. Our approach leverages the Segment Anything Model to extract geometric structures with semantic information, enhancing the algorithm's ability to preserve objects in a manner that aligns more intuitively with human perception. We seek to identify spatial constraints that govern the relationships between various geometric boundaries. Recognizing that multiple geometric boundaries collectively define complete objects, we employ triangular meshes to safeguard not only individual geometric structures but also the overall shapes of objects within the images. Empirical evaluations across multiple image stitching datasets demonstrate that our method establishes a new state-of-the-art benchmark in image stitching. Our implementation and dataset is publicly available at https://github.com/RussRobin/OBJ-GSP .

With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

The recently developed Sora model [1] has exhibited remarkable capabilities in video generation, sparking intense discussions regarding its ability to simulate real-world phenomena. Despite its growing popularity, there is a lack of established metrics to evaluate its fidelity to real-world physics quantitatively. In this paper, we introduce a new benchmark that assesses the quality of the generated videos based on their adherence to real-world physics principles. We employ a method that transforms the generated videos into 3D models, leveraging the premise that the accuracy of 3D reconstruction is heavily contingent on the video quality. From the perspective of 3D reconstruction, we use the fidelity of the geometric constraints satisfied by the constructed 3D models as a proxy to gauge the extent to which the generated videos conform to real-world physics rules. Project page: https://sora-geometrical-consistency.github.io/

With the goal of understanding the visual concepts that CLIP associates with text prompts, we show that the latent space of CLIP can be visualized solely in terms of linear transformations on simple geometric primitives like circles and straight lines. Although existing approaches achieve this by sketch-synthesis-through-optimization, they do so on the space of B\'ezier curves, which exhibit a wastefully large set of structures that they can evolve into, as most of them are non-essential for generating meaningful sketches. We present CLIPDrawX, an algorithm that provides significantly better visualizations for CLIP text embeddings, using only simple primitive shapes like straight lines and circles. This constrains the set of possible outputs to linear transformations on these primitives, thereby exhibiting an inherently simpler mathematical form. The synthesis process of CLIPDrawX can be tracked end-to-end, with each visual concept being explained exclusively in terms of primitives. Implementation will be released upon acceptance. Project Page: https://clipdrawx.github.io/{https://clipdrawx.github.io/}.

Learning 3D shape representation with dense correspondence for deformable objects is a fundamental problem in computer vision. Existing approaches often need additional annotations of specific semantic domain, e.g., skeleton poses for human bodies or animals, which require extra annotation effort and suffer from error accumulation, and they are limited to specific domain. In this paper, we propose a novel self-supervised approach to learn neural implicit shape representation for deformable objects, which can represent shapes with a template shape and dense correspondence in 3D. Our method does not require the priors of skeleton and skinning weight, and only requires a collection of shapes represented in signed distance fields. To handle the large deformation, we constrain the learned template shape in the same latent space with the training shapes, design a new formulation of local rigid constraint that enforces rigid transformation in local region and addresses local reflection issue, and present a new hierarchical rigid constraint to reduce the ambiguity due to the joint learning of template shape and correspondences. Extensive experiments show that our model can represent shapes with large deformations. We also show that our shape representation can support two typical applications, such as texture transfer and shape editing, with competitive performance. The code and models are available at https://iscas3dv.github.io/deformshape

An important feature of many real world facility location problems are capacity limits on the facilities. We show here how capacity constraints make it harder to design strategy proof mechanisms for facility location, but counter-intuitively can improve the guarantees on how well we can approximate the optimal solution.

Geometric diagrams are critical in conveying mathematical and scientific concepts, yet traditional diagram generation methods are often manual and resource-intensive. While text-to-image generation has made strides in photorealistic imagery, creating accurate geometric diagrams remains a challenge due to the need for precise spatial relationships and the scarcity of geometry-specific datasets. This paper presents MagicGeo, a training-free framework for generating geometric diagrams from textual descriptions. MagicGeo formulates the diagram generation process as a coordinate optimization problem, ensuring geometric correctness through a formal language solver, and then employs coordinate-aware generation. The framework leverages the strong language translation capability of large language models, while formal mathematical solving ensures geometric correctness. We further introduce MagicGeoBench, a benchmark dataset of 220 geometric diagram descriptions, and demonstrate that MagicGeo outperforms current methods in both qualitative and quantitative evaluations. This work provides a scalable, accurate solution for automated diagram generation, with significant implications for educational and academic applications.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree tree where the weight of each node is the sum of the weights of its children. A treemap for tree is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of tree and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, polygonal partitions, that use convex polygons, have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio O(depth(tree)). However, depth(tree) still depends on the input tree. Hence we introduce a new type of treemaps, namely orthoconvex treemaps, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio. We also obtain several specialized results for single-level treemaps, that is, treemaps where the input tree has depth~1.

Most 3D object generators focus on aesthetic quality, often neglecting physical constraints necessary in applications. One such constraint is that the 3D object should be self-supporting, i.e., remains balanced under gravity. Prior approaches to generating stable 3D objects used differentiable physics simulators to optimize geometry at test-time, which is slow, unstable, and prone to local optima. Inspired by the literature on aligning generative models to external feedback, we propose Direct Simulation Optimization (DSO), a framework to use the feedback from a (non-differentiable) simulator to increase the likelihood that the 3D generator outputs stable 3D objects directly. We construct a dataset of 3D objects labeled with a stability score obtained from the physics simulator. We can then fine-tune the 3D generator using the stability score as the alignment metric, via direct preference optimization (DPO) or direct reward optimization (DRO), a novel objective, which we introduce, to align diffusion models without requiring pairwise preferences. Our experiments show that the fine-tuned feed-forward generator, using either DPO or DRO objective, is much faster and more likely to produce stable objects than test-time optimization. Notably, the DSO framework works even without any ground-truth 3D objects for training, allowing the 3D generator to self-improve by automatically collecting simulation feedback on its own outputs.

Geometry is a fundamental branch of mathematics and plays a crucial role in evaluating the reasoning capabilities of multimodal large language models (MLLMs). However, existing multimodal mathematics benchmarks mainly focus on plane geometry and largely ignore solid geometry, which requires spatial reasoning and is more challenging than plane geometry. To address this critical gap, we introduce SolidGeo, the first large-scale benchmark specifically designed to evaluate the performance of MLLMs on mathematical reasoning tasks in solid geometry. SolidGeo consists of 3,113 real-world K-12 and competition-level problems, each paired with visual context and annotated with difficulty levels and fine-grained solid geometry categories. Our benchmark covers a wide range of 3D reasoning subjects such as projection, unfolding, spatial measurement, and spatial vector, offering a rigorous testbed for assessing solid geometry. Through extensive experiments, we observe that MLLMs encounter substantial challenges in solid geometry math tasks, with a considerable performance gap relative to human capabilities on SolidGeo. Moreover, we analyze the performance, inference efficiency and error patterns of various models, offering insights into the solid geometric mathematical reasoning capabilities of MLLMs. We hope SolidGeo serves as a catalyst for advancing MLLMs toward deeper geometric reasoning and spatial intelligence.

Geometry problem-solving (GPS), a challenging task requiring both visual comprehension and symbolic reasoning, effectively measures the reasoning capabilities of multimodal large language models (MLLMs). Humans exhibit strong reasoning ability in this task through accurate identification and adaptive application of geometric principles within visual contexts. However, existing benchmarks fail to jointly assess both dimensions of the human-like geometric reasoning mechanism in MLLMs, remaining a critical gap in assessing their ability to tackle GPS. To this end, we introduce GeoSense, the first comprehensive bilingual benchmark designed to systematically evaluate the geometric reasoning abilities of MLLMs through the lens of geometric principles. GeoSense features a five-level hierarchical framework of geometric principles spanning plane and solid geometry, an intricately annotated dataset of 1,789 problems, and an innovative evaluation strategy. Through extensive experiments on GeoSense with various open-source and closed-source MLLMs, we observe that Gemini-2.0-pro-flash performs best, achieving an overall score of 65.3. Our in-depth analysis reveals that the identification and application of geometric principles remain a bottleneck for leading MLLMs, jointly hindering their reasoning abilities. These findings underscore GeoSense's potential to guide future advancements in MLLMs' geometric reasoning capabilities, paving the way for more robust and human-like reasoning in artificial intelligence.

In this work, we introduce CAD-Coder, a novel framework that reformulates text-to-CAD as the generation of CadQuery scripts - a Python-based, parametric CAD language. This representation enables direct geometric validation, a richer modeling vocabulary, and seamless integration with existing LLMs. To further enhance code validity and geometric fidelity, we propose a two-stage learning pipeline: (1) supervised fine-tuning on paired text-CadQuery data, and (2) reinforcement learning with Group Reward Policy Optimization (GRPO), guided by a CAD-specific reward comprising both a geometric reward (Chamfer Distance) and a format reward. We also introduce a chain-of-thought (CoT) planning process to improve model reasoning, and construct a large-scale, high-quality dataset of 110K text-CadQuery-3D model triplets and 1.5K CoT samples via an automated pipeline. Extensive experiments demonstrate that CAD-Coder enables LLMs to generate diverse, valid, and complex CAD models directly from natural language, advancing the state of the art of text-to-CAD generation and geometric reasoning.

The aerodynamic coefficients of aircrafts are significantly impacted by its geometry, especially when the angle of attack (AoA) is large. In the field of aerodynamics, traditional polynomial-based parameterization uses as few parameters as possible to describe the geometry of an airfoil. However, because the 3D geometry of a wing is more complicated than the 2D airfoil, polynomial-based parameterizations have difficulty in accurately representing the entire shape of a wing in 3D space. Existing deep learning-based methods can extract massive latent neural representations for the shape of 2D airfoils or 2D slices of wings. Recent studies highlight that directly taking geometric features as inputs to the neural networks can improve the accuracy of predicted aerodynamic coefficients. Motivated by geometry theory, we propose to incorporate Riemannian geometric features for learning Coefficient of Pressure (CP) distributions on wing surfaces. Our method calculates geometric features (Riemannian metric, connection, and curvature) and further inputs the geometric features, coordinates and flight conditions into a deep learning model to predict the CP distribution. Experimental results show that our method, compared to state-of-the-art Deep Attention Network (DAN), reduces the predicted mean square error (MSE) of CP by an average of 8.41% for the DLR-F11 aircraft test set.

Estimating the pose of an object from a monocular image is an inverse problem fundamental in computer vision. The ill-posed nature of this problem requires incorporating deformation priors to solve it. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a powerful and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach to inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and obtains state-of-the-art performance without the need for offline training.

Machine learning models are often trained to predict the outcome resulting from a human decision. For example, if a doctor decides to test a patient for disease, will the patient test positive? A challenge is that historical decision-making determines whether the outcome is observed: we only observe test outcomes for patients doctors historically tested. Untested patients, for whom outcomes are unobserved, may differ from tested patients along observed and unobserved dimensions. We propose a Bayesian model class which captures this setting. The purpose of the model is to accurately estimate risk for both tested and untested patients. Estimating this model is challenging due to the wide range of possibilities for untested patients. To address this, we propose two domain constraints which are plausible in health settings: a prevalence constraint, where the overall disease prevalence is known, and an expertise constraint, where the human decision-maker deviates from purely risk-based decision-making only along a constrained feature set. We show theoretically and on synthetic data that domain constraints improve parameter inference. We apply our model to a case study of cancer risk prediction, showing that the model's inferred risk predicts cancer diagnoses, its inferred testing policy captures known public health policies, and it can identify suboptimalities in test allocation. Though our case study is in healthcare, our analysis reveals a general class of domain constraints which can improve model estimation in many settings.

