Daily Papers

Implicit Neural Representations (INRs) have emerged as a powerful paradigm for representing signals such as images, 3D shapes, signed distance fields, and radiance fields. While significant progress has been made in architecture design (e.g., SIREN, FFC, KAN-based INRs) and optimization strategies (meta-learning, amortization, distillation), existing approaches still suffer from two core limitations: (1) a representation bottleneck that forces a single MLP to uniformly model heterogeneous local structures, and (2) limited scalability due to the absence of a hierarchical mechanism that dynamically adapts to signal complexity. This work introduces Hyper-Coordinate Implicit Neural Representations (HC-INR), a new class of INRs that break the representational bottleneck by learning signal-adaptive coordinate transformations using a hypernetwork. HC-INR decomposes the representation task into two components: (i) a learned multiscale coordinate transformation module that warps the input domain into a disentangled latent space, and (ii) a compact implicit field network that models the transformed signal with significantly reduced complexity. The proposed model introduces a hierarchical hypernetwork architecture that conditions coordinate transformations on local signal features, enabling dynamic allocation of representation capacity. We theoretically show that HC-INR strictly increases the upper bound of representable frequency bands while maintaining Lipschitz stability. Extensive experiments across image fitting, shape reconstruction, and neural radiance field approximation demonstrate that HC-INR achieves up to 4 times higher reconstruction fidelity than strong INR baselines while using 30--60\% fewer parameters.

The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the regression setting, and use the trained model as a cheap surrogate for the solver. The bottleneck in this scheme is the number of expensive queries of a PDE solver needed to generate the dataset. To alleviate the problem, we propose a computationally cheap augmentation strategy based on general covariance and simple random coordinate transformations. Our approach relies on the fact that physical laws are independent of the coordinate choice, so the change in the coordinate system preserves the type of a parametric PDE and only changes PDE's data (e.g., initial conditions, diffusion coefficient). For tried neural networks and partial differential equations, proposed augmentation improves test error by 23% on average. The worst observed result is a 17% increase in test error for multilayer perceptron, and the best case is a 80% decrease for dilated residual network.

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

In this paper, we propose an alternating numerical scheme for constructing quasi-periodic solutions to a class of nonlinear partial differential equations (PDEs) with bounded perturbations, including the one-dimensional nonlinear Schrödinger (NLS) equations and nonlinear wave (NLW) equations under periodic and Dirichlet boundary conditions. A key observation concerning the resonant Q-equations allows us to establish directly a local diffeomorphism between the amplitudes and the drifted frequencies. This renders the proof considerably cleaner, since it avoids the Birkhoff normal form and the associated coordinate transformations required in classical KAM theory for PDEs.Unlike in the finite-dimensional setting, the lattice boxes must also include the spatial index, so that the multiscale induction has to be carried out in one additional dimension. This introduces two difficulties. First, the measure estimate for the excluded set of ``bad'' frequencies must account for the spatial index. For both the NLS and NLW equations, the linear frequencies and their differences are integers or close to integers; this allows the spatial dependence to be controlled through the modulus of the temporal index, so that the excluded measure remains under control. Second, a singular set may contain sites with the same temporal index but widely separated spatial indices, which obstructs the ``inversion implies localization'' process. For the one-dimensional NLS and NLW equations, however, such sites occur only in pairs with opposite spatial indices. They can therefore be separated by small boxes, after which the global inverse is obtained from the local inverses via the resolvent identity. Numerical experiments illustrate the scheme for nonlinear traveling and standing waves, exhibiting both the computed quasi-periodic solutions and the convergence of the iteration.

Accurate skull-to-array registration is essential for transcranial focused ultrasound (tFUS) because CT-derived skull models must be aligned with the physical-array coordinate system to support patient-specific aberration correction. Current workflows typically use MRI and/or external tracking to establish this spatial relationship, adding imaging infrastructure and coordinate transformations to the treatment workflow. Here, we describe an MRI-free CT-to-ultrasound registration framework using a hybrid hemispherical CMUT tFUS array that combines a sparse therapeutic array with an integrated CMUT US-Cross imaging array. The therapeutic array provides pulse-echo measurements of skull location, while the US-Cross array provides local high-resolution images of the skull surface. The approach was evaluated using a CT-derived outer cortical skull surface across a 25-pose matrix with 10-mm lateral offsets between the skull and array centers and +/-5 degree angular perturbations. Five randomized radial-noise seeds were applied to therapeutic pulse-echo measurements to model measurement or element-placement errors, yielding 125 pose-seed realizations per noise and active-element setting. US-Cross imaging features were generated using Field II and delay-multiply-and-sum plane-wave compounding. Four registration modes were evaluated: Therapeutic only, US-imaging only, Sequential, and Fused registration. Fused registration yielded the best overall performance, with a median total displacement error of 0.271 mm [0.169-0.420] and a median rotation error of 0.904 degrees [0.432-1.639]. These results support an all-ultrasound-based approach using a hybrid CMUT tFUS array for MRI-free CT-US skull registration.

Recovering camera and hand motion in world coordinates from egocentric video is a key capability for activity understanding, robot learning, and augmented reality. Existing systems typically decompose this problem into separate stages for camera motion, depth estimation, hand reconstruction, and trajectory refinement, resulting in substantial computational overhead and preventing the joint modeling of camera and hand motion. We introduce MINT (Minting IN-the-Wild Trajectories), a foundation model for world-space hand motion reconstruction from ego-centric RGB video. From a single shared spatiotemporal video representation, MINT jointly predicts the camera trajectory, field of view (FoV), camera-frame hand states, and per-frame hand observability, and then produces world-space hand motion via explicit coordinate transformations. Training such a model at scale is challenging, since paired world-space camera and hand annotations are scarce. We therefore develop an open-source labeling EGOPIPELINE that converts large collections of public egocentric videos into structured camera-and-hand trajectory supervision. MINT is first pretrained on these large-scale pseudo-labels and then fine-tuned on a small set of high-quality camera-and-hand annotations. Across public benchmarks MINT approaches state-of-the-art accuracy without seeing either benchmark in training, reaching 0.945 frame accuracy, 13.646 mm PA-MPJPE-p and 55.058 px EPE-p for camera-frame bimanual reconstruction on HOT3D, 4.690 mm RPE-T and 0.284 degrees RPE-R for camera trajectory, and a 3.67x end-to-end speedup over the labeling pipeline that supervises it. We release the model, training and inference code, labeling pipeline, and a curated 1,021-hour egocentric trajectory dataset.

We present MetaFind, a scene-aware tri-modal compositional retrieval framework designed to enhance scene generation in the metaverse by retrieving 3D assets from large-scale repositories. MetaFind addresses two core challenges: (i) inconsistent asset retrieval that overlooks spatial, semantic, and stylistic constraints, and (ii) the absence of a standardized retrieval paradigm specifically tailored for 3D asset retrieval, as existing approaches mainly rely on general-purpose 3D shape representation models. Our key innovation is a flexible retrieval mechanism that supports arbitrary combinations of text, image, and 3D modalities as queries, enhancing spatial reasoning and style consistency by jointly modeling object-level features (including appearance) and scene-level layout structures. Methodologically, MetaFind introduces a plug-and-play equivariant layout encoder ESSGNN that captures spatial relationships and object appearance features, ensuring retrieved 3D assets are contextually and stylistically coherent with the existing scene, regardless of coordinate frame transformations. The framework supports iterative scene construction by continuously adapting retrieval results to current scene updates. Empirical evaluations demonstrate the improved spatial and stylistic consistency of MetaFind in various retrieval tasks compared to baseline methods.

Tool calling allows large language models (LLMs) to invoke external computation during problem solving, a useful capability in various fields including AI for mathematics. We study this setting through weighted sum-of-squares (SOS) decomposition, a machine-checkable route to proving polynomial nonnegativity and hence polynomial inequalities. A candidate decomposition can be checked exactly, but finding one requires choosing among non-unique regroupings and coordinating multiple symbolic transformations. We develop an agent that combines algebraic task training, symbolic tools, and verifier-grounded optimization for this task. Rather than training only on the composite SOS task, we construct 1.35 million synthetic examples covering eight supporting polynomial tasks together with weighted-SOS decomposition. We first apply supervised fine-tuning (SFT) to direct algebra problems and simulated symbolic traces, and then use Group Relative Policy Optimization (GRPO) with task-specific symbolic rewards. The SFT corpus contains no native tool-calling messages; at evaluation, the agent uses native SymPy calls for expansion, collection, reordering, and factorization. Every final SOS answer is checked by exact expansion and coefficient comparison. On held-out, same-generator synthetic problems, the full SFT+GRPO+tools system is the strongest of four evaluated configurations, reaching 78.96% verified success on weighted SOS, compared with 44.73% for the base model with the same tools, and 91.75% macro accuracy across nine polynomial tasks. Within this controlled setting, our work provides a case study of combining domain-specific skill training, executable tools, and verifier feedback, and may inform the design of tool-calling agents in other domains with exactly checkable outputs.

With the rapid growth of the low-altitude economy, UAVs have become crucial for measurement and tracking in patrol systems. However, in GNSS-denied areas, satellite-based localization methods are prone to failure. This paper presents a cross-view UAV localization framework that performs map matching via object detection, aimed at effectively addressing cross-temporal, cross-view, heterogeneous aerial image matching. In typical pipelines, UAV visual localization is formulated as an image-retrieval problem: features are extracted to build a localization map, and the pose of a query image is estimated by matching it to a reference database with known poses. Because publicly available UAV localization datasets are limited, many approaches recast localization as a classification task and rely on scene labels in these datasets to ensure accuracy. Other methods seek to reduce cross-domain differences using polar-coordinate reprojection, perspective transformations, or generative adversarial networks; however, they can suffer from misalignment, content loss, and limited realism. In contrast, we leverage modern object detection to accurately extract salient instances from UAV and satellite images, and integrate a graph neural network to reason about inter-image and intra-image node relationships. Using a fine-grained, graph-based node-similarity metric, our method achieves strong retrieval and localization performance. Extensive experiments on public and real-world datasets show that our approach handles heterogeneous appearance differences effectively and generalizes well, making it applicable to scenarios with larger modality gaps, such as infrared-visible image matching. Our dataset will be publicly available at the following URL: https://github.com/liutao23/ODGNNLoc.git.

How can prior knowledge on the transformation invariances of a domain be incorporated into the architecture of a neural network? We propose Equivariant Transformers (ETs), a family of differentiable image-to-image mappings that improve the robustness of models towards pre-defined continuous transformation groups. Through the use of specially-derived canonical coordinate systems, ETs incorporate functions that are equivariant by construction with respect to these transformations. We show empirically that ETs can be flexibly composed to improve model robustness towards more complicated transformation groups in several parameters. On a real-world image classification task, ETs improve the sample efficiency of ResNet classifiers, achieving relative improvements in error rate of up to 15% in the limited data regime while increasing model parameter count by less than 1%.