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

We study the problem of constrained efficient global optimization, where both the objective and constraints are expensive black-box functions that can be learned with Gaussian processes. We propose CONFIG (CONstrained efFIcient Global Optimization), a simple and effective algorithm to solve it. Under certain regularity assumptions, we show that our algorithm enjoys the same cumulative regret bound as that in the unconstrained case and similar cumulative constraint violation upper bounds. For commonly used Matern and Squared Exponential kernels, our bounds are sublinear and allow us to derive a convergence rate to the optimal solution of the original constrained problem. In addition, our method naturally provides a scheme to declare infeasibility when the original black-box optimization problem is infeasible. Numerical experiments on sampled instances from the Gaussian process, artificial numerical problems, and a black-box building controller tuning problem all demonstrate the competitive performance of our algorithm. Compared to the other state-of-the-art methods, our algorithm significantly improves the theoretical guarantees, while achieving competitive empirical performance.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Image-based 3D generation has vast applications in robotics and gaming, where high-quality, diverse outputs and consistent 3D representations are crucial. However, existing methods have limitations: 3D diffusion models are limited by dataset scarcity and the absence of strong pre-trained priors, while 2D diffusion-based approaches struggle with geometric consistency. We propose a method that leverages 2D diffusion models' implicit 3D reasoning ability while ensuring 3D consistency via Gaussian-splatting-based geometric distillation. Specifically, the proposed Gaussian Splatting Decoder enforces 3D consistency by transforming SV3D latent outputs into an explicit 3D representation. Unlike SV3D, which only relies on implicit 2D representations for video generation, Gaussian Splatting explicitly encodes spatial and appearance attributes, enabling multi-view consistency through geometric constraints. These constraints correct view inconsistencies, ensuring robust geometric consistency. As a result, our approach simultaneously generates high-quality, multi-view-consistent images and accurate 3D models, providing a scalable solution for single-image-based 3D generation and bridging the gap between 2D Diffusion diversity and 3D structural coherence. Experimental results demonstrate state-of-the-art multi-view consistency and strong generalization across diverse datasets. The code will be made publicly available upon acceptance.

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance varepsilon, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as varepsilon varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from varepsilon=0, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of varepsilon. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.

Learning to manipulate dynamic and deformable objects from a single demonstration video holds great promise in terms of scalability. Previous approaches have predominantly focused on either replaying object relationships or actor trajectories. The former often struggles to generalize across diverse tasks, while the latter suffers from data inefficiency. Moreover, both methodologies encounter challenges in capturing invisible physical attributes, such as forces. In this paper, we propose to interpret video demonstrations through Parameterized Symbolic Abstraction Graphs (PSAG), where nodes represent objects and edges denote relationships between objects. We further ground geometric constraints through simulation to estimate non-geometric, visually imperceptible attributes. The augmented PSAG is then applied in real robot experiments. Our approach has been validated across a range of tasks, such as Cutting Avocado, Cutting Vegetable, Pouring Liquid, Rolling Dough, and Slicing Pizza. We demonstrate successful generalization to novel objects with distinct visual and physical properties.

This paper proposes a novel column generation framework for combinatorial software testing. In particular, it combines Mathematical Programming and Constraint Programming in a hybrid decomposition to generate covering arrays. The approach allows generating parameterized test cases with coverage guarantees between parameter interactions of a given application. Compared to exhaustive testing, combinatorial test case generation reduces the number of tests to run significantly. Our column generation algorithm is generic and can accommodate mixed coverage arrays over heterogeneous alphabets. The algorithm is realized in practice as a cloud service and recognized as one of the five winners of the company-wide cloud application challenge at Oracle. The service is currently helping software developers from a range of different product teams in their testing efforts while exposing declarative constraint models and hybrid optimization techniques to a broader audience.

The field of 3D reconstruction from images has rapidly evolved in the past few years, first with the introduction of Neural Radiance Field (NeRF) and more recently with 3D Gaussian Splatting (3DGS). The latter provides a significant edge over NeRF in terms of the training and inference speed, as well as the reconstruction quality. Although 3DGS works well for dense input images, the unstructured point-cloud like representation quickly overfits to the more challenging setup of extremely sparse input images (e.g., 3 images), creating a representation that appears as a jumble of needles from novel views. To address this issue, we propose regularized optimization and depth-based initialization. Our key idea is to introduce a structured Gaussian representation that can be controlled in 2D image space. We then constraint the Gaussians, in particular their position, and prevent them from moving independently during optimization. Specifically, we introduce single and multiview constraints through an implicit convolutional decoder and a total variation loss, respectively. With the coherency introduced to the Gaussians, we further constrain the optimization through a flow-based loss function. To support our regularized optimization, we propose an approach to initialize the Gaussians using monocular depth estimates at each input view. We demonstrate significant improvements compared to the state-of-the-art sparse-view NeRF-based approaches on a variety of scenes.

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics. This is a chapter contribution to the book Machine learning and Algebraic Geometry, edited by A. Kasprzyk et al.

Mathematical geometric reasoning is essential for scientific discovery and educational development, requiring precise logic and rigorous formal verification. While recent advances in Multimodal Large Language Models (MLLMs) have improved reasoning tasks, existing models typically struggle with formal geometric reasoning, particularly when dynamically constructing and verifying auxiliary geometric elements. To address these challenges, we introduce Geoint-R1, a multimodal reasoning framework designed to generate formally verifiable geometric solutions from textual descriptions and visual diagrams. Geoint-R1 uniquely integrates auxiliary elements construction, formal reasoning represented via Lean4, and interactive visualization. To systematically evaluate and advance formal geometric reasoning, we propose the Geoint benchmark, comprising 1,885 rigorously annotated geometry problems across diverse topics such as plane, spatial, and solid geometry. Each problem includes structured textual annotations, precise Lean4 code for auxiliary constructions, and detailed solution steps verified by experts. Extensive experiments demonstrate that Geoint-R1 significantly surpasses existing multimodal and math-specific reasoning models, particularly on challenging problems requiring explicit auxiliary element constructions.

This work presents DAVINCI, a unified architecture for single-stage Computer-Aided Design (CAD) sketch parameterization and constraint inference directly from raster sketch images. By jointly learning both outputs, DAVINCI minimizes error accumulation and enhances the performance of constrained CAD sketch inference. Notably, DAVINCI achieves state-of-the-art results on the large-scale SketchGraphs dataset, demonstrating effectiveness on both precise and hand-drawn raster CAD sketches. To reduce DAVINCI's reliance on large-scale annotated datasets, we explore the efficacy of CAD sketch augmentations. We introduce Constraint-Preserving Transformations (CPTs), i.e. random permutations of the parametric primitives of a CAD sketch that preserve its constraints. This data augmentation strategy allows DAVINCI to achieve reasonable performance when trained with only 0.1% of the SketchGraphs dataset. Furthermore, this work contributes a new version of SketchGraphs, augmented with CPTs. The newly introduced CPTSketchGraphs dataset includes 80 million CPT-augmented sketches, thus providing a rich resource for future research in the CAD sketch domain.

Affine correspondences have received significant attention due to their benefits in tasks like image matching and pose estimation. Existing methods for extracting affine correspondences still have many limitations in terms of performance; thus, exploring a new paradigm is crucial. In this paper, we present a new pipeline designed for extracting accurate affine correspondences by integrating dense matching and geometric constraints. Specifically, a novel extraction framework is introduced, with the aid of dense matching and a novel keypoint scale and orientation estimator. For this purpose, we propose loss functions based on geometric constraints, which can effectively improve accuracy by supervising neural networks to learn feature geometry. The experimental show that the accuracy and robustness of our method outperform the existing ones in image matching tasks. To further demonstrate the effectiveness of the proposed method, we applied it to relative pose estimation. Affine correspondences extracted by our method lead to more accurate poses than the baselines on a range of real-world datasets. The code is available at https://github.com/stilcrad/DenseAffine.

This paper presents the computational challenge on differential geometry and topology that happened within the ICLR 2021 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide creative contributions to the fields of computational geometry and topology through the open-source repositories Geomstats and Giotto-TDA. The challenge attracted 16 teams in its two month duration. This paper describes the design of the challenge and summarizes its main findings.

Character animation in real-world scenarios necessitates a variety of constraints, such as trajectories, key-frames, interactions, etc. Existing methodologies typically treat single or a finite set of these constraint(s) as separate control tasks. They are often specialized, and the tasks they address are rarely extendable or customizable. We categorize these as solutions to the close-set motion control problem. In response to the complexity of practical motion control, we propose and attempt to solve the open-set motion control problem. This problem is characterized by an open and fully customizable set of motion control tasks. To address this, we introduce a new paradigm, programmable motion generation. In this paradigm, any given motion control task is broken down into a combination of atomic constraints. These constraints are then programmed into an error function that quantifies the degree to which a motion sequence adheres to them. We utilize a pre-trained motion generation model and optimize its latent code to minimize the error function of the generated motion. Consequently, the generated motion not only inherits the prior of the generative model but also satisfies the required constraints. Experiments show that we can generate high-quality motions when addressing a wide range of unseen tasks. These tasks encompass motion control by motion dynamics, geometric constraints, physical laws, interactions with scenes, objects or the character own body parts, etc. All of these are achieved in a unified approach, without the need for ad-hoc paired training data collection or specialized network designs. During the programming of novel tasks, we observed the emergence of new skills beyond those of the prior model. With the assistance of large language models, we also achieved automatic programming. We hope that this work will pave the way for the motion control of general AI agents.

This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time R^3times R, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually R^3 times {0}. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions.

Mathematical geometric problem solving (GPS) often requires effective integration of multimodal information and verifiable logical coherence. Despite the fast development of large language models in general problem solving, it remains unresolved regarding with both methodology and benchmarks, especially given the fact that exiting synthetic GPS benchmarks are often not self-verified and contain noise and self-contradicted information due to the illusion of LLMs. In this paper, we propose a scalable data engine called TrustGeoGen for problem generation, with formal verification to provide a principled benchmark, which we believe lays the foundation for the further development of methods for GPS. The engine synthesizes geometric data through four key innovations: 1) multimodal-aligned generation of diagrams, textual descriptions, and stepwise solutions; 2) formal verification ensuring rule-compliant reasoning paths; 3) a bootstrapping mechanism enabling complexity escalation via recursive state generation and 4) our devised GeoExplore series algorithms simultaneously produce multi-solution variants and self-reflective backtracking traces. By formal logical verification, TrustGeoGen produces GeoTrust-200K dataset with guaranteed modality integrity, along with GeoTrust-test testset. Experiments reveal the state-of-the-art models achieve only 49.17\% accuracy on GeoTrust-test, demonstrating its evaluation stringency. Crucially, models trained on GeoTrust achieve OOD generalization on GeoQA, significantly reducing logical inconsistencies relative to pseudo-label annotated by OpenAI-o1. Our code is available at https://github.com/Alpha-Innovator/TrustGeoGen

We report on a new, simple, modular, and flexible approach for automated generation of illustrations for (readable) synthetic geometry proofs. The underlying proofs are generated using the Larus automated prover for coherent logic, and corresponding illustrations are generated in the GCLC language. Animated illustrations are also supported.

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

A growing literature in computational neuroscience leverages gradient descent and learning algorithms that approximate it to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying assumption: the choice of distance for synaptic changes - i.e. the geometry of synaptic plasticity. Gradient descent assumes that the distance is Euclidean, but many other distances are possible, and there is no reason that biology necessarily uses Euclidean geometry. Here, using the theoretical tools provided by mirror descent, we show that the distribution of synaptic weights will depend on the geometry of synaptic plasticity. We use these results to show that experimentally-observed log-normal weight distributions found in several brain areas are not consistent with standard gradient descent (i.e. a Euclidean geometry), but rather with non-Euclidean distances. Finally, we show that it should be possible to experimentally test for different synaptic geometries by comparing synaptic weight distributions before and after learning. Overall, our work shows that the current paradigm in theoretical work on synaptic plasticity that assumes Euclidean synaptic geometry may be misguided and that it should be possible to experimentally determine the true geometry of synaptic plasticity in the brain.

In this paper, we present Gaussian Splatting based text-to-3D generation (GSGEN), a novel approach for generating high-quality 3D objects. Previous methods suffer from inaccurate geometry and limited fidelity due to the absence of 3D prior and proper representation. We leverage 3D Gaussian Splatting, a recent state-of-the-art representation, to address existing shortcomings by exploiting the explicit nature that enables the incorporation of 3D prior. Specifically, our method adopts a progressive optimization strategy, which includes a geometry optimization stage and an appearance refinement stage. In geometry optimization, a coarse representation is established under a 3D geometry prior along with the ordinary 2D SDS loss, ensuring a sensible and 3D-consistent rough shape. Subsequently, the obtained Gaussians undergo an iterative refinement to enrich details. In this stage, we increase the number of Gaussians by compactness-based densification to enhance continuity and improve fidelity. With these designs, our approach can generate 3D content with delicate details and more accurate geometry. Extensive evaluations demonstrate the effectiveness of our method, especially for capturing high-frequency components. Video results are provided at https://gsgen3d.github.io. Our code is available at https://github.com/gsgen3d/gsgen

Large language models have shown impressive results for multi-hop mathematical reasoning when the input question is only textual. Many mathematical reasoning problems, however, contain both text and image. With the ever-increasing adoption of vision language models (VLMs), understanding their reasoning abilities for such problems is crucial. In this paper, we evaluate the reasoning capabilities of VLMs along various axes through the lens of geometry problems. We procedurally create a synthetic dataset of geometry questions with controllable difficulty levels along multiple axes, thus enabling a systematic evaluation. The empirical results obtained using our benchmark for state-of-the-art VLMs indicate that these models are not as capable in subjects like geometry (and, by generalization, other topics requiring similar reasoning) as suggested by previous benchmarks. This is made especially clear by the construction of our benchmark at various depth levels, since solving higher-depth problems requires long chains of reasoning rather than additional memorized knowledge. We release the dataset for further research in this area.

This paper aims to develop an accurate 3D geometry representation of satellite images using satellite-ground image pairs. Our focus is on the challenging problem of 3D-aware ground-views synthesis from a satellite image. We draw inspiration from the density field representation used in volumetric neural rendering and propose a new approach, called Sat2Density. Our method utilizes the properties of ground-view panoramas for the sky and non-sky regions to learn faithful density fields of 3D scenes in a geometric perspective. Unlike other methods that require extra depth information during training, our Sat2Density can automatically learn accurate and faithful 3D geometry via density representation without depth supervision. This advancement significantly improves the ground-view panorama synthesis task. Additionally, our study provides a new geometric perspective to understand the relationship between satellite and ground-view images in 3D space.

Plane geometry problem solving (PGPS) has recently gained significant attention as a benchmark to assess the multi-modal reasoning capabilities of large vision-language models. Despite the growing interest in PGPS, the research community still lacks a comprehensive overview that systematically synthesizes recent work in PGPS. To fill this gap, we present a survey of existing PGPS studies. We first categorize PGPS methods into an encoder-decoder framework and summarize the corresponding output formats used by their encoders and decoders. Subsequently, we classify and analyze these encoders and decoders according to their architectural designs. Finally, we outline major challenges and promising directions for future research. In particular, we discuss the hallucination issues arising during the encoding phase within encoder-decoder architectures, as well as the problem of data leakage in current PGPS benchmarks.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

Multimodal large language models (MLLMs) have made rapid progress in recent years, yet continue to struggle with low-level visual perception (LLVP) -- particularly the ability to accurately describe the geometric details of an image. This capability is crucial for applications in areas such as robotics, medical image analysis, and manufacturing. In this paper, we first introduce Geoperception, a benchmark designed to evaluate an MLLM's ability to accurately transcribe 2D geometric information from an image. Using this benchmark, we demonstrate the limitations of leading MLLMs, and then conduct a comprehensive empirical study to explore strategies for improving their performance on geometric tasks. Our findings highlight the benefits of certain model architectures, training techniques, and data strategies, including the use of high-fidelity synthetic data and multi-stage training with a data curriculum. Notably, we find that a data curriculum enables models to learn challenging geometry understanding tasks which they fail to learn from scratch. Leveraging these insights, we develop Euclid, a family of models specifically optimized for strong low-level geometric perception. Although purely trained on synthetic multimodal data, Euclid shows strong generalization ability to novel geometry shapes. For instance, Euclid outperforms the best closed-source model, Gemini-1.5-Pro, by up to 58.56% on certain Geoperception benchmark tasks and 10.65% on average across all tasks.

Multimodal large language models (MLLMs) have made significant progress in integrating visual and linguistic understanding. Existing benchmarks typically focus on high-level semantic capabilities, such as scene understanding and visual reasoning, but often overlook a crucial, foundational ability: geometric perception. Geometric perception involves understanding geometric shapes, structures, and spatial relationships, which are essential for supporting higher-level semantic tasks. Despite its importance, this capability remains underexplored in current MLLM research. To address this gap, we introduce GePBench, a novel benchmark designed to assess the geometric perception abilities of MLLMs. Our extensive evaluations reveal that current state-of-the-art MLLMs exhibit significant deficiencies in geometric perception tasks. Furthermore, we show that models trained with GePBench data demonstrate substantial improvements on a wide range of benchmark tasks, highlighting the critical role of geometric perception in enabling advanced multimodal applications. Our code and datasets will be publicly available.

Existing Large Multimodal Models (LMMs) struggle with mathematical geometric reasoning due to a lack of high-quality image-text paired data. Current geometric data generation approaches, which apply preset templates to generate geometric data or use Large Language Models (LLMs) to rephrase questions and answers (Q&A), unavoidably limit data accuracy and diversity. To synthesize higher-quality data, we propose a two-stage Reverse Chain-of-Thought (R-CoT) geometry problem generation pipeline. First, we introduce GeoChain to produce high-fidelity geometric images and corresponding descriptions highlighting relations among geometric elements. We then design a Reverse A&Q method that reasons step-by-step based on the descriptions and generates questions in reverse from the reasoning results. Experiments demonstrate that the proposed method brings significant and consistent improvements on multiple LMM baselines, achieving new performance records in the 2B, 7B, and 8B settings. Notably, R-CoT-8B significantly outperforms previous state-of-the-art open-source mathematical models by 16.6% on MathVista and 9.2% on GeoQA, while also surpassing the closed-source model GPT-4o by an average of 13% across both datasets. The code is available at https://github.com/dle666/R-CoT.

This paper introduces an approach to endow generative diffusion processes the ability to satisfy and certify compliance with constraints and physical principles. The proposed method recast the traditional sampling process of generative diffusion models as a constrained optimization problem, steering the generated data distribution to remain within a specified region to ensure adherence to the given constraints. These capabilities are validated on applications featuring both convex and challenging, non-convex, constraints as well as ordinary differential equations, in domains spanning from synthesizing new materials with precise morphometric properties, generating physics-informed motion, optimizing paths in planning scenarios, and human motion synthesis.

This paper introduces a new parameterization of deep neural networks (both fully-connected and convolutional) with guaranteed ell^2 Lipschitz bounds, i.e. limited sensitivity to input perturbations. The Lipschitz guarantees are equivalent to the tightest-known bounds based on certification via a semidefinite program (SDP). We provide a ``direct'' parameterization, i.e., a smooth mapping from mathbb R^N onto the set of weights satisfying the SDP-based bound. Moreover, our parameterization is complete, i.e. a neural network satisfies the SDP bound if and only if it can be represented via our parameterization. This enables training using standard gradient methods, without any inner approximation or computationally intensive tasks (e.g. projections or barrier terms) for the SDP constraint. The new parameterization can equivalently be thought of as either a new layer type (the sandwich layer), or a novel parameterization of standard feedforward networks with parameter sharing between neighbouring layers. A comprehensive set of experiments on image classification shows that sandwich layers outperform previous approaches on both empirical and certified robust accuracy. Code is available at https://github.com/acfr/LBDN.

Computer-aided design (CAD) significantly enhances the efficiency, accuracy, and innovation of design processes by enabling precise 2D and 3D modeling, extensive analysis, and optimization. Existing methods for creating CAD models rely on latent vectors or point clouds, which are difficult to obtain and costly to store. Recent advances in Multimodal Large Language Models (MLLMs) have inspired researchers to use natural language instructions and images for CAD model construction. However, these models still struggle with inferring accurate 3D spatial location and orientation, leading to inaccuracies in determining the spatial 3D starting points and extrusion directions for constructing geometries. This work introduces CAD-GPT, a CAD synthesis method with spatial reasoning-enhanced MLLM that takes either a single image or a textual description as input. To achieve precise spatial inference, our approach introduces a 3D Modeling Spatial Mechanism. This method maps 3D spatial positions and 3D sketch plane rotation angles into a 1D linguistic feature space using a specialized spatial unfolding mechanism, while discretizing 2D sketch coordinates into an appropriate planar space to enable precise determination of spatial starting position, sketch orientation, and 2D sketch coordinate translations. Extensive experiments demonstrate that CAD-GPT consistently outperforms existing state-of-the-art methods in CAD model synthesis, both quantitatively and qualitatively.

Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in R^d. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a rm polylog(Delta)-approximation algorithm for embedding n-point ultrametrics into R^d with minimum distortion, where Delta denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

In this paper, we deal with the gluing of two surfaces, where the gluing locus is assumed to be a curve. We consider a moving frame along the gluing locus, and define developable surfaces with respect to the frame. Considering geometric properties of these developable surfaces, we study the geometry of gluing two surfaces.