Human-object interaction (HOI) video generation has garnered increasing attention due to its promising applications in digital humans, e-commerce, advertising, and robotics imitation learning. However, existing methods face two critical limitations: (1) a lack of effective mechanisms to inject multi-view information of the object into the model, leading to poor cross-view consistency, and (2) heavy reliance on fine-grained hand mesh annotations for modeling interaction occlusions. To address these challenges, we introduce ByteLoom, a Diffusion Transformer (DiT)-based framework that generates realistic HOI videos with geometrically consistent object illustration, using simplified human conditioning and 3D object inputs. We first propose an RCM-cache mechanism that leverages Relative Coordinate Maps (RCM) as a universal representation to maintain object's geometry consistency and precisely control 6-DoF object transformations in the meantime. To compensate HOI dataset scarcity and leverage existing datasets, we further design a training curriculum that enhances model capabilities in a progressive style and relaxes the demand of hand mesh. Extensive experiments demonstrate that our method faithfully preserves human identity and the object's multi-view geometry, while maintaining smooth motion and object manipulation.

Feed-forward view synthesis models predict a novel view in a single pass with minimal 3D inductive bias. Existing works encode cameras as Plücker ray maps, which tie predictions to the arbitrary world coordinate gauge and make them sensitive to small camera transformations, thereby undermining geometric consistency. In this paper, we ask what inputs best condition a model for robust and consistent view synthesis. We propose projective conditioning, which replaces raw camera parameters with a target-view projective cue that provides a stable 2D input. This reframes the task from a brittle geometric regression problem in ray space to a well-conditioned target-view image-to-image translation problem. Additionally, we introduce a masked autoencoding pretraining strategy tailored to this cue, enabling the use of large-scale uncalibrated data for pretraining. Our method shows improved fidelity and stronger cross-view consistency compared to ray-conditioned baselines on our view-consistency benchmark. It also achieves state-of-the-art quality on standard novel view synthesis benchmarks.

Representational similarity methods compare the geometries of neural representations, but they do not measure how consistently the geometry of a single representation is recovered from subsets of its feature coordinates. We call this property geometric stability and introduce Shesha, which estimates it by correlating representational dissimilarity matrices from complementary random feature subsets. Shesha is not invariant to orthogonal rotations: representations with identical Gram matrices, and therefore identical linear CKA, can have different geometric stability. Controlled transformations further separate the quantities. Across 2{,}463 encoder configurations spanning seven domains, similarity and stability are positively associated across non-PCA transformations (ρ=+0.75) but negatively associated under PCA-coordinate compression (ρ=-0.47). We further evaluate 170 pretrained vision models across six datasets. DINOv2 combines strong transfer performance with bottom-quartile stability on five of six datasets, showing that transferability and feature-split stability need not coincide. Across random feature subsets, the marginal relationship between Shesha and linear-probe variability is dataset-dependent; after controlling for task alignment with LogME, higher Shesha is associated with lower variability on five of six datasets. These results identify geometric stability as a basis-dependent property that complements representational similarity and task alignment.

Local image watermarking embeds an invisible signal into selected image regions rather than spreading it across the entire image, enabling payload recovery from specific objects or regions without perceptibly altering the image. Existing studies evaluate the robustness of payload recovery and localization under image transformations, but they often focus on their own proposed method, resulting in narrow evaluations with inconsistent choices of transformations, datasets, and metrics. These inconsistencies across studies limit direct comparisons across methods and muddle the overall picture of local watermark robustness. To address this gap, we present the first systematic robustness benchmark for local watermarks across 55 image transformations, including (i) signal distortions, (ii) changes in image coordinate alignment, (iii) indirect local edits, and (iv) direct watermark edits. The benchmark evaluates MaskWM, WAM, OmniGuard, TrustMark, and PixelSeal, all methods that either provide native localization or require minimal adaptation to support it. Our results show that all evaluated methods are vulnerable to some transformation, with MaskWM standing out as offering the strongest payload recovery and localization, although it has the lowest image quality in the clean setting. Synchronization further improves MaskWM's payload recovery under several geometric transformations, albeit at an additional cost to image quality. A key finding is that local watermark robustness depends strongly on the nature of the transformation: signal distortions are often tolerated by the strongest methods, while geometric misalignment and generative local edits, such as inpainting and outpainting, can completely impair payload recovery. We observe that payload recovery and localization are related but not interchangeable, and both strongly depend on the transformation's impact on the watermark region.

In this paper we construct the canonical harmonic coordinates of the Kerr metric within the multipolar post-Minkowskian (MPM) formalism to the fourth post-Minkowskian (4PM) order. Based on the well known Geroch--Hansen moments of Kerr metric and Gürsel's theorem, we derive the exact canonical MPM moments M_L,S_L, which are free of any gauge moments. With these moments, we iteratively compute the gothic metric perturbation h^{μν}_{can} up to 4PM order and compute the 4PM canonical metric g_{μν}^{can}. The resulting spatial and time components of these metrics are even functions of the spin parameter a while the mixed components are odd. This parity property distinguishes the canonical coordinates from other harmonic coordinates. To contrast this minimal-gauge construction, we also extract the 1PM source moments of the Kerr metric in the Jiang--Lin coordinates. We find that the Jiang--Lin representation possesses non-vanishing gauge moments starting from the 1PM order, whereas in the canonical representation gauge moments vanish to all orders. This comparison highlights the canonical coordinates as the most gauge-pure representation of the Kerr metric in the MPM framework. The complete canonical metric for the Schwarzschild case is also computed to all PM orders. A recent independent construction by Damgaard et al. using momentum-space recursion yields 4PM equivalent results expressed as a power series in a, providing a cross-validation of our closed-form 4PM canonical metric. The coordinate transformation linking the canonical Kerr coordinates to previously known harmonic Kerr coordinates remains an open problem.

In aerodynamics, accurately modeling subsonic compressible flow over airfoils is critical for aircraft design. However, solving the governing nonlinear perturbation velocity potential equation presents computational challenges. Traditional approaches often rely on linearized equations or finite, truncated domains, which introduce non-negligible errors and limit applicability in real-world scenarios. In this study, we propose a novel framework utilizing Physics-Informed Neural Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded (infinite) domain. We address the unbounded-domain and convergence challenges inherent in standard PINNs by incorporating a coordinate transformation and embedding physical asymptotic constraints directly into the network architecture. Furthermore, we employ a Multi-Stage PINN (MS-PINN) approach to iteratively minimize residuals, achieving solution accuracy approaching machine precision. We validate this framework by simulating flow over circular and elliptical geometries, comparing our results against traditional finite-domain and linearized solutions. Our findings quantify the noticeable discrepancies introduced by domain truncation and linearization, particularly at higher Mach numbers, and demonstrate that this new framework is a robust, high-fidelity tool for computational fluid dynamics.

Numerical simulations of combustion processes rely on chemical kinetic models, whose predictive capability is typically assessed by comparing measurements of quantities of interest (QoIs) in idealized experimental configurations with the corresponding simulated values. Sensitivity analysis quantifies the influence of model parameters on QoIs, making it a key tool for model development and optimization. Given the large number of parameters involved in detailed kinetic models compared to the number of QoIs, adjoint methods are a promising approach for efficient sensitivity analysis. Ignition delay times (IDTs) and extinction strain rates (ESRs) are among the QoIs of primary importance. However, applying adjoint methods to these quantities is not straightforward and is addressed in this work. The system of governing equations is reformulated via a coordinate transformation that makes the QoI an explicit component of the solution vector. This reformulation enables the application of adjoint sensitivity analysis for both IDT and ESR. A computational complexity analysis demonstrates that the proposed formulation significantly improves algorithmic efficiency compared to conventional approaches. The methodology is verified by comparing adjoint sensitivity coefficients with those obtained from other methods. Execution time measurements show computational cost reductions of up to two orders of magnitude relative to standard direct sensitivity analysis. A benchmark study is conducted across kinetic models of increasing size to assess how far theoretical efficiency gains are realized in practice. While the results show that implementation details can limit performance for smaller problem sizes, the expected asymptotic scaling is observed for larger models. The benchmark thus provides a reference framework for identifying key performance bottlenecks and guiding subsequent optimization efforts.

Grid-based structures are commonly used to encode explicit features for graphics primitives such as images, signed distance functions (SDF), and neural radiance fields (NeRF) due to their simple implementation. However, in n-dimensional space, calculating the value of a sampled point requires interpolating the values of its 2^n neighboring vertices. The exponential scaling with dimension leads to significant computational overheads. To address this issue, we propose a simplex-based approach for encoding graphics primitives. The number of vertices in a simplex-based structure increases linearly with dimension, making it a more efficient and generalizable alternative to grid-based representations. Using the non-axis-aligned simplicial structure property, we derive and prove a coordinate transformation, simplicial subdivision, and barycentric interpolation scheme for efficient sampling, which resembles transformation procedures in the simplex noise algorithm. Finally, we use hash tables to store multiresolution features of all interest points in the simplicial grid, which are passed into a tiny fully connected neural network to parameterize graphics primitives. We implemented a detailed simplex-based structure encoding algorithm in C++ and CUDA using the methods outlined in our approach. In the 2D image fitting task, the proposed method is capable of fitting a giga-pixel image with 9.4% less time compared to the baseline method proposed by instant-ngp, while maintaining the same quality and compression rate. In the volumetric rendering setup, we observe a maximum 41.2% speedup when the samples are dense enough.

3D spatial reasoning in dynamic, audio-visual environments is a cornerstone of human cognition yet remains largely unexplored by existing Audio-Visual Large Language Models (AV-LLMs) and benchmarks, which predominantly focus on static or 2D scenes. We introduce SAVVY-Bench, the first benchmark for 3D spatial reasoning in dynamic scenes with synchronized spatial audio. SAVVY-Bench is comprised of thousands of relationships involving static and moving objects, and requires fine-grained temporal grounding, consistent 3D localization, and multi-modal annotation. To tackle this challenge, we propose SAVVY, a novel training-free reasoning pipeline that consists of two stages: (i) Egocentric Spatial Tracks Estimation, which leverages AV-LLMs as well as other audio-visual methods to track the trajectories of key objects related to the query using both visual and spatial audio cues, and (ii) Dynamic Global Map Construction, which aggregates multi-modal queried object trajectories and converts them into a unified global dynamic map. Using the constructed map, a final QA answer is obtained through a coordinate transformation that aligns the global map with the queried viewpoint. Empirical evaluation demonstrates that SAVVY substantially enhances performance of state-of-the-art AV-LLMs, setting a new standard and stage for approaching dynamic 3D spatial reasoning in AV-LLMs.