We introduce a diffusion-based framework that performs aligned novel view image and geometry generation via a warping-and-inpainting methodology. Unlike prior methods that require dense posed images or pose-embedded generative models limited to in-domain views, our method leverages off-the-shelf geometry predictors to predict partial geometries viewed from reference images, and formulates novel-view synthesis as an inpainting task for both image and geometry. To ensure accurate alignment between generated images and geometry, we propose cross-modal attention distillation, where attention maps from the image diffusion branch are injected into a parallel geometry diffusion branch during both training and inference. This multi-task approach achieves synergistic effects, facilitating geometrically robust image synthesis as well as well-defined geometry prediction. We further introduce proximity-based mesh conditioning to integrate depth and normal cues, interpolating between point cloud and filtering erroneously predicted geometry from influencing the generation process. Empirically, our method achieves high-fidelity extrapolative view synthesis on both image and geometry across a range of unseen scenes, delivers competitive reconstruction quality under interpolation settings, and produces geometrically aligned colored point clouds for comprehensive 3D completion. Project page is available at https://cvlab-kaist.github.io/MoAI.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

Learning about the three-dimensional world from two-dimensional images is a fundamental problem in computer vision. An ideal neural network architecture for such tasks would leverage the fact that objects can be rotated and translated in three dimensions to make predictions about novel images. However, imposing SO(3)-equivariance on two-dimensional inputs is difficult because the group of three-dimensional rotations does not have a natural action on the two-dimensional plane. Specifically, it is possible that an element of SO(3) will rotate an image out of plane. We show that an algorithm that learns a three-dimensional representation of the world from two dimensional images must satisfy certain geometric consistency properties which we formulate as SO(2)-equivariance constraints. We use the induced and restricted representations of SO(2) on SO(3) to construct and classify architectures which satisfy these geometric consistency constraints. We prove that any architecture which respects said consistency constraints can be realized as an instance of our construction. We show that three previously proposed neural architectures for 3D pose prediction are special cases of our construction. We propose a new algorithm that is a learnable generalization of previously considered methods. We test our architecture on three pose predictions task and achieve SOTA results on both the PASCAL3D+ and SYMSOL pose estimation tasks.

This note describes the development of an exact solver for Minimal Directed Feedback Vertex Set as part of the PACE 2022 competition. The solver is powered largely by aggressively trying to reduce the DFVS problem to a Minimal Cover problem, and applying reduction rules adapted from Vertex Cover literature. The resulting problem is solved as an Integer Linear Program (ILP) using SCIP. The resulting solver performed the second-best in the competition, although a bug at submission time disqualified it. As an additional note, we describe a new vertex cover reduction generalizing the Desk reduction rule.

Mesh generation has become a critical topic in recent years, forming the foundation of all 3D objects used across various applications, such as virtual reality, gaming, and 3D printing. With advancements in computational resources and machine learning, neural networks have emerged as powerful tools for generating high-quality 3D object representations, enabling accurate scene and object reconstructions. Despite these advancements, many methods produce meshes that lack realism or exhibit geometric and textural flaws, necessitating additional processing to improve their quality. This research introduces a convex optimization programming called disciplined convex programming to enhance existing meshes by refining their texture and geometry with a conic solver. By focusing on a sparse set of point clouds from both the original and target meshes, this method demonstrates significant improvements in mesh quality with minimal data requirements. To evaluate the approach, the classical dolphin mesh dataset from Facebook AI was used as a case study, with optimization performed using the CVXPY library. The results reveal promising potential for streamlined and effective mesh refinement.

Geometric deep learning, i.e., designing neural networks to handle the ubiquitous geometric data such as point clouds and graphs, have achieved great successes in the last decade. One critical inductive bias is that the model can maintain invariance towards various transformations such as translation, rotation, and scaling. The existing graph neural network (GNN) approaches can only maintain permutation-invariance, failing to guarantee invariance with respect to other transformations. Besides GNNs, other works design sophisticated transformation-invariant layers, which are computationally expensive and difficult to be extended. To solve this problem, we revisit why the existing neural networks cannot maintain transformation invariance when handling geometric data. Our findings show that transformation-invariant and distance-preserving initial representations are sufficient to achieve transformation invariance rather than needing sophisticated neural layer designs. Motivated by these findings, we propose Transformation Invariant Neural Networks (TinvNN), a straightforward and general framework for geometric data. Specifically, we realize transformation-invariant and distance-preserving initial point representations by modifying multi-dimensional scaling before feeding the representations into neural networks. We prove that TinvNN can strictly guarantee transformation invariance, being general and flexible enough to be combined with the existing neural networks. Extensive experimental results on point cloud analysis and combinatorial optimization demonstrate the effectiveness and general applicability of our proposed method. Based on the experimental results, we advocate that TinvNN should be considered a new starting point and an essential baseline for further studies of transformation-invariant geometric deep learning.

We solve a challenging yet practically useful variant of 3D Bin Packing Problem (3D-BPP). In our problem, the agent has limited information about the items to be packed into the bin, and an item must be packed immediately after its arrival without buffering or readjusting. The item's placement also subjects to the constraints of collision avoidance and physical stability. We formulate this online 3D-BPP as a constrained Markov decision process. To solve the problem, we propose an effective and easy-to-implement constrained deep reinforcement learning (DRL) method under the actor-critic framework. In particular, we introduce a feasibility predictor to predict the feasibility mask for the placement actions and use it to modulate the action probabilities output by the actor during training. Such supervisions and transformations to DRL facilitate the agent to learn feasible policies efficiently. Our method can also be generalized e.g., with the ability to handle lookahead or items with different orientations. We have conducted extensive evaluation showing that the learned policy significantly outperforms the state-of-the-art methods. A user study suggests that our method attains a human-level performance.

We introduce PhysGaussian, a new method that seamlessly integrates physically grounded Newtonian dynamics within 3D Gaussians to achieve high-quality novel motion synthesis. Employing a custom Material Point Method (MPM), our approach enriches 3D Gaussian kernels with physically meaningful kinematic deformation and mechanical stress attributes, all evolved in line with continuum mechanics principles. A defining characteristic of our method is the seamless integration between physical simulation and visual rendering: both components utilize the same 3D Gaussian kernels as their discrete representations. This negates the necessity for triangle/tetrahedron meshing, marching cubes, "cage meshes," or any other geometry embedding, highlighting the principle of "what you see is what you simulate (WS^2)." Our method demonstrates exceptional versatility across a wide variety of materials--including elastic entities, metals, non-Newtonian fluids, and granular materials--showcasing its strong capabilities in creating diverse visual content with novel viewpoints and movements. Our project page is at: https://xpandora.github.io/PhysGaussian/

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

Modern reinforcement learning (RL) systems capture deep truths about general, human problem-solving. In domains where new data can be simulated cheaply, these systems uncover sequential decision-making policies that far exceed the ability of any human. Society faces many problems whose solutions require this skill, but they are often in domains where new data cannot be cheaply simulated. In such scenarios, we can learn simulators from existing data, but these will only ever be approximately correct, and can be pathologically incorrect when queried outside of their training distribution. As a result, a misalignment between the environments in which we train our agents and the real-world in which we wish to deploy our agents is inevitable. Dealing with this misalignment is the primary concern of zero-shot reinforcement learning, a problem setting where the agent must generalise to a new task or domain with zero practice shots. Whilst impressive progress has been made on methods that perform zero-shot RL in idealised settings, new work is needed if these results are to be replicated in real-world settings. In this thesis, we argue that doing so requires us to navigate (at least) three constraints. First, the data quality constraint: real-world datasets are small and homogeneous. Second, the observability constraint: states, dynamics and rewards in the real-world are often only partially observed. And third, the data availability constraint: a priori access to data cannot always be assumed. This work proposes a suite of methods that perform zero-shot RL subject to these constraints. In a series of empirical studies we expose the failings of existing methods, and justify our techniques for remedying them. We believe these designs take us a step closer to RL methods that can be deployed to solve real-world problems.

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.

We present an algorithm for reconstructing dense, geometrically consistent depth for all pixels in a monocular video. We leverage a conventional structure-from-motion reconstruction to establish geometric constraints on pixels in the video. Unlike the ad-hoc priors in classical reconstruction, we use a learning-based prior, i.e., a convolutional neural network trained for single-image depth estimation. At test time, we fine-tune this network to satisfy the geometric constraints of a particular input video, while retaining its ability to synthesize plausible depth details in parts of the video that are less constrained. We show through quantitative validation that our method achieves higher accuracy and a higher degree of geometric consistency than previous monocular reconstruction methods. Visually, our results appear more stable. Our algorithm is able to handle challenging hand-held captured input videos with a moderate degree of dynamic motion. The improved quality of the reconstruction enables several applications, such as scene reconstruction and advanced video-based visual effects.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Recent progress in Sign Language Translation (SLT) has focussed primarily on improving the representational capacity of large language models to incorporate Sign Language features. This work explores an alternative direction: enhancing the geometric properties of skeletal representations themselves. We propose Geo-Sign, a method that leverages the properties of hyperbolic geometry to model the hierarchical structure inherent in sign language kinematics. By projecting skeletal features derived from Spatio-Temporal Graph Convolutional Networks (ST-GCNs) into the Poincar\'e ball model, we aim to create more discriminative embeddings, particularly for fine-grained motions like finger articulations. We introduce a hyperbolic projection layer, a weighted Fr\'echet mean aggregation scheme, and a geometric contrastive loss operating directly in hyperbolic space. These components are integrated into an end-to-end translation framework as a regularisation function, to enhance the representations within the language model. This work demonstrates the potential of hyperbolic geometry to improve skeletal representations for Sign Language Translation, improving on SOTA RGB methods while preserving privacy and improving computational efficiency. Code available here: https://github.com/ed-fish/geo-sign.

The quantum geometric tensor (QGT) is a fundamental quantity for characterizing the geometric properties of quantum states and plays an essential role in elucidating various physical phenomena. The traditional QGT, defined only for pure states, has limited applicability in realistic scenarios where mixed states are common. To address this limitation, we generalize the definition of the QGT to mixed states using the purification bundle and the covariant derivative. Notably, our proposed definition reduces to the traditional QGT when mixed states approach pure states. In our framework, the real and imaginary parts of this generalized QGT correspond to the Bures metric and the mean gauge curvature, respectively, endowing it with a broad range of potential applications. Additionally, using our proposed mixed-state QGT (MSQGT), we derive the geodesic equation applicable to mixed states. This work establishes a unified framework for the geometric analysis of both pure and mixed states, thereby deepening our understanding of the geometric properties of quantum states.

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly available tool to manipulate knot diagrams in a real-time, interactive way is yet to be developed. We introduce a method of operating on the geometry of the knot diagram itself without any underlying three-dimensional structure that can underpin such an application. This allows us to directly interact with vector graphics knot diagrams while at the same time computing knot invariants in ways proposed by previous work. An implementation of this method is provided.

Effective scheduling under tight resource, timing, and operational constraints underpins large-scale planning across sectors such as capital projects, manufacturing, logistics, and IT fleet transitions. However, the reliability of large language models (LLMs) when reasoning under high-constraint regimes is insufficiently characterized. To address this gap, we present R-ConstraintBench, a scalable framework that evaluates models on Resource-Constrained Project Scheduling Problems (RCPSP), an NP-Complete feasibility class, while difficulty increases via linear growth in constraints. R-ConstraintBench incrementally increases non-redundant precedence constraints in Directed Acyclic Graphs (DAGs) and then introduces downtime, temporal windows, and disjunctive constraints. As an illustrative example, we instantiate the benchmark in a data center migration setting and evaluate multiple LLMs using feasibility and error analysis, identifying degradation thresholds and constraint types most associated with failure. Empirically, strong models are near-ceiling on precedence-only DAGs, but feasibility performance collapses when downtime, temporal windows, and disjunctive constraints interact, implicating constraint interaction, not graph depth, as the principal bottleneck. Performance on clean synthetic ramps also does not guarantee transfer to domain-grounded scenarios, underscoring limited generalization.

The abundance of data has given machine learning considerable momentum in natural sciences and engineering, though modeling of physical processes is often difficult. A particularly tough problem is the efficient representation of geometric boundaries. Triangularized geometric boundaries are well understood and ubiquitous in engineering applications. However, it is notoriously difficult to integrate them into machine learning approaches due to their heterogeneity with respect to size and orientation. In this work, we introduce an effective theory to model particle-boundary interactions, which leads to our new Boundary Graph Neural Networks (BGNNs) that dynamically modify graph structures to obey boundary conditions. The new BGNNs are tested on complex 3D granular flow processes of hoppers, rotating drums and mixers, which are all standard components of modern industrial machinery but still have complicated geometry. BGNNs are evaluated in terms of computational efficiency as well as prediction accuracy of particle flows and mixing entropies. BGNNs are able to accurately reproduce 3D granular flows within simulation uncertainties over hundreds of thousands of simulation timesteps. Most notably, in our experiments, particles stay within the geometric objects without using handcrafted conditions or restrictions.

Multi-camera 3D object detection for autonomous driving is a challenging problem that has garnered notable attention from both academia and industry. An obstacle encountered in vision-based techniques involves the precise extraction of geometry-conscious features from RGB images. Recent approaches have utilized geometric-aware image backbones pretrained on depth-relevant tasks to acquire spatial information. However, these approaches overlook the critical aspect of view transformation, resulting in inadequate performance due to the misalignment of spatial knowledge between the image backbone and view transformation. To address this issue, we propose a novel geometric-aware pretraining framework called GAPretrain. Our approach incorporates spatial and structural cues to camera networks by employing the geometric-rich modality as guidance during the pretraining phase. The transference of modal-specific attributes across different modalities is non-trivial, but we bridge this gap by using a unified bird's-eye-view (BEV) representation and structural hints derived from LiDAR point clouds to facilitate the pretraining process. GAPretrain serves as a plug-and-play solution that can be flexibly applied to multiple state-of-the-art detectors. Our experiments demonstrate the effectiveness and generalization ability of the proposed method. We achieve 46.2 mAP and 55.5 NDS on the nuScenes val set using the BEVFormer method, with a gain of 2.7 and 2.1 points, respectively. We also conduct experiments on various image backbones and view transformations to validate the efficacy of our approach. Code will be released at https://github.com/OpenDriveLab/BEVPerception-Survey-Recipe.

We tackle the problem of estimating a Manhattan frame, i.e. three orthogonal vanishing points, and the unknown focal length of the camera, leveraging a prior vertical direction. The direction can come from an Inertial Measurement Unit that is a standard component of recent consumer devices, e.g., smartphones. We provide an exhaustive analysis of minimal line configurations and derive two new 2-line solvers, one of which does not suffer from singularities affecting existing solvers. Additionally, we design a new non-minimal method, running on an arbitrary number of lines, to boost the performance in local optimization. Combining all solvers in a hybrid robust estimator, our method achieves increased accuracy even with a rough prior. Experiments on synthetic and real-world datasets demonstrate the superior accuracy of our method compared to the state of the art, while having comparable runtimes. We further demonstrate the applicability of our solvers for relative rotation estimation. The code is available at https://github.com/cvg/VP-Estimation-with-Prior-Gravity.

Twenty-seven years ago, E. Freuder highlighted that "Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it". Nowadays, CP users have great modeling tools available (like Minizinc and CPMpy), allowing them to formulate the problem and then let a solver do the rest of the job, getting closer to the stated goal. However, this still requires the CP user to know the formalism and respect it. Another significant challenge lies in the expertise required to effectively model combinatorial problems. All this limits the wider adoption of CP. In this position paper, we investigate a possible approach to leverage pre-trained Large Language Models to extract models from textual problem descriptions. More specifically, we take inspiration from the Natural Language Processing for Optimization (NL4OPT) challenge and present early results with a decomposition-based prompting approach to GPT Models.

Computer-aided design (CAD) tools are utilized in the manufacturing industry for modeling everything from cups to spacecraft. These programs are complex to use and typically require years of training and experience to master. Structured and well-constrained 2D sketches and 3D constructions are crucial components of CAD modeling. A well-executed CAD model can be seamlessly integrated into the manufacturing process, thereby enhancing production efficiency. Deep generative models of 3D shapes and 3D object reconstruction models have garnered significant research interest. However, most of these models produce discrete forms of 3D objects that are not editable. Moreover, the few models based on CAD operations often have substantial input restrictions. In this work, we fine-tuned pre-trained models to create OpenECAD models (0.55B, 0.89B, 2.4B and 3.1B), leveraging the visual, logical, coding, and general capabilities of visual language models. OpenECAD models can process images of 3D designs as input and generate highly structured 2D sketches and 3D construction commands, ensuring that the designs are editable. These outputs can be directly used with existing CAD tools' APIs to generate project files. To train our network, we created a series of OpenECAD datasets. These datasets are derived from existing public CAD datasets, adjusted and augmented to meet the specific requirements of vision language model (VLM) training. Additionally, we have introduced an approach that utilizes dependency relationships to define and generate sketches, further enriching the content and functionality of the datasets.

Adversarial threats against LLMs are escalating faster than current defenses can adapt. We expose a critical geometric blind spot in alignment: adversarial prompts exploit latent camouflage, embedding perilously close to the safe representation manifold while encoding unsafe intent thereby evading surface level defenses like Direct Preference Optimization (DPO), which remain blind to the latent geometry. We introduce ALKALI, the first rigorously curated adversarial benchmark and the most comprehensive to date spanning 9,000 prompts across three macro categories, six subtypes, and fifteen attack families. Evaluation of 21 leading LLMs reveals alarmingly high Attack Success Rates (ASRs) across both open and closed source models, exposing an underlying vulnerability we term latent camouflage, a structural blind spot where adversarial completions mimic the latent geometry of safe ones. To mitigate this vulnerability, we introduce GRACE - Geometric Representation Aware Contrastive Enhancement, an alignment framework coupling preference learning with latent space regularization. GRACE enforces two constraints: latent separation between safe and adversarial completions, and adversarial cohesion among unsafe and jailbreak behaviors. These operate over layerwise pooled embeddings guided by a learned attention profile, reshaping internal geometry without modifying the base model, and achieve up to 39% ASR reduction. Moreover, we introduce AVQI, a geometry aware metric that quantifies latent alignment failure via cluster separation and compactness. AVQI reveals when unsafe completions mimic the geometry of safe ones, offering a principled lens into how models internally encode safety. We make the code publicly available at https://anonymous.4open.science/r/alkali-B416/README.md.

Constraint Programming (CP) is a declarative programming paradigm that allows for modeling and solving combinatorial optimization problems, such as the Job-Shop Scheduling Problem (JSSP). While CP solvers manage to find optimal or near-optimal solutions for small instances, they do not scale well to large ones, i.e., they require long computation times or yield low-quality solutions. Therefore, real-world scheduling applications often resort to fast, handcrafted, priority-based dispatching heuristics to find a good initial solution and then refine it using optimization methods. This paper proposes a novel end-to-end approach to solving scheduling problems by means of CP and Reinforcement Learning (RL). In contrast to previous RL methods, tailored for a given problem by including procedural simulation algorithms, complex feature engineering, or handcrafted reward functions, our neural-network architecture and training algorithm merely require a generic CP encoding of some scheduling problem along with a set of small instances. Our approach leverages existing CP solvers to train an agent learning a Priority Dispatching Rule (PDR) that generalizes well to large instances, even from separate datasets. We evaluate our method on seven JSSP datasets from the literature, showing its ability to find higher-quality solutions for very large instances than obtained by static PDRs and by a CP solver within the same time limit.

In computer-aided design (CAD) systems, 2D line drawings are commonly used to illustrate 3D object designs. To reconstruct the 3D models depicted by a single 2D line drawing, an important key is finding the edge loops in the line drawing which correspond to the actual faces of the 3D object. In this paper, we approach the classical problem of face identification from a novel data-driven point of view. We cast it as a sequence generation problem: starting from an arbitrary edge, we adopt a variant of the popular Transformer model to predict the edges associated with the same face in a natural order. This allows us to avoid searching the space of all possible edge loops with various hand-crafted rules and heuristics as most existing methods do, deal with challenging cases such as curved surfaces and nested edge loops, and leverage additional cues such as face types. We further discuss how possibly imperfect predictions can be used for 3D object reconstruction.

Recent advances in Hierarchical Multi-label Classification (HMC), particularly neurosymbolic-based approaches, have demonstrated improved consistency and accuracy by enforcing constraints on a neural model during training. However, such work assumes the existence of such constraints a-priori. In this paper, we relax this strong assumption and present an approach based on Error Detection Rules (EDR) that allow for learning explainable rules about the failure modes of machine learning models. We show that these rules are not only effective in detecting when a machine learning classifier has made an error but also can be leveraged as constraints for HMC, thereby allowing the recovery of explainable constraints even if they are not provided. We show that our approach is effective in detecting machine learning errors and recovering constraints, is noise tolerant, and can function as a source of knowledge for neurosymbolic models on multiple datasets, including a newly introduced military vehicle recognition dataset.

Since a building's floorplans are easily accessible, consistent over time, and inherently robust to changes in visual appearance, self-localization within the floorplan has attracted researchers' interest. However, since floorplans are minimalist representations of a building's structure, modal and geometric differences between visual perceptions and floorplans pose challenges to this task. While existing methods cleverly utilize 2D geometric features and pose filters to achieve promising performance, they fail to address the localization errors caused by frequent visual changes and view occlusions due to variously shaped 3D objects. To tackle these issues, this paper views the 2D Floorplan Localization (FLoc) problem from a higher dimension by injecting 3D geometric priors into the visual FLoc algorithm. For the 3D geometric prior modeling, we first model geometrically aware view invariance using multi-view constraints, i.e., leveraging imaging geometric principles to provide matching constraints between multiple images that see the same points. Then, we further model the view-scene aligned geometric priors, enhancing the cross-modal geometry-color correspondences by associating the scene's surface reconstruction with the RGB frames of the sequence. Both 3D priors are modeled through self-supervised contrastive learning, thus no additional geometric or semantic annotations are required. These 3D priors summarized in extensive realistic scenes bridge the modal gap while improving localization success without increasing the computational burden on the FLoc algorithm. Sufficient comparative studies demonstrate that our method significantly outperforms state-of-the-art methods and substantially boosts the FLoc accuracy. All data and code will be released after the anonymous review.