Generating human videos with realistic and controllable motions is a challenging task. While existing methods can generate visually compelling videos, they lack separate control over four key video elements: foreground subject, background video, human trajectory and action patterns. In this paper, we propose a decomposed human motion control and video generation framework that explicitly decouples motion from appearance, subject from background, and action from trajectory, enabling flexible mix-and-match composition of these elements. Concretely, we first build a ground-aware 3D world coordinate system and perform motion editing directly in the 3D space. Trajectory control is implemented by unprojecting edited 2D trajectories into 3D with focal-length calibration and coordinate transformation, followed by speed alignment and orientation adjustment; actions are supplied by a motion bank or generated via text-to-motion methods. Then, based on modern text-to-video diffusion transformer models, we inject the subject as tokens for full attention, concatenate the background along the channel dimension, and add motion (trajectory and action) control signals by addition. Such a design opens up the possibility for us to generate realistic videos of anyone doing anything anywhere. Extensive experiments on benchmark datasets and real-world cases demonstrate that our method achieves state-of-the-art performance on both element-wise controllability and overall video quality.

Recently, camera-controlled video generation has seen rapid development, offering more precise control over video generation. However, existing methods predominantly focus on camera control in perspective projection video generation, while geometrically consistent panoramic video generation remains challenging. This limitation is primarily due to the inherent complexities in panoramic pose representation and spherical projection. To address this issue, we propose CamPVG, the first diffusion-based framework for panoramic video generation guided by precise camera poses. We achieve camera position encoding for panoramic images and cross-view feature aggregation based on spherical projection. Specifically, we propose a panoramic Plücker embedding that encodes camera extrinsic parameters through spherical coordinate transformation. This pose encoder effectively captures panoramic geometry, overcoming the limitations of traditional methods when applied to equirectangular projections. Additionally, we introduce a spherical epipolar module that enforces geometric constraints through adaptive attention masking along epipolar lines. This module enables fine-grained cross-view feature aggregation, substantially enhancing the quality and consistency of generated panoramic videos. Extensive experiments demonstrate that our method generates high-quality panoramic videos consistent with camera trajectories, far surpassing existing methods in panoramic video generation.

Visual analysis and reconstruction of pipeline inner walls remain challenging in industrial inspection scenarios. This paper presents a dedicated reconstruction system for pipeline inner walls via industrial endoscopes, which is built on panoramic image stitching technology. Equipped with a custom graphical user interface (GUI), the system extracts key frames from endoscope video footage, and integrates polar coordinate transformation with image stitching techniques to unwrap annular video frames of pipeline inner walls into planar panoramic images. Experimental results demonstrate that the proposed method enables efficient processing of industrial endoscope videos, and the generated panoramic stitched images preserve all detailed features of pipeline inner walls in their entirety. This provides intuitive and accurate visual support for defect detection and condition assessment of pipeline inner walls. In comparison with the traditional frame-by-frame video review method, the proposed approach significantly elevates the efficiency of pipeline inner wall reconstruction and exhibits considerable engineering application value.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Visual-token compression is effective for improving the efficiency of vision-language models, but under extreme compression budgets, token pruning can break visual grounding while learned resamplers increase parameter count, attention cost, and training complexity. We revisit compression through a token parameterization lens, separating (i) basis transformation and structured truncation (retained subspace/compressibility) from (ii) coordinate organization (optimization and cross-modal alignment). This view yields two coupled objectives, compressibility and learnability, which we formalize as unified functionals. Guided by these objectives, we design Braco, a lightweight four-step coder that combines transform-basis truncation, input-independent basis-coordinate embeddings, budget-dependent orthogonal re-parameterization, and learned spatial residual tokens from lightweight pooling. Experiments show that Braco forms the favorable empirical accuracy-efficiency frontier under 23times--64times compression and remains competitive at 144times, reaching 95.2% accuracy while reducing prefill FLOPs by 84.2%--86.7% relative to the uncompressed upper bound. Against prior methods, Braco matches or improves accuracy while achieving up to approximately 36% end-to-end speedup and using 16.6times/78.8times lower compressor latency/FLOPs.

Cooperative perception offers an optimal solution to overcome the perception limitations of single-agent systems by leveraging Vehicle-to-Everything (V2X) communication for data sharing and fusion across multiple agents. However, most existing approaches focus on single-modality data exchange, limiting the potential of both homogeneous and heterogeneous fusion across agents. This overlooks the opportunity to utilize multi-modality data per agent, restricting the system's performance. In the automotive industry, manufacturers adopt diverse sensor configurations, resulting in heterogeneous combinations of sensor modalities across agents. To harness the potential of every possible data source for optimal performance, we design a robust LiDAR and camera cross-modality fusion module, Radian-Glue-Attention (RG-Attn), applicable to both intra-agent cross-modality fusion and inter-agent cross-modality fusion scenarios, owing to the convenient coordinate conversion by transformation matrix and the unified sampling/inversion mechanism. We also propose two different architectures, named Paint-To-Puzzle (PTP) and Co-Sketching-Co-Coloring (CoS-CoCo), for conducting cooperative perception. PTP aims for maximum precision performance and achieves smaller data packet size by limiting cross-agent fusion to a single instance, but requiring all participants to be equipped with LiDAR. In contrast, CoS-CoCo supports agents with any configuration-LiDAR-only, camera-only, or LiDAR-camera-both, presenting more generalization ability. Our approach achieves state-of-the-art (SOTA) performance on both real and simulated cooperative perception datasets. The code is now available at GitHub.

Coordinate networks are widely used in computer vision due to their ability to represent signals as compressed, continuous entities. However, training these networks with first-order optimizers can be slow, hindering their use in real-time applications. Recent works have opted for shallow voxel-based representations to achieve faster training, but this sacrifices memory efficiency. This work proposes a solution that leverages second-order optimization methods to significantly reduce training times for coordinate networks while maintaining their compressibility. Experiments demonstrate the effectiveness of this approach on various signal modalities, such as audio, images, videos, shape reconstruction, and neural radiance fields.

Incorporating spatial information, particularly those influenced by climate, weather, and demographic factors, is crucial for improving underwriting precision and enhancing risk management in insurance. However, spatial data are often unstructured, high-dimensional, and difficult to integrate into predictive models. Embedding methods are needed to convert spatial data into meaningful representations for modelling tasks. We propose a novel multi-view contrastive learning framework for generating spatial embeddings that combine information from multiple spatial data sources. To train the model, we construct a spatial dataset that merges satellite imagery and OpenStreetMap features across Europe. The framework aligns these spatial views with coordinate-based encodings, producing low-dimensional embeddings that capture both spatial structure and contextual similarity. Once trained, the model generates embeddings directly from latitude-longitude pairs, enabling any dataset with coordinates to be enriched with meaningful spatial features without requiring access to the original spatial inputs. In a case study on French real estate prices, we compare models trained on raw coordinates against those using our spatial embeddings as inputs. The embeddings consistently improve predictive accuracy across generalised linear, additive, and boosting models, while providing interpretable spatial effects and demonstrating transferability to unseen regions.

We introduce a novel grasp representation named the Unified Gripper Coordinate Space (UGCS) for grasp synthesis and grasp transfer. Our representation leverages spherical coordinates to create a shared coordinate space across different robot grippers, enabling it to synthesize and transfer grasps for both novel objects and previously unseen grippers. The strength of this representation lies in the ability to map palm and fingers of a gripper and the unified coordinate space. Grasp synthesis is formulated as predicting the unified spherical coordinates on object surface points via a conditional variational autoencoder. The predicted unified gripper coordinates establish exact correspondences between the gripper and object points, which is used to optimize grasp pose and joint values. Grasp transfer is facilitated through the point-to-point correspondence between any two (potentially unseen) grippers and solved via a similar optimization. Extensive simulation and real-world experiments showcase the efficacy of the unified grasp representation for grasp synthesis in generating stable and diverse grasps. Similarly, we showcase real-world grasp transfer from human demonstrations across different objects.

Geographic coordinates are common inputs in applied machine learning, but latitude and longitude describe position rather than the surrounding environment. Image-grounded location encoders learn this context from satellite or ground-level imagery; structured open map data offers another source of supervision whose utility for continuous coordinate encoders remains less explored. We introduce CartoVec, a coordinate encoder trained to predict Overture-derived land cover, land use, transportation, place and connectivity targets at multiple spatial scales, without image inputs or downstream-task labels. We evaluate frozen embeddings on Census outcomes, Airbnb prices and Philadelphia crime in the US Northeast using spatial block cross-validation. CartoVec has the highest mean performance on most Census outcomes and the highest average across the full suite, with several comparisons close across folds. The findings support open map data as a practical source of reusable location representations with a coordinate-to-vector interface that requires no map lookup at inference. The open-weight model is available at https://huggingface.co/gursoyfurkan/CartoVec.

GUI grounding, the task of mapping natural-language instructions to pixel coordinates, is crucial for autonomous agents, yet remains difficult for current VLMs. The core bottleneck is reliable patch-to-pixel mapping, which breaks when extrapolating to high-resolution displays unseen during training. Current approaches generate coordinates as text tokens directly from visual features, forcing the model to infer complex position-to-pixel mappings implicitly; as a result, accuracy degrades and failures proliferate on new resolutions. We address this with two complementary innovations. First, RULER tokens serve as explicit coordinate markers, letting the model reference positions similar to gridlines on a map and adjust rather than generate coordinates from scratch. Second, Interleaved MRoPE (I-MRoPE) improves spatial encoding by ensuring that width and height dimensions are represented equally, addressing the asymmetry of standard positional schemes. Experiments on ScreenSpot, ScreenSpot-V2, and ScreenSpot-Pro show consistent gains in grounding accuracy, with the largest improvements on high-resolution interfaces. By providing explicit spatial guidance rather than relying on implicit learning, our approach enables more reliable GUI automation across diverse resolutions and platforms.

Modern feed-forward 3D reconstruction methods like VGGT predict pixel-aligned pointmaps in camera-centric coordinate frames. However, this choice of coordinate frame is not always optimal. We propose instead to predict pointmaps in upright, gravity-aligned frames that exploit strong structural cues present in many real-world scenes. Unlike camera-centric frames, gravity-aligned frames share a common vertical axis across viewpoints, reducing the rotational degrees of freedom needed to relate pointmaps to one another. To this end, we introduce the Gravity Grounded Geometry Transformer (G3T), fine-tuned from existing models on gravity-aligned 3D data. G3T produces highly accurate gravity-aware predictions, including upright pointmaps and camera-to-gravity poses. We further introduce G3T-Long, a submap-based incremental 3D reconstruction pipeline that leverages the reduced rotational degrees of freedom afforded by upright frames to achieve significantly improved reconstruction accuracy.