3D textured face reconstruction from sketches applicable in many scenarios such as animation, 3D avatars, artistic design, missing people search, etc., is a highly promising but underdeveloped research topic. On the one hand, the stylistic diversity of sketches leads to existing sketch-to-3D-face methods only being able to handle pose-limited and realistically shaded sketches. On the other hand, texture plays a vital role in representing facial appearance, yet sketches lack this information, necessitating additional texture control in the reconstruction process. This paper proposes a novel method for reconstructing controllable textured and detailed 3D faces from sketches, named S2TD-Face. S2TD-Face introduces a two-stage geometry reconstruction framework that directly reconstructs detailed geometry from the input sketch. To keep geometry consistent with the delicate strokes of the sketch, we propose a novel sketch-to-geometry loss that ensures the reconstruction accurately fits the input features like dimples and wrinkles. Our training strategies do not rely on hard-to-obtain 3D face scanning data or labor-intensive hand-drawn sketches. Furthermore, S2TD-Face introduces a texture control module utilizing text prompts to select the most suitable textures from a library and seamlessly integrate them into the geometry, resulting in a 3D detailed face with controllable texture. S2TD-Face surpasses existing state-of-the-art methods in extensive quantitative and qualitative experiments. Our project is available at https://github.com/wang-zidu/S2TD-Face .

Recent advances in data-driven geometric multi-view 3D reconstruction foundation models (e.g., DUSt3R) have shown remarkable performance across various 3D vision tasks, facilitated by the release of large-scale, high-quality 3D datasets. However, as we observed, constrained by their matching-based principles, the reconstruction quality of existing models suffers significant degradation in challenging regions with limited matching cues, particularly in weakly textured areas and low-light conditions. To mitigate these limitations, we propose to harness the inherent robustness of monocular geometry estimation to compensate for the inherent shortcomings of matching-based methods. Specifically, we introduce a monocular-guided refinement module that integrates monocular geometric priors into multi-view reconstruction frameworks. This integration substantially enhances the robustness of multi-view reconstruction systems, leading to high-quality feed-forward reconstructions. Comprehensive experiments across multiple benchmarks demonstrate that our method achieves substantial improvements in both mutli-view camera pose estimation and point cloud accuracy.

Neural surface representation has demonstrated remarkable success in the areas of novel view synthesis and 3D reconstruction. However, assessing the geometric quality of 3D reconstructions in the absence of ground truth mesh remains a significant challenge, due to its rendering-based optimization process and entangled learning of appearance and geometry with photometric losses. In this paper, we present a novel framework, i.e, GURecon, which establishes a geometric uncertainty field for the neural surface based on geometric consistency. Different from existing methods that rely on rendering-based measurement, GURecon models a continuous 3D uncertainty field for the reconstructed surface, and is learned by an online distillation approach without introducing real geometric information for supervision. Moreover, in order to mitigate the interference of illumination on geometric consistency, a decoupled field is learned and exploited to finetune the uncertainty field. Experiments on various datasets demonstrate the superiority of GURecon in modeling 3D geometric uncertainty, as well as its plug-and-play extension to various neural surface representations and improvement on downstream tasks such as incremental reconstruction. The code and supplementary material are available on the project website: https://zju3dv.github.io/GURecon/.

We present Image Sculpting, a new framework for editing 2D images by incorporating tools from 3D geometry and graphics. This approach differs markedly from existing methods, which are confined to 2D spaces and typically rely on textual instructions, leading to ambiguity and limited control. Image Sculpting converts 2D objects into 3D, enabling direct interaction with their 3D geometry. Post-editing, these objects are re-rendered into 2D, merging into the original image to produce high-fidelity results through a coarse-to-fine enhancement process. The framework supports precise, quantifiable, and physically-plausible editing options such as pose editing, rotation, translation, 3D composition, carving, and serial addition. It marks an initial step towards combining the creative freedom of generative models with the precision of graphics pipelines.

Deep neural networks are prone to adversarial examples that maliciously alter the network's outcome. Due to the increasing popularity of 3D sensors in safety-critical systems and the vast deployment of deep learning models for 3D point sets, there is a growing interest in adversarial attacks and defenses for such models. So far, the research has focused on the semantic level, namely, deep point cloud classifiers. However, point clouds are also widely used in a geometric-related form that includes encoding and reconstructing the geometry. In this work, we are the first to consider the problem of adversarial examples at a geometric level. In this setting, the question is how to craft a small change to a clean source point cloud that leads, after passing through an autoencoder model, to the reconstruction of a different target shape. Our attack is in sharp contrast to existing semantic attacks on 3D point clouds. While such works aim to modify the predicted label by a classifier, we alter the entire reconstructed geometry. Additionally, we demonstrate the robustness of our attack in the case of defense, where we show that remnant characteristics of the target shape are still present at the output after applying the defense to the adversarial input. Our code is publicly available at https://github.com/itailang/geometric_adv.

We present a system for generating indoor scenes in response to text prompts. The prompts are not limited to a fixed vocabulary of scene descriptions, and the objects in generated scenes are not restricted to a fixed set of object categories -- we call this setting indoor scene generation. Unlike most prior work on indoor scene generation, our system does not require a large training dataset of existing 3D scenes. Instead, it leverages the world knowledge encoded in pre-trained large language models (LLMs) to synthesize programs in a domain-specific layout language that describe objects and spatial relations between them. Executing such a program produces a specification of a constraint satisfaction problem, which the system solves using a gradient-based optimization scheme to produce object positions and orientations. To produce object geometry, the system retrieves 3D meshes from a database. Unlike prior work which uses databases of category-annotated, mutually-aligned meshes, we develop a pipeline using vision-language models (VLMs) to retrieve meshes from massive databases of un-annotated, inconsistently-aligned meshes. Experimental evaluations show that our system outperforms generative models trained on 3D data for traditional, closed-universe scene generation tasks; it also outperforms a recent LLM-based layout generation method on open-universe scene generation.

We use tools from geometric statistics to analyze the usual estimation procedure of a template shape. This applies to shapes from landmarks, curves, surfaces, images etc. We demonstrate the asymptotic bias of the template shape estimation using the stratified geometry of the shape space. We give a Taylor expansion of the bias with respect to a parameter sigma describing the measurement error on the data. We propose two bootstrap procedures that quantify the bias and correct it, if needed. They are applicable for any type of shape data. We give a rule of thumb to provide intuition on whether the bias has to be corrected. This exhibits the parameters that control the bias' magnitude. We illustrate our results on simulated and real shape data.

We introduce a novel dataset designed to benchmark the physical and spatial reasoning capabilities of Large Language Models (LLM) based on topology optimization, a method for computing optimal material distributions within a design space under prescribed loads and supports. In this dataset, LLMs are provided with conditions such as 2D boundary, applied forces and supports, and must reason about the resulting optimal material distribution. The dataset includes a variety of tasks, ranging from filling in masked regions within partial structures to predicting complete material distributions. Solving these tasks requires understanding the flow of forces and the required material distribution under given constraints, without access to simulation tools or explicit physical models, challenging models to reason about structural stability and spatial organization. Our dataset targets the evaluation of spatial and physical reasoning abilities in 2D settings, offering a complementary perspective to traditional language and logic benchmarks.

We propose a novel mixed-integer programming (MIP) formulation for generating precise sparse correspondences for highly non-rigid shapes. To this end, we introduce a projected Laplace-Beltrami operator (PLBO) which combines intrinsic and extrinsic geometric information to measure the deformation quality induced by predicted correspondences. We integrate the PLBO, together with an orientation-aware regulariser, into a novel MIP formulation that can be solved to global optimality for many practical problems. In contrast to previous methods, our approach is provably invariant to rigid transformations and global scaling, initialisation-free, has optimality guarantees, and scales to high resolution meshes with (empirically observed) linear time. We show state-of-the-art results for sparse non-rigid matching on several challenging 3D datasets, including data with inconsistent meshing, as well as applications in mesh-to-point-cloud matching.

The recent developments in neural fields have brought phenomenal capabilities to the field of shape generation, but they lack crucial properties, such as incremental control - a fundamental requirement for artistic work. Triangular meshes, on the other hand, are the representation of choice for most geometry related tasks, offering efficiency and intuitive control, but do not lend themselves to neural optimization. To support downstream tasks, previous art typically proposes a two-step approach, where first a shape is generated using neural fields, and then a mesh is extracted for further processing. Instead, in this paper we introduce a hybrid approach that maintains both a mesh and a Signed Distance Field (SDF) representations consistently. Using this representation, we introduce MagicClay - an artist friendly tool for sculpting regions of a mesh according to textual prompts while keeping other regions untouched. Our framework carefully and efficiently balances consistency between the representations and regularizations in every step of the shape optimization; Relying on the mesh representation, we show how to render the SDF at higher resolutions and faster. In addition, we employ recent work in differentiable mesh reconstruction to adaptively allocate triangles in the mesh where required, as indicated by the SDF. Using an implemented prototype, we demonstrate superior generated geometry compared to the state-of-the-art, and novel consistent control, allowing sequential prompt-based edits to the same mesh for the first time.

The two-stage object pose estimation paradigm first detects semantic keypoints on the image and then estimates the 6D pose by minimizing reprojection errors. Despite performing well on standard benchmarks, existing techniques offer no provable guarantees on the quality and uncertainty of the estimation. In this paper, we inject two fundamental changes, namely conformal keypoint detection and geometric uncertainty propagation, into the two-stage paradigm and propose the first pose estimator that endows an estimation with provable and computable worst-case error bounds. On one hand, conformal keypoint detection applies the statistical machinery of inductive conformal prediction to convert heuristic keypoint detections into circular or elliptical prediction sets that cover the groundtruth keypoints with a user-specified marginal probability (e.g., 90%). Geometric uncertainty propagation, on the other, propagates the geometric constraints on the keypoints to the 6D object pose, leading to a Pose UnceRtainty SEt (PURSE) that guarantees coverage of the groundtruth pose with the same probability. The PURSE, however, is a nonconvex set that does not directly lead to estimated poses and uncertainties. Therefore, we develop RANdom SAmple averaGing (RANSAG) to compute an average pose and apply semidefinite relaxation to upper bound the worst-case errors between the average pose and the groundtruth. On the LineMOD Occlusion dataset we demonstrate: (i) the PURSE covers the groundtruth with valid probabilities; (ii) the worst-case error bounds provide correct uncertainty quantification; and (iii) the average pose achieves better or similar accuracy as representative methods based on sparse keypoints.

Geometric alignment appears in a variety of applications, ranging from domain adaptation, optimal transport, and normalizing flows in machine learning; optical flow and learned augmentation in computer vision and deformable registration within biomedical imaging. A recurring challenge is the alignment of domains whose topology is not the same; a problem that is routinely ignored, potentially introducing bias in downstream analysis. As a first step towards solving such alignment problems, we propose an unsupervised algorithm for the detection of changes in image topology. The model is based on a conditional variational auto-encoder and detects topological changes between two images during the registration step. We account for both topological changes in the image under spatial variation and unexpected transformations. Our approach is validated on two tasks and datasets: detection of topological changes in microscopy images of cells, and unsupervised anomaly detection brain imaging.