Multi-person 3D mesh recovery from videos is a critical first step towards automatic perception of group behavior in virtual reality, physical therapy and beyond. However, existing approaches rely on multi-stage paradigms, where the person detection and tracking stages are performed in a multi-person setting, while temporal dynamics are only modeled for one person at a time. Consequently, their performance is severely limited by the lack of inter-person interactions in the spatial-temporal mesh recovery, as well as by detection and tracking defects. To address these challenges, we propose the Coordinate transFormer (CoordFormer) that directly models multi-person spatial-temporal relations and simultaneously performs multi-mesh recovery in an end-to-end manner. Instead of partitioning the feature map into coarse-scale patch-wise tokens, CoordFormer leverages a novel Coordinate-Aware Attention to preserve pixel-level spatial-temporal coordinate information. Additionally, we propose a simple, yet effective Body Center Attention mechanism to fuse position information. Extensive experiments on the 3DPW dataset demonstrate that CoordFormer significantly improves the state-of-the-art, outperforming the previously best results by 4.2%, 8.8% and 4.7% according to the MPJPE, PAMPJPE, and PVE metrics, respectively, while being 40% faster than recent video-based approaches. The released code can be found at https://github.com/Li-Hao-yuan/CoordFormer.

We present, for the first time, a novel theoretical approach to address the problem of correspondence free multivector cloud registration in conformal geometric algebra. Such formalism achieves several favorable properties. Primarily, it forms an orthogonal automorphism that extends beyond the typical vector space to the entire conformal geometric algebra while respecting the multivector grading. Concretely, the registration can be viewed as an orthogonal transformation (\it i.e., scale, translation, rotation) belonging to SO(4,1) - group of special orthogonal transformations in conformal geometric algebra. We will show that such formalism is able to: (i) perform the registration without directly accessing the input multivectors. Instead, we use primitives or geometric objects provided by the conformal model - the multivectors, (ii) the geometric objects are obtained by solving a multilinear eigenvalue problem to find sets of eigenmultivectors. In this way, we can explicitly avoid solving the correspondences in the registration process. Most importantly, this offers rotation and translation equivariant properties between the input multivectors and the eigenmultivectors. Experimental evaluation is conducted in datasets commonly used in point cloud registration, to testify the usefulness of the approach with emphasis to ambiguities arising from high levels of noise. The code is available at https://github.com/Numerical-Geometric-Algebra/RegistrationGA . This work was submitted to the International Journal of Computer Vision and is currently under review.

Dimensionality reduction techniques are fundamental for analyzing and visualizing high-dimensional data. With established methods like t-SNE and PCA presenting a trade-off between representational power and interpretability. This paper introduces a novel approach that bridges this gap by combining the interpretability of linear methods with the expressiveness of non-linear transformations. The proposed algorithm constructs a non-linear mapping between high-dimensional and low-dimensional spaces through a combination of linear transformations, each weighted by Gaussian functions. This architecture enables complex non-linear transformations while preserving the interpretability advantages of linear methods, as each transformation can be analyzed independently. The resulting model provides both powerful dimensionality reduction and transparent insights into the transformed space. Techniques for interpreting the learned transformations are presented, including methods for identifying suppressed dimensions and how space is expanded and contracted. These tools enable practitioners to understand how the algorithm preserves and modifies geometric relationships during dimensionality reduction. To ensure the practical utility of this algorithm, the creation of user-friendly software packages is emphasized, facilitating its adoption in both academia and industry.

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.

In this paper, we adopt the Universal Manifold Embedding (UME) framework for the estimation of rigid transformations and extend it, so that it can accommodate scenarios involving partial overlap and differently sampled point clouds. UME is a methodology designed for mapping observations of the same object, related by rigid transformations, into a single low-dimensional linear subspace. This process yields a transformation-invariant representation of the observations, with its matrix form representation being covariant (i.e. equivariant) with the transformation. We extend the UME framework by introducing a UME-compatible feature extraction method augmented with a unique UME contrastive loss and a sampling equalizer. These components are integrated into a comprehensive and robust registration pipeline, named UMERegRobust. We propose the RotKITTI registration benchmark, specifically tailored to evaluate registration methods for scenarios involving large rotations. UMERegRobust achieves better than state-of-the-art performance on the KITTI benchmark, especially when strict precision of (1{\deg}, 10cm) is considered (with an average gain of +9%), and notably outperform SOTA methods on the RotKITTI benchmark (with +45% gain compared the most recent SOTA method).

In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.

Recent techniques for text-to-4D generation synthesize dynamic 3D scenes using supervision from pre-trained text-to-video models. However, existing representations for motion, such as deformation models or time-dependent neural representations, are limited in the amount of motion they can generate-they cannot synthesize motion extending far beyond the bounding box used for volume rendering. The lack of a more flexible motion model contributes to the gap in realism between 4D generation methods and recent, near-photorealistic video generation models. Here, we propose TC4D: trajectory-conditioned text-to-4D generation, which factors motion into global and local components. We represent the global motion of a scene's bounding box using rigid transformation along a trajectory parameterized by a spline. We learn local deformations that conform to the global trajectory using supervision from a text-to-video model. Our approach enables the synthesis of scenes animated along arbitrary trajectories, compositional scene generation, and significant improvements to the realism and amount of generated motion, which we evaluate qualitatively and through a user study. Video results can be viewed on our website: https://sherwinbahmani.github.io/tc4d.

In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this to topological concepts such as homeomorphism and embedding. We then investigate what are continuous and discontinuous representations for 2D, 3D, and n-dimensional rotations. We demonstrate that for 3D rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions. Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. We also present continuous representations for the general case of the n-dimensional rotation group SO(n). While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms. We finally present empirical results, which show that our continuous rotation representations outperform discontinuous ones for several practical problems in graphics and vision, including a simple autoencoder sanity test, a rotation estimator for 3D point clouds, and an inverse kinematics solver for 3D human poses.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Generating complete 3D assets from sparse, unposed images requires preserving observed geometry while completing unseen regions. Joint reconstruction and generation offers a way to address both goals: reconstruction recovers the geometry observed in the input images, while generation contributes learned shape priors. However, combining camera-relative reconstruction with canonical-space generation introduces a coordinate mismatch that makes direct geometric interaction more difficult. Moreover, conditioning each 3D location on all views without accounting for their relevance may mix useful geometric evidence with unrelated image content, hindering local detail recovery. We present UniForge3D, which jointly models reconstruction and generation in a shared anchor-guided space, resolving their coordinate mismatch. This space follows the anchor view's azimuth while keeping the object upright. The two branches exchange features bidirectionally to predict per-view point maps and a complete sparse voxel structure. We further condition point-map prediction on the generated structure to establish pixel-to-voxel correspondences and select the most informative input view for each voxel. Features from the selected views then guide detailed shape generation, helping preserve observed geometric details. Our method achieves state-of-the-art appearance and geometry results on Toys4K and GSO, generating complete 3D assets with improved fidelity. Project webpage: https://xishuxishu.github.io/uniforge3d-project/

Learning feature representations of geographical space is vital for any machine learning model that integrates geolocated data, spanning application domains such as remote sensing, ecology, or epidemiology. Recent work mostly embeds coordinates using sine and cosine projections based on Double Fourier Sphere (DFS) features -- these embeddings assume a rectangular data domain even on global data, which can lead to artifacts, especially at the poles. At the same time, relatively little attention has been paid to the exact design of the neural network architectures these functional embeddings are combined with. This work proposes a novel location encoder for globally distributed geographic data that combines spherical harmonic basis functions, natively defined on spherical surfaces, with sinusoidal representation networks (SirenNets) that can be interpreted as learned Double Fourier Sphere embedding. We systematically evaluate the cross-product of positional embeddings and neural network architectures across various classification and regression benchmarks and synthetic evaluation datasets. In contrast to previous approaches that require the combination of both positional encoding and neural networks to learn meaningful representations, we show that both spherical harmonics and sinusoidal representation networks are competitive on their own but set state-of-the-art performances across tasks when combined. We provide source code at www.github.com/marccoru/locationencoder

Scene coordinate regression (SCR) methods are a family of visual localization methods that directly regress 2D-3D matches for camera pose estimation. They are effective in small-scale scenes but face significant challenges in large-scale scenes that are further amplified in the absence of ground truth 3D point clouds for supervision. Here, the model can only rely on reprojection constraints and needs to implicitly triangulate the points. The challenges stem from a fundamental dilemma: The network has to be invariant to observations of the same landmark at different viewpoints and lighting conditions, etc., but at the same time discriminate unrelated but similar observations. The latter becomes more relevant and severe in larger scenes. In this work, we tackle this problem by introducing the concept of co-visibility to the network. We propose GLACE, which integrates pre-trained global and local encodings and enables SCR to scale to large scenes with only a single small-sized network. Specifically, we propose a novel feature diffusion technique that implicitly groups the reprojection constraints with co-visibility and avoids overfitting to trivial solutions. Additionally, our position decoder parameterizes the output positions for large-scale scenes more effectively. Without using 3D models or depth maps for supervision, our method achieves state-of-the-art results on large-scale scenes with a low-map-size model. On Cambridge landmarks, with a single model, we achieve 17% lower median position error than Poker, the ensemble variant of the state-of-the-art SCR method ACE. Code is available at: https://github.com/cvg/glace.

Recent methods for arbitrary-skeleton motion capture from monocular video follow a factorized pipeline, where a Video-to-Pose network predicts joint positions and an analytical inverse-kinematics (IK) stage recovers joint rotations. While effective, this design is inherently limited, since joint positions do not fully determine rotations and leave degrees of freedom such as bone-axis twist ambiguous, and the non-differentiable IK stage prevents the system from adapting to noisy predictions or optimizing for the final animation objective. In this work, we present the first fully end-to-end framework in which both Video-to-Pose and Pose-to-Rotation are learnable and jointly optimized. We observe that the ambiguity in pose-to-rotation mapping arises from missing coordinate system information: the same joint positions can correspond to different rotations under different rest poses and local axis conventions. To resolve this, we introduce a reference pose-rotation pair from the target asset, which, together with the rest pose, not only anchors the mapping but also defines the underlying rotation coordinate system. This formulation turns rotation prediction into a well-constrained conditional problem and enables effective learning. In addition, our model predicts joint positions directly from video without relying on mesh intermediates, improving both robustness and efficiency. Both stages share a skeleton-aware Global-Local Graph-guided Multi-Head Attention (GL-GMHA) module for joint-level local reasoning and global coordination. Experiments on Truebones Zoo and Objaverse show that our method reduces rotation error from ~17 degrees to ~10 degrees, and to 6.54 degrees on unseen skeletons, while achieving ~20x faster inference than mesh-based pipelines. Project page: https://animotionlab.github.io/MoCapAnythingV2/

We study the problem of extracting accurate correspondences for point cloud registration. Recent keypoint-free methods bypass the detection of repeatable keypoints which is difficult in low-overlap scenarios, showing great potential in registration. They seek correspondences over downsampled superpoints, which are then propagated to dense points. Superpoints are matched based on whether their neighboring patches overlap. Such sparse and loose matching requires contextual features capturing the geometric structure of the point clouds. We propose Geometric Transformer to learn geometric feature for robust superpoint matching. It encodes pair-wise distances and triplet-wise angles, making it robust in low-overlap cases and invariant to rigid transformation. The simplistic design attains surprisingly high matching accuracy such that no RANSAC is required in the estimation of alignment transformation, leading to 100 times acceleration. Our method improves the inlier ratio by 17{sim}30 percentage points and the registration recall by over 7 points on the challenging 3DLoMatch benchmark. Our code and models are available at https://github.com/qinzheng93/GeoTransformer.