Biometric authentication systems that make use of signature verification methods often render optimum performance only under limited and restricted conditions. Such methods utilize several training samples so as to achieve high accuracy. Moreover, several constraints are imposed on the end-user so that the system may work optimally, and as expected. For example, the user is made to sign within a small box, in order to limit their signature to a predefined set of dimensions, thus eliminating scaling. Moreover, the angular rotation with respect to the referenced signature that will be inadvertently introduced as human error, hampers performance of biometric signature verification systems. To eliminate this, traditionally, a user is asked to sign exactly on top of a reference line. In this paper, we propose a robust system that optimizes the signature obtained from the user for a large range of variation in Rotation-Scaling-Translation (RST) and resolves these error parameters in the user signature according to the reference signature stored in the database.

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimization challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Combinatorial problems are present in a wide range of industries. Constraint Programming (CP) is a well-suited problem-solving paradigm, but its core process, namely constraint modelling, is a bottleneck for wider adoption. Aiming to alleviate this bottleneck, recent studies have explored using Large Language Models (LLMs) as modelling assistants, transforming combinatorial problem descriptions to executable constraint models, similar to coding assistants. However, the existing evaluation datasets for constraint modelling are often limited to small, homogeneous, or domain-specific instances, which do not capture the diversity of real-world scenarios. This work addresses this gap by introducing CP-Bench, a novel benchmark dataset that includes a diverse set of well-known combinatorial problem classes sourced from the CP community, structured explicitly for evaluating LLM-driven CP modelling. With this dataset, and given the variety of constraint modelling frameworks, we compare and evaluate the modelling capabilities of LLMs for three distinct constraint modelling systems, which vary in abstraction level and underlying syntax: the high-level MiniZinc language and Python-based CPMpy library, and the lower-level Python interface of the OR-Tools CP-SAT solver. In order to enhance the ability of LLMs to produce valid constraint models, we systematically evaluate the use of prompt-based and inference-time compute methods adapted from existing LLM-based code generation research. Our results underscore the modelling convenience provided by Python-based frameworks, as well as the effectiveness of documentation-rich system prompts, which, augmented with repeated sampling and self-verification, achieve further improvements, reaching up to 70\% accuracy on this new, highly challenging benchmark.

We propose a novel, object-agnostic method for learning a universal policy for dexterous object grasping from realistic point cloud observations and proprioceptive information under a table-top setting, namely UniDexGrasp++. To address the challenge of learning the vision-based policy across thousands of object instances, we propose Geometry-aware Curriculum Learning (GeoCurriculum) and Geometry-aware iterative Generalist-Specialist Learning (GiGSL) which leverage the geometry feature of the task and significantly improve the generalizability. With our proposed techniques, our final policy shows universal dexterous grasping on thousands of object instances with 85.4% and 78.2% success rate on the train set and test set which outperforms the state-of-the-art baseline UniDexGrasp by 11.7% and 11.3%, respectively.

Deep Generative Models (DGMs) have been shown to be powerful tools for generating tabular data, as they have been increasingly able to capture the complex distributions that characterize them. However, to generate realistic synthetic data, it is often not enough to have a good approximation of their distribution, as it also requires compliance with constraints that encode essential background knowledge on the problem at hand. In this paper, we address this limitation and show how DGMs for tabular data can be transformed into Constrained Deep Generative Models (C-DGMs), whose generated samples are guaranteed to be compliant with the given constraints. This is achieved by automatically parsing the constraints and transforming them into a Constraint Layer (CL) seamlessly integrated with the DGM. Our extensive experimental analysis with various DGMs and tasks reveals that standard DGMs often violate constraints, some exceeding 95% non-compliance, while their corresponding C-DGMs are never non-compliant. Then, we quantitatively demonstrate that, at training time, C-DGMs are able to exploit the background knowledge expressed by the constraints to outperform their standard counterparts with up to 6.5% improvement in utility and detection. Further, we show how our CL does not necessarily need to be integrated at training time, as it can be also used as a guardrail at inference time, still producing some improvements in the overall performance of the models. Finally, we show that our CL does not hinder the sample generation time of the models.

Neural volume rendering became increasingly popular recently due to its success in synthesizing novel views of a scene from a sparse set of input images. So far, the geometry learned by neural volume rendering techniques was modeled using a generic density function. Furthermore, the geometry itself was extracted using an arbitrary level set of the density function leading to a noisy, often low fidelity reconstruction. The goal of this paper is to improve geometry representation and reconstruction in neural volume rendering. We achieve that by modeling the volume density as a function of the geometry. This is in contrast to previous work modeling the geometry as a function of the volume density. In more detail, we define the volume density function as Laplace's cumulative distribution function (CDF) applied to a signed distance function (SDF) representation. This simple density representation has three benefits: (i) it provides a useful inductive bias to the geometry learned in the neural volume rendering process; (ii) it facilitates a bound on the opacity approximation error, leading to an accurate sampling of the viewing ray. Accurate sampling is important to provide a precise coupling of geometry and radiance; and (iii) it allows efficient unsupervised disentanglement of shape and appearance in volume rendering. Applying this new density representation to challenging scene multiview datasets produced high quality geometry reconstructions, outperforming relevant baselines. Furthermore, switching shape and appearance between scenes is possible due to the disentanglement of the two.

Recent advancements in implicit 3D representations and generative models have markedly propelled the field of 3D object generation forward. However, it remains a significant challenge to accurately model geometries with defined sharp features under parametric controls, which is crucial in fields like industrial design and manufacturing. To bridge this gap, we introduce a framework that employs Large Language Models (LLMs) to generate text-driven 3D shapes, manipulating 3D software via program synthesis. We present 3D-PreMise, a dataset specifically tailored for 3D parametric modeling of industrial shapes, designed to explore state-of-the-art LLMs within our proposed pipeline. Our work reveals effective generation strategies and delves into the self-correction capabilities of LLMs using a visual interface. Our work highlights both the potential and limitations of LLMs in 3D parametric modeling for industrial applications.

Some results of author's work in a non-geometrical approach to quantum gravity are reviewed here, among them: a quantum mechanism of classical gravity giving a possibility to compute the Newton constant; asymptotic freedom at short distances; interaction of photons with the graviton background leading to the important cosmological consequences; the time delay of photons due to interactions with gravitons; deceleration of massive bodies in the graviton background which may be connected with the Pioneer anomaly and with the problem of dark matter.

Following the advent of NeRFs, 3D Gaussian Splatting (3D-GS) has paved the way to real-time neural rendering overcoming the computational burden of volumetric methods. Following the pioneering work of 3D-GS, several methods have attempted to achieve compressible and high-fidelity performance alternatives. However, by employing a geometry-agnostic optimization scheme, these methods neglect the inherent 3D structure of the scene, thereby restricting the expressivity and the quality of the representation, resulting in various floating points and artifacts. In this work, we propose a structure-aware Gaussian Splatting method (SAGS) that implicitly encodes the geometry of the scene, which reflects to state-of-the-art rendering performance and reduced storage requirements on benchmark novel-view synthesis datasets. SAGS is founded on a local-global graph representation that facilitates the learning of complex scenes and enforces meaningful point displacements that preserve the scene's geometry. Additionally, we introduce a lightweight version of SAGS, using a simple yet effective mid-point interpolation scheme, which showcases a compact representation of the scene with up to 24times size reduction without the reliance on any compression strategies. Extensive experiments across multiple benchmark datasets demonstrate the superiority of SAGS compared to state-of-the-art 3D-GS methods under both rendering quality and model size. Besides, we demonstrate that our structure-aware method can effectively mitigate floating artifacts and irregular distortions of previous methods while obtaining precise depth maps. Project page https://eververas.github.io/SAGS/.

The latest regularized Neural Radiance Field (NeRF) approaches produce poor geometry and view extrapolation for multiview stereo (MVS) benchmarks such as ETH3D. In this paper, we aim to create 3D models that provide accurate geometry and view synthesis, partially closing the large geometric performance gap between NeRF and traditional MVS methods. We propose a patch-based approach that effectively leverages monocular surface normal and relative depth predictions. The patch-based ray sampling also enables the appearance regularization of normalized cross-correlation (NCC) and structural similarity (SSIM) between randomly sampled virtual and training views. We further show that "density restrictions" based on sparse structure-from-motion points can help greatly improve geometric accuracy with a slight drop in novel view synthesis metrics. Our experiments show 4x the performance of RegNeRF and 8x that of FreeNeRF on average F1@2cm for ETH3D MVS benchmark, suggesting a fruitful research direction to improve the geometric accuracy of NeRF-based models, and sheds light on a potential future approach to enable NeRF-based optimization to eventually outperform traditional MVS.

In this work, we aim to improve the 3D reasoning ability of Transformers in multi-view 3D human pose estimation. Recent works have focused on end-to-end learning-based transformer designs, which struggle to resolve geometric information accurately, particularly during occlusion. Instead, we propose a novel hybrid model, MVGFormer, which has a series of geometric and appearance modules organized in an iterative manner. The geometry modules are learning-free and handle all viewpoint-dependent 3D tasks geometrically which notably improves the model's generalization ability. The appearance modules are learnable and are dedicated to estimating 2D poses from image signals end-to-end which enables them to achieve accurate estimates even when occlusion occurs, leading to a model that is both accurate and generalizable to new cameras and geometries. We evaluate our approach for both in-domain and out-of-domain settings, where our model consistently outperforms state-of-the-art methods, and especially does so by a significant margin in the out-of-domain setting. We will release the code and models: https://github.com/XunshanMan/MVGFormer.

We introduce GeoDANO, a geometric vision-language model (VLM) with a domain-agnostic vision encoder, for solving plane geometry problems. Although VLMs have been employed for solving geometry problems, their ability to recognize geometric features remains insufficiently analyzed. To address this gap, we propose a benchmark that evaluates the recognition of visual geometric features, including primitives such as dots and lines, and relations such as orthogonality. Our preliminary study shows that vision encoders often used in general-purpose VLMs, e.g., OpenCLIP, fail to detect these features and struggle to generalize across domains. We develop GeoCLIP, a CLIP based model trained on synthetic geometric diagram-caption pairs to overcome the limitation. Benchmark results show that GeoCLIP outperforms existing vision encoders in recognizing geometric features. We then propose our VLM, GeoDANO, which augments GeoCLIP with a domain adaptation strategy for unseen diagram styles. GeoDANO outperforms specialized methods for plane geometry problems and GPT-4o on MathVerse.

Recent facial image synthesis methods have been mainly based on conditional generative models. Sketch-based conditions can effectively describe the geometry of faces, including the contours of facial components, hair structures, as well as salient edges (e.g., wrinkles) on face surfaces but lack effective control of appearance, which is influenced by color, material, lighting condition, etc. To have more control of generated results, one possible approach is to apply existing disentangling works to disentangle face images into geometry and appearance representations. However, existing disentangling methods are not optimized for human face editing, and cannot achieve fine control of facial details such as wrinkles. To address this issue, we propose DeepFaceEditing, a structured disentanglement framework specifically designed for face images to support face generation and editing with disentangled control of geometry and appearance. We adopt a local-to-global approach to incorporate the face domain knowledge: local component images are decomposed into geometry and appearance representations, which are fused consistently using a global fusion module to improve generation quality. We exploit sketches to assist in extracting a better geometry representation, which also supports intuitive geometry editing via sketching. The resulting method can either extract the geometry and appearance representations from face images, or directly extract the geometry representation from face sketches. Such representations allow users to easily edit and synthesize face images, with decoupled control of their geometry and appearance. Both qualitative and quantitative evaluations show the superior detail and appearance control abilities of our method compared to state-of-the-art methods.

Empirically validating new 3D-printing related algorithms and implementations requires testing data representative of inputs encountered in the wild. An ideal benchmarking dataset should not only draw from the same distribution of shapes people print in terms of class (e.g., toys, mechanisms, jewelry), representation type (e.g., triangle soup meshes) and complexity (e.g., number of facets), but should also capture problems and artifacts endemic to 3D printing models (e.g., self-intersections, non-manifoldness). We observe that the contextual and geometric characteristics of 3D printing models differ significantly from those used for computer graphics applications, not to mention standard models (e.g., Stanford bunny, Armadillo, Fertility). We present a new dataset of 10,000 models collected from an online 3D printing model-sharing database. Via analysis of both geometric (e.g., triangle aspect ratios, manifoldness) and contextual (e.g., licenses, tags, classes) characteristics, we demonstrate that this dataset represents a more concise summary of real-world models used for 3D printing compared to existing datasets. To facilitate future research endeavors, we also present an online query interface to select subsets of the dataset according to project-specific characteristics. The complete dataset and per-model statistical data are freely available to the public.

We address the problem of finite-horizon control of a discrete-time linear system, where the initial state distribution follows a Gaussian mixture model, the terminal state must follow a specified Gaussian distribution, and the state and control inputs must obey chance constraints. We show that, throughout the time horizon, the state and control distributions are fully characterized by Gaussian mixtures. We then formulate the cost, distributional terminal constraint, and affine/2-norm chance constraints on the state and control, as convex functions of the decision variables. This is leveraged to formulate the chance-constrained path planning problem as a single convex optimization problem. A numerical example demonstrates the effectiveness of the proposed method.

We propose a system for rearranging objects in a scene to achieve a desired object-scene placing relationship, such as a book inserted in an open slot of a bookshelf. The pipeline generalizes to novel geometries, poses, and layouts of both scenes and objects, and is trained from demonstrations to operate directly on 3D point clouds. Our system overcomes challenges associated with the existence of many geometrically-similar rearrangement solutions for a given scene. By leveraging an iterative pose de-noising training procedure, we can fit multi-modal demonstration data and produce multi-modal outputs while remaining precise and accurate. We also show the advantages of conditioning on relevant local geometric features while ignoring irrelevant global structure that harms both generalization and precision. We demonstrate our approach on three distinct rearrangement tasks that require handling multi-modality and generalization over object shape and pose in both simulation and the real world. Project website, code, and videos: https://anthonysimeonov.github.io/rpdiff-multi-modal/

Generative models for 3D object synthesis have seen significant advancements with the incorporation of prior knowledge distilled from 2D diffusion models. Nevertheless, challenges persist in the form of multi-view geometric inconsistencies and slow generation speeds within the existing 3D synthesis frameworks. This can be attributed to two factors: firstly, the deficiency of abundant geometric a priori knowledge in optimization, and secondly, the entanglement issue between geometry and texture in conventional 3D generation methods.In response, we introduce MetaDreammer, a two-stage optimization approach that leverages rich 2D and 3D prior knowledge. In the first stage, our emphasis is on optimizing the geometric representation to ensure multi-view consistency and accuracy of 3D objects. In the second stage, we concentrate on fine-tuning the geometry and optimizing the texture, thereby achieving a more refined 3D object. Through leveraging 2D and 3D prior knowledge in two stages, respectively, we effectively mitigate the interdependence between geometry and texture. MetaDreamer establishes clear optimization objectives for each stage, resulting in significant time savings in the 3D generation process. Ultimately, MetaDreamer can generate high-quality 3D objects based on textual prompts within 20 minutes, and to the best of our knowledge, it is the most efficient text-to-3D generation method. Furthermore, we introduce image control into the process, enhancing the controllability of 3D generation. Extensive empirical evidence confirms that our method is not only highly efficient but also achieves a quality level that is at the forefront of current state-of-the-art 3D generation techniques.

We study the Dirichlet problem for the weighted Schr\"odinger operator \[-\Delta u +Vu = \lambda \rho u,\] where rho is a positive weighting function and V is a potential. Such equations appear naturally in conformal geometry and in the composite membrane problem. Our primary goal is to establish concavity estimates for the principle eigenfunction with respect to conformal connections. Doing so, we obtain new bounds on the fundamental gap problem, which is the difference between the first and second eigenvalues. In particular, we partially resolve a conjecture of Nguyen, Stancu and Wei [IMRN 2022] on the fundamental gap of horoconvex domains. In addition, we obtain a power convexity estimate for solutions to the torsion problem in spherical geometry on convex domains which are not too large.

While 3D body reconstruction methods have made remarkable progress recently, it remains difficult to acquire the sufficiently accurate and numerous 3D supervisions required for training. In this paper, we propose KNOWN, a framework that effectively utilizes body KNOWledge and uNcertainty modeling to compensate for insufficient 3D supervisions. KNOWN exploits a comprehensive set of generic body constraints derived from well-established body knowledge. These generic constraints precisely and explicitly characterize the reconstruction plausibility and enable 3D reconstruction models to be trained without any 3D data. Moreover, existing methods typically use images from multiple datasets during training, which can result in data noise (e.g., inconsistent joint annotation) and data imbalance (e.g., minority images representing unusual poses or captured from challenging camera views). KNOWN solves these problems through a novel probabilistic framework that models both aleatoric and epistemic uncertainty. Aleatoric uncertainty is encoded in a robust Negative Log-Likelihood (NLL) training loss, while epistemic uncertainty is used to guide model refinement. Experiments demonstrate that KNOWN's body reconstruction outperforms prior weakly-supervised approaches, particularly on the challenging minority images.

Existing reasoning datasets saturate and fail to test abstract, multi-step problems, especially pathfinding and complex rule constraint satisfaction. We introduce SPaRC (Spatial Pathfinding Reasoning Challenge), a dataset of 1,000 2D grid pathfinding puzzles to evaluate spatial and symbolic reasoning, requiring step-by-step planning with arithmetic and geometric rules. Humans achieve near-perfect accuracy (98.0%; 94.5% on hard puzzles), while the best reasoning models, such as o4-mini, struggle (15.8%; 1.1% on hard puzzles). Models often generate invalid paths (>50% of puzzles for o4-mini), and reasoning tokens reveal they make errors in navigation and spatial logic. Unlike humans, who take longer on hard puzzles, models fail to scale test-time compute with difficulty. Allowing models to make multiple solution attempts improves accuracy, suggesting potential for better spatial reasoning with improved training and efficient test-time scaling methods. SPaRC can be used as a window into models' spatial reasoning limitations and drive research toward new methods that excel in abstract, multi-step problem-solving.

It is common in graphic design humans visually arrange various elements according to their design intent and semantics. For example, a title text almost always appears on top of other elements in a document. In this work, we generate graphic layouts that can flexibly incorporate such design semantics, either specified implicitly or explicitly by a user. We optimize using the latent space of an off-the-shelf layout generation model, allowing our approach to be complementary to and used with existing layout generation models. Our approach builds on a generative layout model based on a Transformer architecture, and formulates the layout generation as a constrained optimization problem where design constraints are used for element alignment, overlap avoidance, or any other user-specified relationship. We show in the experiments that our approach is capable of generating realistic layouts in both constrained and unconstrained generation tasks with a single model. The code is available at https://github.com/ktrk115/const_layout .

Parametric Computer-Aided Design (CAD) is central to contemporary mechanical design. However, it encounters challenges in achieving precise parametric sketch modeling and lacks practical evaluation metrics suitable for mechanical design. We harness the capabilities of pre-trained foundation models, renowned for their successes in natural language processing and computer vision, to develop generative models specifically for CAD. These models are adept at understanding complex geometries and design reasoning, a crucial advancement in CAD technology. In this paper, we propose CadVLM, an end-to-end vision language model for CAD generation. Our approach involves adapting pre-trained foundation models to manipulate engineering sketches effectively, integrating both sketch primitive sequences and sketch images. Extensive experiments demonstrate superior performance on multiple CAD sketch generation tasks such as CAD autocompletion, CAD autoconstraint, and image conditional generation. To our knowledge, this is the first instance of a multimodal Large Language Model (LLM) being successfully applied to parametric CAD generation, representing a pioneering step in the field of computer-aided mechanical design.

In this paper, we introduce Geometry-Inverse-Meet-Pixel-Insert, short for GEO, an exceptionally versatile image editing technique designed to cater to customized user requirements at both local and global scales. Our approach seamlessly integrates text prompts and image prompts to yield diverse and precise editing outcomes. Notably, our method operates without the need for training and is driven by two key contributions: (i) a novel geometric accumulation loss that enhances DDIM inversion to faithfully preserve pixel space geometry and layout, and (ii) an innovative boosted image prompt technique that combines pixel-level editing for text-only inversion with latent space geometry guidance for standard classifier-free reversion. Leveraging the publicly available Stable Diffusion model, our approach undergoes extensive evaluation across various image types and challenging prompt editing scenarios, consistently delivering high-fidelity editing results for real images.

Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework that unifies generative modeling and partial differential equation fulfillment by introducing a first-principle-based loss term that enforces generated samples to fulfill the underlying physical constraints. Our approach reduces the residual error by up to two orders of magnitude compared to previous work in a fluid flow case study and outperforms task-specific frameworks in relevant metrics for structural topology optimization. We also present numerical evidence that our extended training objective acts as a natural regularization mechanism against overfitting. Our framework is simple to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.

In this paper, we propose a new mixed-integer linear programming (MILP) model ontology and a novel constraint typology of MILP formulations. MILP is a commonly used mathematical programming technique for modelling and solving real-life scheduling, routing, planning, resource allocation, and timetabling optimization problems providing optimized business solutions for industry sectors such as manufacturing, agriculture, defence, healthcare, medicine, energy, finance, and transportation. Despite the numerous real-life Combinatorial Optimization Problems found and solved and millions yet to be discovered and formulated, the number of types of constraints (the building blocks of a MILP) is relatively small. In the search for a suitable machine-readable knowledge representation structure for MILPs, we propose an optimization modelling tree built based upon an MILP model ontology that can be used as a guide for automated systems to elicit an MILP model from end-users on their combinatorial business optimization problems. Our ultimate aim is to develop a machine-readable knowledge representation for MILP that allows us to map an end-user's natural language description of the business optimization problem to an MILP formal specification as a first step towards automated mathematical modelling.