When manipulating three-dimensional data, it is possible to ensure that rotational and translational symmetries are respected by applying so-called SE(3)-equivariant models. Protein structure prediction is a prominent example of a task which displays these symmetries. Recent work in this area has successfully made use of an SE(3)-equivariant model, applying an iterative SE(3)-equivariant attention mechanism. Motivated by this application, we implement an iterative version of the SE(3)-Transformer, an SE(3)-equivariant attention-based model for graph data. We address the additional complications which arise when applying the SE(3)-Transformer in an iterative fashion, compare the iterative and single-pass versions on a toy problem, and consider why an iterative model may be beneficial in some problem settings. We make the code for our implementation available to the community.

We address the task of estimating camera parameters from a set of images depicting a scene. Popular feature-based structure-from-motion (SfM) tools solve this task by incremental reconstruction: they repeat triangulation of sparse 3D points and registration of more camera views to the sparse point cloud. We re-interpret incremental structure-from-motion as an iterated application and refinement of a visual relocalizer, that is, of a method that registers new views to the current state of the reconstruction. This perspective allows us to investigate alternative visual relocalizers that are not rooted in local feature matching. We show that scene coordinate regression, a learning-based relocalization approach, allows us to build implicit, neural scene representations from unposed images. Different from other learning-based reconstruction methods, we do not require pose priors nor sequential inputs, and we optimize efficiently over thousands of images. Our method, ACE0 (ACE Zero), estimates camera poses to an accuracy comparable to feature-based SfM, as demonstrated by novel view synthesis. Project page: https://nianticlabs.github.io/acezero/

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 present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters. These symmetries arise from the presence of one or more planes/axis of symmetry in the system's morphology, resulting in harmonious duplication and distribution of body parts. Significantly, we characterize how morphological symmetries extend to symmetries in the system's dynamics, optimal control policies, and in all proprioceptive and exteroceptive measurements related to the system's dynamics evolution. In the context of data-driven methods, symmetry represents an inductive bias that justifies the use of data augmentation or symmetric function approximators. To tackle this, we present a theoretical and practical framework for identifying the system's morphological symmetry group G and characterizing the symmetries in proprioceptive and exteroceptive data measurements. We then exploit these symmetries using data augmentation and G-equivariant neural networks. Our experiments on both synthetic and real-world applications provide empirical evidence of the advantageous outcomes resulting from the exploitation of these symmetries, including improved sample efficiency, enhanced generalization, and reduction of trainable parameters.

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 construction of a theory of quantum gravity is an outstanding problem that can benefit from better understanding the laws of nature that are expected to hold in regimes currently inaccessible to experiment. Such fundamental laws can be found by considering the classical counterparts of a quantum theory. For example, conservation laws in a quantum theory often stem from conservation laws of the corresponding classical theory. In order to construct such laws, this thesis is concerned with the interplay between symmetries and conservation laws of classical field theories and their application to asymptotically flat spacetimes. This work begins with an explanation of symmetries in field theories with a focus on variational symmetries and their associated conservation laws. Boundary conditions for general relativity are then formulated on three-dimensional asymptotically flat spacetimes at null infinity using the method of conformal completion. Conserved quantities related to asymptotic symmetry transformations are derived and their properties are studied. This is done in a manifestly coordinate independent manner. In a separate step a coordinate system is introduced, such that the results can be compared to existing literature. Next, asymptotically flat spacetimes which contain both future as well as past null infinity are considered. Asymptotic symmetries occurring at these disjoint regions of three-dimensional asymptotically flat spacetimes are linked and the corresponding conserved quantities are matched. Finally, it is shown how asymptotic symmetries lead to the notion of distinct Minkowski spaces that can be differentiated by conserved quantities.

Transformer representations evolve through learned additive transformations that either preserve their current direction or redirect it. We study this evolution as a functional geometry, decomposing learned updates into parallel and perpendicular components. Across pretrained models, we find substantial parallel components beyond the residual identity path. We then apply the decomposition in two spaces: to attention and MLP updates relative to the hidden state, and to attention value aggregation relative to the current token's value. Targeted edits reveal a strongly space-dependent asymmetry: exclude-self value-space parallel manipulation is markedly more robust than residual-space and perpendicular counterparts, preserving the direct self message while scaling only the non-self aggregate. The same decomposition gives a component-resolved description of compression-induced update error: perpendicular error separates compression methods more clearly than parallel error. Extensive experiments further demonstrate that full-aggregate parallel suppression during from-scratch pretraining lowers validation-loss trajectories and improves downstream averages, with the value-space variant strongest. Together, these results connect representation geometry to editing robustness, compression diagnosis, and training-time intervention. Code is available in the https://github.com/Shwai-He/Transformer-Geometry{project repository}.

Point cloud registration is challenging in the presence of heavy outlier correspondences. This paper focuses on addressing the robust correspondence-based registration problem with gravity prior that often arises in practice. The gravity directions are typically obtained by inertial measurement units (IMUs) and can reduce the degree of freedom (DOF) of rotation from 3 to 1. We propose a novel transformation decoupling strategy by leveraging screw theory. This strategy decomposes the original 4-DOF problem into three sub-problems with 1-DOF, 2-DOF, and 1-DOF, respectively, thereby enhancing the computation efficiency. Specifically, the first 1-DOF represents the translation along the rotation axis and we propose an interval stabbing-based method to solve it. The second 2-DOF represents the pole which is an auxiliary variable in screw theory and we utilize a branch-and-bound method to solve it. The last 1-DOF represents the rotation angle and we propose a global voting method for its estimation. The proposed method sequentially solves three consensus maximization sub-problems, leading to efficient and deterministic registration. In particular, it can even handle the correspondence-free registration problem due to its significant robustness. Extensive experiments on both synthetic and real-world datasets demonstrate that our method is more efficient and robust than state-of-the-art methods, even when dealing with outlier rates exceeding 99%.

Testing equality of two high-dimensional covariance matrices is challenging when many entries differ only slightly. Dependence among sample covariance entries can also affect the finite-sample size and power of tests that aggregate their differences. We propose a studentized ell_2-type statistic that averages squared, marginally standardized differences between corresponding entries of the two sample covariance matrices. Before constructing the statistic, we apply the same nonsingular linear transformation to both samples. In the transformed coordinates, the covariance-equality hypothesis is unchanged, whereas the standardized differences and correlations among their estimators generally change. The transformation can therefore incorporate structural or scientific information without reducing dimension. We approximate the null distribution using a Gaussian multiplier bootstrap that uses coordinatewise studentization and avoids forming or inverting the full covariance matrix of the vectorized sample covariance entries. For deterministic transformations, we derive nonasymptotic Gaussian and bootstrap approximation bounds and establish asymptotic size validity and power consistency. We also show that replacing a population transformation by a same-sample estimator leaves the statistic and bootstrap critical value asymptotically unchanged in relative terms under an operator-norm convergence condition. Simulations show that suitable transformations improve size accuracy and power under weak dependence. An analysis of breast cancer gene-expression data illustrates the method.

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.

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/}.

We generalize quasi-arithmetic means beyond scalars by considering the gradient map of a Legendre type real-valued function. The gradient map of a Legendre type function is proven strictly comonotone with a global inverse. It thus yields a generalization of strictly mononotone and differentiable functions generating scalar quasi-arithmetic means. Furthermore, the Legendre transformation gives rise to pairs of dual quasi-arithmetic averages via the convex duality. We study the invariance and equivariance properties under affine transformations of quasi-arithmetic averages via the lens of dually flat spaces of information geometry. We show how these quasi-arithmetic averages are used to express points on dual geodesics and sided barycenters in the dual affine coordinate systems. We then consider quasi-arithmetic mixtures and describe several parametric and non-parametric statistical models which are closed under the quasi-arithmetic mixture operation.

Recent 4D Gaussian Splatting (4DGS) methods achieve impressive dynamic scene reconstruction but often rely on piecewise linear velocity approximations and short temporal windows. This disjointed modeling leads to severe temporal fragmentation, forcing primitives to be repeatedly eliminated and regenerated to track complex nonlinear dynamics. This makeshift approximation eliminates the long-term temporal identity of objects and causes an inevitable proliferation of Gaussians, hindering scalability to extended video sequences. To address this, we propose TRiGS, a novel 4D representation that utilizes unified, continuous geometric transformations. By integrating SE(3) transformations, hierarchical Bezier residuals, and learnable local anchors, TRiGS models geometrically consistent rigid motions for individual primitives. This continuous formulation preserves temporal identity and effectively mitigates unbounded memory growth. Extensive experiments demonstrate that TRiGS achieves high fidelity rendering on standard benchmarks while uniquely scaling to extended video sequences (e.g., 600 to 1200 frames) without severe memory bottlenecks, significantly outperforming prior works in temporal stability.

We address the problem of performing backpropagation for computation graphs involving 3D transformation groups SO(3), SE(3), and Sim(3). 3D transformation groups are widely used in 3D vision and robotics, but they do not form vector spaces and instead lie on smooth manifolds. The standard backpropagation approach, which embeds 3D transformations in Euclidean spaces, suffers from numerical difficulties. We introduce a new library, which exploits the group structure of 3D transformations and performs backpropagation in the tangent spaces of manifolds. We show that our approach is numerically more stable, easier to implement, and beneficial to a diverse set of tasks. Our plug-and-play PyTorch library is available at https://github.com/princeton-vl/lietorch.

We present PhysiFormer, a diffusion transformer for physically-plausible 3D object motion. Unlike video world models that operate in view-dependent pixel space, PhysiFormer represents objects as 3D meshes expressed in world coordinates. Given the initial vertex positions and velocities, as well as object material type, rigid or elastic, the model samples future vertex trajectories. While related neural physics approaches build on ad-hoc latent spaces or explicitly enforce rigidity and causality, PhysiFormer shows that excellent results can be obtained without any such inductive biases, by casting vertex trajectory prediction as a single denoising diffusion process directly in world coordinates. The probabilistic formulation captures uncertainty in the learned dynamics, enabling diverse plausible futures from initial conditions, making this framework potentially useful for applications with unobserved uncertainty. The model features attention factorised over time, space, and objects for efficiency, enabling permutation-invariant multi-object reasoning without needing explicit object encoding. Trained on over 100k simulated trajectories, PhysiFormer generates rigid and elastic mechanics, and generalises to mixed-material settings, unseen real-world geometries, and larger object counts. It substantially outperforms autoregressive baselines in trajectory accuracy, rigidity preservation, and momentum-based physical consistency. Our results position coordinate-space diffusion as a promising step toward view-invariant, geometry-aware world modelling for robotics, graphics, and physical design. Visualisations, code, and models are available at https://yimingc9.github.io/physiformer.

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/

Single-view 3D generative models have achieved impressive visual quality, yet they are not designed to satisfy structural or functional requirements, and in practice, often fall short. Symmetry is one such requirement: violations, even subtle ones, on symmetry can render a model physically unusable. We present SymTRELLIS, a method that enforces arbitrary finite point group symmetries (rotational, reflectional, and polyhedral) during the flow-based 3D generation of TRELLIS.2, without retraining the underlying VAE or flow model. Our key idea is to approximate the latent-space action of spatial transformations as a learned linear operator on voxel latents, implemented as a lightweight spatial-transform latent mapper trained on generic, non-symmetric 3D data. At generation time, we enforce symmetry by averaging predicted flow velocities across all symmetry-equivalent transformations at each ODE step, a process we call velocity symmetrization. The symmetry specification can be estimated automatically from an initial TRELLIS.2 generation or supplied by the user, enabling deliberate fold manipulation beyond what the input image suggests. On a curated benchmark of 266 strictly symmetric objects spanning 2- to 20-fold rotations and polyhedral symmetry groups, SymTRELLIS substantially reduces all symmetry error metrics compared to TRELLIS.2, Hunyuan3D-2.1, and TripoSG, while maintaining reconstruction accuracy comparable to the base model.

This paper concerns the research problem of point cloud registration to find the rigid transformation to optimally align the source point set with the target one. Learning robust point cloud registration models with deep neural networks has emerged as a powerful paradigm, offering promising performance in predicting the global geometric transformation for a pair of point sets. Existing methods firstly leverage an encoder to regress a latent shape embedding, which is then decoded into a shape-conditioned transformation via concatenation-based conditioning. However, different regions of a 3D shape vary in their geometric structures which makes it more sense that we have a region-conditioned transformation instead of the shape-conditioned one. In this paper we present a Region-Aware point cloud Registration, denoted as RAR, to predict transformation for pairwise point sets in the self-supervised learning fashion. More specifically, we develop a novel region-aware decoder (RAD) module that is formed with an implicit neural region representation parameterized by neural networks. The implicit neural region representation is learned with a self-supervised 3D shape reconstruction loss without the need for region labels. Consequently, the region-aware decoder (RAD) module guides the training of the region-aware transformation (RAT) module and region-aware weight (RAW) module, which predict the transforms and weights for different regions respectively. The global geometric transformation from source point set to target one is then formed by the weighted fusion of region-aware transforms. Compared to the state-of-the-art approaches, our experiments show that our RAR achieves superior registration performance over various benchmark datasets (e.g. ModelNet40).

The next Point-of-Interest (POI) recommendation task aims to predict users' next destinations based on their historical movement data and plays a key role in location-based services and personalized applications. Accurate next POI recommendation depends on effectively modeling geographic information and POI transition relations, which are crucial for capturing spatial dependencies and user movement patterns. While Large Language Models (LLMs) exhibit strong capabilities in semantic understanding and contextual reasoning, applying them to spatial tasks like next POI recommendation remains challenging. First, the infrequent nature of specific GPS coordinates makes it difficult for LLMs to model precise spatial contexts. Second, the lack of knowledge about POI transitions limits their ability to capture potential POI-POI relationships. To address these issues, we propose GA-LLM (Geography-Aware Large Language Model), a novel framework that enhances LLMs with two specialized components. The Geographic Coordinate Injection Module (GCIM) transforms GPS coordinates into spatial representations using hierarchical and Fourier-based positional encoding, enabling the model to understand geographic features from multiple perspectives. The POI Alignment Module (PAM) incorporates POI transition relations into the LLM's semantic space, allowing it to infer global POI relationships and generalize to unseen POIs. Experiments on three real-world datasets demonstrate the state-of-the-art performance of GA-LLM.

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.

Coordinate-based neural representations have shown significant promise as an alternative to discrete, array-based representations for complex low dimensional signals. However, optimizing a coordinate-based network from randomly initialized weights for each new signal is inefficient. We propose applying standard meta-learning algorithms to learn the initial weight parameters for these fully-connected networks based on the underlying class of signals being represented (e.g., images of faces or 3D models of chairs). Despite requiring only a minor change in implementation, using these learned initial weights enables faster convergence during optimization and can serve as a strong prior over the signal class being modeled, resulting in better generalization when only partial observations of a given signal are available. We explore these benefits across a variety of tasks, including representing 2D images, reconstructing CT scans, and recovering 3D shapes and scenes from 2D image observations.

We present a novel method for recovering world-grounded human motion from monocular video. The main challenge lies in the ambiguity of defining the world coordinate system, which varies between sequences. Previous approaches attempt to alleviate this issue by predicting relative motion in an autoregressive manner, but are prone to accumulating errors. Instead, we propose estimating human poses in a novel Gravity-View (GV) coordinate system, which is defined by the world gravity and the camera view direction. The proposed GV system is naturally gravity-aligned and uniquely defined for each video frame, largely reducing the ambiguity of learning image-pose mapping. The estimated poses can be transformed back to the world coordinate system using camera rotations, forming a global motion sequence. Additionally, the per-frame estimation avoids error accumulation in the autoregressive methods. Experiments on in-the-wild benchmarks demonstrate that our method recovers more realistic motion in both the camera space and world-grounded settings, outperforming state-of-the-art methods in both accuracy and speed. The code is available at https://zju3dv.github.io/gvhmr/.

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

Feedforward 3D reconstruction has largely followed an {appearance-driven paradigm}: multi-view images are encoded into visual representations, cross-view correspondence is implicitly inferred from learned features, and 3D primitives are subsequently predicted from these representations. However, calibrated cameras and sparse metric measurements already provide explicit constraints on where the same physical point should appear across views. We propose a novel paradigm, Geometry-Conditioned Feedforward Reconstruction, in which metric geometry establishes cross-view correspondence before the 3D representation is formed, while learned visual features refine rather than discover this correspondence. Based on this paradigm, we introduce GeoForward. Sparse LiDAR points are unprojected into 3D and analytically projected across calibrated cameras to provide metric correspondence anchors. A bounded continuous local search then refines these anchors using visual features, and the resulting geometry-conditioned representation is fused with global multi-view context from VGGT. The shared representation drives separate metric-depth and Gaussian-attribute prediction, establishing a direct path from explicit geometry to both Gaussian centers and spatial attributes. A sparse 3D refinement module subsequently improves local attribute compatibility while preserving the geometrically determined centers. For temporal reconstruction, endpoint Gaussians are composed using vehicle masks and motion priors. On nuScenes, GeoForward achieves the lowest cross-camera depth-consistency error and Chamfer distance among the compared methods while maintaining competitive reconstruction quality. These results demonstrate that explicitly conditioning correspondence formation on metric geometry provides an effective alternative to purely appearance-driven feedforward 3D reconstruction.

Point cloud completion aims to recover accurate global geometry and preserve fine-grained local details from partial point clouds. Conventional methods typically predict unseen points directly from 3D point cloud coordinates or use self-projected multi-view depth maps to ease this task. However, these gray-scale depth maps cannot reach multi-view consistency, consequently restricting the performance. In this paper, we introduce a GeoFormer that simultaneously enhances the global geometric structure of the points and improves the local details. Specifically, we design a CCM Feature Enhanced Point Generator to integrate image features from multi-view consistent canonical coordinate maps (CCMs) and align them with pure point features, thereby enhancing the global geometry feature. Additionally, we employ the Multi-scale Geometry-aware Upsampler module to progressively enhance local details. This is achieved through cross attention between the multi-scale features extracted from the partial input and the features derived from previously estimated points. Extensive experiments on the PCN, ShapeNet-55/34, and KITTI benchmarks demonstrate that our GeoFormer outperforms recent methods, achieving the state-of-the-art performance. Our code is available at https://github.com/Jinpeng-Yu/GeoFormer{https://github.com/Jinpeng-Yu/GeoFormer}.

Processing information on 3D objects requires methods stable to rigid-body transformations, in particular rotations, of the input data. In image processing tasks, convolutional neural networks achieve this property using rotation-equivariant operations. However, contrary to images, graphs generally have irregular topology. This makes it challenging to define a rotation-equivariant convolution operation on these structures. In this work, we propose Spherical Graph Convolutional Network (S-GCN) that processes 3D models of proteins represented as molecular graphs. In a protein molecule, individual amino acids have common topological elements. This allows us to unambiguously associate each amino acid with a local coordinate system and construct rotation-equivariant spherical filters that operate on angular information between graph nodes. Within the framework of the protein model quality assessment problem, we demonstrate that the proposed spherical convolution method significantly improves the quality of model assessment compared to the standard message-passing approach. It is also comparable to state-of-the-art methods, as we demonstrate on Critical Assessment of Structure Prediction (CASP) benchmarks. The proposed technique operates only on geometric features of protein 3D models. This makes it universal and applicable to any other geometric-learning task where the graph structure allows constructing local coordinate systems.

In visual computing, 3D geometry is represented in many different forms including meshes, point clouds, voxel grids, level sets, and depth images. Each representation is suited for different tasks thus making the transformation of one representation into another (forward map) an important and common problem. We propose Omnidirectional Distance Fields (ODFs), a new 3D shape representation that encodes geometry by storing the depth to the object's surface from any 3D position in any viewing direction. Since rays are the fundamental unit of an ODF, it can be used to easily transform to and from common 3D representations like meshes or point clouds. Different from level set methods that are limited to representing closed surfaces, ODFs are unsigned and can thus model open surfaces (e.g., garments). We demonstrate that ODFs can be effectively learned with a neural network (NeuralODF) despite the inherent discontinuities at occlusion boundaries. We also introduce efficient forward mapping algorithms for transforming ODFs to and from common 3D representations. Specifically, we introduce an efficient Jumping Cubes algorithm for generating meshes from ODFs. Experiments demonstrate that NeuralODF can learn to capture high-quality shape by overfitting to a single object, and also learn to generalize on common shape categories.

Standard sequence models encode position by token rank rather than physical coordinate, typically forcing practitioners to destructively resample continuous physical signals such as spectra observed across heterogeneous instruments. We identify this coordinate mismatch as a structural limitation in cross-instrument representation learning, and propose OmniSpectra, a wavelength-native foundation model for spectral representation learning: a continuous-coordinate masked autoencoder that discards the integer pixel indexing of sequence modeling. Mapping physical coordinates to a logarithmic basis turns the multiplicative scaling induced by cosmological redshift into a uniform additive shift. In continuous rotary position embeddings, this shared shift cancels from relative token distances rather than producing grid-dependent positional distortion. We further counter the spectral bias inherent to implicit neural decoders with a physics-grounded, resolution-gated Fourier decoder that provides anti-aliasing. Pretrained on eight surveys and benchmarked against seven baselines, our model reaches state-of-the-art performance with only 12.2M parameters while outperforming substantially larger models, indicating that encoding physical structure can substitute for parameter scale.

Visual grounding is a central challenge for GUI agents, requiring an instruction to be mapped to a target within a crowded screenshot. Coordinate generation with single-point supervision reduces a spatially extended target to one position, despite the presence of many valid click locations. Region-based patch scoring alleviates this ambiguity, but its spatial decisions remain tied to visual-token footprints that may span the target, neighboring controls, and background. To address these limitations, we propose Seg2Click, which reformulates GUI grounding as executable-region prediction: identifying where a click can validly execute the given instruction. Using bounding-box supervision, Seg2Click predicts an executable-region mask and computes the final click point through a deterministic rule. This formulation captures the spatial extent of valid click locations and enables finer localization beyond the visual-token grid, making it well suited to small, densely arranged controls without requiring a verifier or zoom-in pass. Experiments across five backbone configurations from two model families demonstrate the effectiveness of Seg2Click. Seg2Click-9B achieves 92.06\%, 94.10\%, and 55.22\% on ScreenSpot, ScreenSpot-v2, and ScreenSpot-Pro, respectively, while matched-backbone comparisons show improvements over both coordinate generation and patch-based grounding. Code will be available at https://github.com/jcwang0602/Seg2Click.

Contours or closed planar curves are common in many domains. For example, they appear as object boundaries in computer vision, isolines in meteorology, and the orbits of rotating machinery. In many cases when learning from contour data, planar rotations of the input will result in correspondingly rotated outputs. It is therefore desirable that deep learning models be rotationally equivariant. In addition, contours are typically represented as an ordered sequence of edge points, where the choice of starting point is arbitrary. It is therefore also desirable for deep learning methods to be equivariant under cyclic shifts. We present RotaTouille, a deep learning framework for learning from contour data that achieves both rotation and cyclic shift equivariance through complex-valued circular convolution. We further introduce and characterize equivariant non-linearities, coarsening layers, and global pooling layers to obtain invariant representations for downstream tasks. Finally, we demonstrate the effectiveness of RotaTouille through experiments in shape classification, reconstruction, and contour regression.

Let λ_{mathbb K}(m) denote the maximal absolute projection constant among m-dimensional Banach spaces over mathbb K=mathbb R or mathbb C. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for λ_{mathbb R}(m). Numerical evidence suggests that these bounds may be sharp in dimensions 6 and 8. We also discuss weighted spherical (2,2)-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions 4 and 5 indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

This work presents a novel online model-predictive trajectory planner for robotic manipulators called BoundMPC. This planner allows the collision-free following of Cartesian reference paths in the end-effector's position and orientation, including via-points, within desired asymmetric bounds of the orthogonal path error. The path parameter synchronizes the position and orientation reference paths. The decomposition of the path error into the tangential direction, describing the path progress, and the orthogonal direction, which represents the deviation from the path, is well known for the position from the path-following control in the literature. This paper extends this idea to the orientation by utilizing the Lie theory of rotations. Moreover, the orthogonal error plane is further decomposed into basis directions to define asymmetric Cartesian error bounds easily. Using piecewise linear position and orientation reference paths with via-points is computationally very efficient and allows replanning the pose trajectories during the robot's motion. This feature makes it possible to use this planner for dynamically changing environments and varying goals. The flexibility and performance of BoundMPC are experimentally demonstrated by two scenarios on a 7-DoF Kuka LBR iiwa 14 R820 robot. The first scenario shows the transfer of a larger object from a start to a goal pose through a confined space where the object must be tilted. The second scenario deals with grasping an object from a table where the grasping point changes during the robot's motion, and collisions with other obstacles in the scene must be avoided.

Hyperbolic neural networks have shown great potential for modeling complex data. However, existing hyperbolic networks are not completely hyperbolic, as they encode features in a hyperbolic space yet formalize most of their operations in the tangent space (a Euclidean subspace) at the origin of the hyperbolic space. This hybrid method greatly limits the modeling ability of networks. In this paper, we propose a fully hyperbolic framework to build hyperbolic networks based on the Lorentz model by adapting the Lorentz transformations (including boost and rotation) to formalize essential operations of neural networks. Moreover, we also prove that linear transformation in tangent spaces used by existing hyperbolic networks is a relaxation of the Lorentz rotation and does not include the boost, implicitly limiting the capabilities of existing hyperbolic networks. The experimental results on four NLP tasks show that our method has better performance for building both shallow and deep networks. Our code will be released to facilitate follow-up research.

Representing 3D surfaces as level sets of continuous functions over R^3 is the common denominator of neural implicit representations, which recently enabled remarkable progress in geometric deep learning and computer vision tasks. In order to represent 3D motion within this framework, it is often assumed (either explicitly or implicitly) that the transformations which a surface may undergo are homeomorphic: this is not necessarily true, for instance, in the case of fluid dynamics. In order to represent more general classes of deformations, we propose to apply this theoretical framework as regularizers for the optimization of simple 4D implicit functions (such as signed distance fields). We show that our representation is capable of capturing both homeomorphic and topology-changing deformations, while also defining correspondences over the continuously-reconstructed surfaces.

Polygon representation learning is essential for diverse applications, encompassing tasks such as shape coding, building pattern classification, and geographic question answering. While recent years have seen considerable advancements in this field, much of the focus has been on single polygons, overlooking the intricate inner- and inter-polygonal relationships inherent in multipolygons. To address this gap, our study introduces a comprehensive framework specifically designed for learning representations of polygonal geometries, particularly multipolygons. Central to our approach is the incorporation of a heterogeneous visibility graph, which seamlessly integrates both inner- and inter-polygonal relationships. To enhance computational efficiency and minimize graph redundancy, we implement a heterogeneous spanning tree sampling method. Additionally, we devise a rotation-translation invariant geometric representation, ensuring broader applicability across diverse scenarios. Finally, we introduce Multipolygon-GNN, a novel model tailored to leverage the spatial and semantic heterogeneity inherent in the visibility graph. Experiments on five real-world and synthetic datasets demonstrate its ability to capture informative representations for polygonal geometries. Code and data are available at https://github.com/dyu62/PolyGNN{github.com/dyu62/PolyGNN}.

Generative visual models fundamentally struggle with precise spatial control. This arises from a core disconnect: models can process textual descriptions of space but cannot directly map numerical coordinates onto the 2D image canvas. We introduce MetaPoint, a method that bridges this gap by representing a continuous 2D coordinate as a single, special token. Crucially, MetaPoint requires no new architectural components; it directly leverages the model's inherent positional encoding schemes to interpret these coordinates, treating our token as a virtual point on the canvas. This lightweight approach enables pixel-level control of an object's position with one token or its bounding box with two, all without requiring architectural changes or bespoke attention masking. The MetaPoint tokens are designed to be compositional, serving as spatial primitives. This allows a planner agent to decompose a high-level user request into a structured sequence of primitives for the generator. By providing a simple, precise, and scalable building block for spatial control, MetaPoint unlocks more powerful compositional generative agents and enables intuitive, interactive editing systems.

Before acting on an object, a robot needs to anticipate how its hand motion will change the physical world and what the resulting scene will look like. Existing approaches typically entangle action-conditioned dynamics, camera motion, and visual synthesis within a video generation model, or represent dynamics as point trajectories without explicit object structure. We introduce DeXplicit, a dexterous manipulation world model that instead predicts how the world evolves explicitly in metric 3D before rendering its visual consequences. Given an initial scene and a future hand trajectory, a lightweight 3D dynamics network predicts object-centric motion and deformation, providing structured and geometrically grounded future world states. These predictions then serve as a strong scaffold for video generation: appearance is transported through the predicted 3D motion and projected into future views, allowing the renderer to focus on completing uncertain and newly visible content rather than implicitly learning world dynamics from pixels. This explicit-3D-first decomposition also enables efficient causal generation, requiring only two denoising steps per block while conditioning on generated history. Across HOT3D, OakInk2, and PhysTwin, DeXplicit reduces moved-point error by 27-40% relative to PointWorld and improves object fidelity IoU+ by 47% on HOT3D under a shared renderer. Our design with two-step causal generation achieves a 57x speedup from our 14B bi-directional configuration in video synthesis.

Many dynamical processes unfold on the sphere but the default scientific machine learning architectures are Euclidean. Applying these architectures on a regular lat-lon grid causes problems: Cartesian convolutions become distorted at high latitude; 2D FFTs in Fourier neural operators incorrectly assume double periodicity; Cartesian positional encodings in ViTs distort spherical geodesic distances. Recent work moves towards natively spherical primitives, including spherical convolutions (e.g., DeepSphere or DISCO), Spherical Fourier Neural Operators (SFNOs), and geodesic attention. Here we propose Dandelion, a spherical version of Flower, a warp-based neural PDE solver. Layers of Dandelion predict a tangent-plane displacement and transport features along great circles. We obtain a U-Net-like structure by implementing hierarchical pooling entirely in the spherical-harmonic domain. There are thus no convolutions: spatial mixing is achieved only through spherical coordinate changes, or warps. To compare Dandelion with existing spherical architectures, we release an evolving benchmark suite of challenging, natively-spherical PDE datasets including a modified Galewsky jet, anomalous chained turbulence, Cahn-Hilliard decomposition, spherical Riemann shocks, Held-Suarez dry atmospheric transport and global ocean dynamics. This new benchmark fills the gap in existing spherical datasets which are either too small and stylized, or much too large (ERA5) for model iteration. Dandelion is best or second-best on every dataset, and the gap to non-warp baselines widens with resolution: at 256times 512, Dandelion and Flower2D occupy the top two slots in both single-step prediction and rollout.

3D shape generation aims to produce innovative 3D content adhering to specific conditions and constraints. Existing methods often decompose 3D shapes into a sequence of localized components, treating each element in isolation without considering spatial consistency. As a result, these approaches exhibit limited versatility in 3D data representation and shape generation, hindering their ability to generate highly diverse 3D shapes that comply with the specified constraints. In this paper, we introduce a novel spatial-aware 3D shape generation framework that leverages 2D plane representations for enhanced 3D shape modeling. To ensure spatial coherence and reduce memory usage, we incorporate a hybrid shape representation technique that directly learns a continuous signed distance field representation of the 3D shape using orthogonal 2D planes. Additionally, we meticulously enforce spatial correspondences across distinct planes using a transformer-based autoencoder structure, promoting the preservation of spatial relationships in the generated 3D shapes. This yields an algorithm that consistently outperforms state-of-the-art 3D shape generation methods on various tasks, including unconditional shape generation, multi-modal shape completion, single-view reconstruction, and text-to-shape synthesis.

Generative models have shown great promise in generating 3D geometric systems, which is a fundamental problem in many natural science domains such as molecule and protein design. However, existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. In this work, we propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories. Modeling such distribution is challenging as it requires capturing both the complex spatial interactions with physical symmetries and temporal correspondence encapsulated in the dynamics. We theoretically justify that diffusion models with equivariant temporal kernels can lead to density with desired symmetry, and develop a novel transition kernel leveraging SE(3)-equivariant spatial convolution and temporal attention. Furthermore, to induce an expressive trajectory distribution for conditional generation, we introduce a generalized learnable geometric prior into the forward diffusion process to enhance temporal conditioning. We conduct extensive experiments on both unconditional and conditional generation in various scenarios, including physical simulation, molecular dynamics, and pedestrian motion. Empirical results on a wide suite of metrics demonstrate that GeoTDM can generate realistic geometric trajectories with significantly higher quality.

Sparse-view 3D modeling represents a fundamental tension between reconstruction fidelity and generative plausibility. While feed-forward reconstruction excels in efficiency and input alignment, it often lacks the global priors needed for structural completeness. Conversely, diffusion-based generation provides rich geometric details but struggles with multi-view consistency. We present UniRecGen, a unified framework that integrates these two paradigms into a single cooperative system. To overcome inherent conflicts in coordinate spaces, 3D representations, and training objectives, we align both models within a shared canonical space. We employ disentangled cooperative learning, which maintains stable training while enabling seamless collaboration during inference. Specifically, the reconstruction module is adapted to provide canonical geometric anchors, while the diffusion generator leverages latent-augmented conditioning to refine and complete the geometric structure. Experimental results demonstrate that UniRecGen achieves superior fidelity and robustness, outperforming existing methods in creating complete and consistent 3D models from sparse observations.

In this paper, we introduce a novel approach to fine-grained cross-view geo-localization. Our method aligns a warped ground image with a corresponding GPS-tagged satellite image covering the same area using homography estimation. We first employ a differentiable spherical transform, adhering to geometric principles, to accurately align the perspective of the ground image with the satellite map. This transformation effectively places ground and aerial images in the same view and on the same plane, reducing the task to an image alignment problem. To address challenges such as occlusion, small overlapping range, and seasonal variations, we propose a robust correlation-aware homography estimator to align similar parts of the transformed ground image with the satellite image. Our method achieves sub-pixel resolution and meter-level GPS accuracy by mapping the center point of the transformed ground image to the satellite image using a homography matrix and determining the orientation of the ground camera using a point above the central axis. Operating at a speed of 30 FPS, our method outperforms state-of-the-art techniques, reducing the mean metric localization error by 21.3% and 32.4% in same-area and cross-area generalization tasks on the VIGOR benchmark, respectively, and by 34.4% on the KITTI benchmark in same-area evaluation.

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free 2^nd-order optimizers for deep learning in low precision settings. Code: https://github.com/yorkerlin/StructuredNGD-DL

Leveraging diverse robotic data for pretraining remains a critical challenge. Existing methods typically model the dataset's action distribution using simple observations as inputs. However, these inputs are often incomplete, resulting in a dispersed conditional action distribution-an issue we refer to as coordinate system chaos and state chaos. This inconsistency significantly hampers pretraining efficiency. To address this, we propose 4D-VLA, a novel approach that effectively integrates 4D information into the input to mitigate these sources of chaos. Our model introduces depth and temporal information into visual features with sequential RGB-D inputs, aligning the coordinate systems of the robot and the scene. This alignment endows the model with strong spatiotemporal reasoning capabilities while minimizing training overhead. Additionally, we introduce memory bank sampling, a frame sampling strategy designed to extract informative frames from historical images, further improving effectiveness and efficiency. Experimental results demonstrate that our pretraining method and architectural components substantially enhance model performance. In both simulated and real-world experiments, our model achieves a significant increase in success rate over OpenVLA. To further assess spatial perception and generalization to novel views, we introduce MV-Bench, a multi-view simulation benchmark. Our model consistently outperforms existing methods, demonstrating stronger spatial understanding and adaptability.

Learning an accurate model of the environment is essential for model-based control tasks. Existing methods in robotic visuomotor control usually learn from data with heavily labelled actions, object entities or locations, which can be demanding in many cases. To cope with this limitation, we propose a method, dubbed DMotion, that trains a forward model from video data only, via disentangling the motion of controllable agent to model the transition dynamics. An object extractor and an interaction learner are trained in an end-to-end manner without supervision. The agent's motions are explicitly represented using spatial transformation matrices containing physical meanings. In the experiments, DMotion achieves superior performance on learning an accurate forward model in a Grid World environment, as well as a more realistic robot control environment in simulation. With the accurate learned forward models, we further demonstrate their usage in model predictive control as an effective approach for robotic manipulations.

Game world models have made rapid progress toward real-time interactive generation. Yet action control is still predominantly implemented as feature conditioning, leaving the model to learn from training data how each action should move the visual world. This formulation is poorly matched to a broad class of planar-camera worlds, such as top-down and isometric games. In these worlds, player motion induces a geometrically well-defined planar camera displacement. We observe that the key to spatially consistent control in such worlds is to represent the effect of each action directly in world coordinates. To this end, we introduce ActionRoPE, a parameter-free action interface that converts action-driven camera displacement into world-coordinate rotary positions. As a result, tokens corresponding to the same world location retain the same positional address across time. These world coordinates further expose a spatial asymmetry in generation: regions covered by the initial observation are geometrically known, while newly revealed regions remain to be synthesized. We therefore propose Coordinate-Conditioned Noise Scheduling (CNS), which assigns token-wise denoising schedules according to this known-new structure. It allows observed regions to stabilize earlier, while new content is generated with reference to them. Compared with four representative action-conditioning mechanisms, our approach follows the commanded motion more faithfully and reduces the revisit drift by 66% over the strongest baseline. The same interface transfers across video backbones from 1.3B to 33B parameters and to out-of-distribution scenes. Project Page: https://teawhite.cn/ActionRoPE/

This paper introduces a dataset of enriched geographic coordinates retrieved from Diderot and d'Alembert's eighteenth-century Encyclopedie. Automatically recovering geographic coordinates from historical texts is a complex task, as they are expressed in a variety of ways and with varying levels of precision. To improve retrieval of coordinates from similar digitized early modern texts, we have created a gold standard dataset, trained models, published the resulting inferred and normalized coordinate data, and experimented applying these models to new texts. From 74,000 total articles in each of the digitized versions of the Encyclopedie from ARTFL and ENCCRE, we examined 15,278 geographical entries, manually identifying 4,798 containing coordinates, and 10,480 with descriptive but non-numerical references. Leveraging our gold standard annotations, we trained transformer-based models to retrieve and normalize coordinates. The pipeline presented here combines a classifier to identify coordinate-bearing entries and a second model for retrieval, tested across encoder-decoder and decoder architectures. Cross-validation yielded an 86% EM score. On an out-of-domain eighteenth-century Trevoux dictionary (also in French), our fine-tuned model had a 61% EM score, while for the nineteenth-century, 7th edition of the Encyclopaedia Britannica in English, the EM was 77%. These findings highlight the gold standard dataset's usefulness as training data, and our two-step method's cross-lingual, cross-domain generalizability.

We introduce Point2Skeleton, an unsupervised method to learn skeletal representations from point clouds. Existing skeletonization methods are limited to tubular shapes and the stringent requirement of watertight input, while our method aims to produce more generalized skeletal representations for complex structures and handle point clouds. Our key idea is to use the insights of the medial axis transform (MAT) to capture the intrinsic geometric and topological natures of the original input points. We first predict a set of skeletal points by learning a geometric transformation, and then analyze the connectivity of the skeletal points to form skeletal mesh structures. Extensive evaluations and comparisons show our method has superior performance and robustness. The learned skeletal representation will benefit several unsupervised tasks for point clouds, such as surface reconstruction and segmentation.

Synthesizing controllable 6-DOF object manipulation trajectories in 3D environments is essential for enabling robots to interact with complex scenes, yet remains challenging due to the need for accurate spatial reasoning, physical feasibility, and multimodal scene understanding. Existing approaches often rely on 2D or partial 3D representations, limiting their ability to capture full scene geometry and constraining trajectory precision. We present GMT, a multimodal transformer framework that generates realistic and goal-directed object trajectories by jointly leveraging 3D bounding box geometry, point cloud context, semantic object categories, and target end poses. The model represents trajectories as continuous 6-DOF pose sequences and employs a tailored conditioning strategy that fuses geometric, semantic, contextual, and goaloriented information. Extensive experiments on synthetic and real-world benchmarks demonstrate that GMT outperforms state-of-the-art human motion and human-object interaction baselines, such as CHOIS and GIMO, achieving substantial gains in spatial accuracy and orientation control. Our method establishes a new benchmark for learningbased manipulation planning and shows strong generalization to diverse objects and cluttered 3D environments. Project page: https://huajian- zeng.github. io/projects/gmt/.

The cascade of 2D geometric transformations were exploited to model relations between entities in a knowledge graph (KG), leading to an effective KG embedding (KGE) model, CompoundE. Furthermore, the rotation in the 3D space was proposed as a new KGE model, Rotate3D, by leveraging its non-commutative property. Inspired by CompoundE and Rotate3D, we leverage 3D compound geometric transformations, including translation, rotation, scaling, reflection, and shear and propose a family of KGE models, named CompoundE3D, in this work. CompoundE3D allows multiple design variants to match rich underlying characteristics of a KG. Since each variant has its own advantages on a subset of relations, an ensemble of multiple variants can yield superior performance. The effectiveness and flexibility of CompoundE3D are experimentally verified on four popular link prediction datasets.

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.

Learning-based visual localization methods that use scene coordinate regression (SCR) offer the advantage of smaller map sizes. However, on datasets with complex illumination changes or image-level ambiguities, it remains a less robust alternative to feature matching methods. This work aims to close the gap. We introduce a covisibility graph-based global encoding learning and data augmentation strategy, along with a depth-adjusted reprojection loss to facilitate implicit triangulation. Additionally, we revisit the network architecture and local feature extraction module. Our method achieves state-of-the-art on challenging large-scale datasets without relying on network ensembles or 3D supervision. On Aachen Day-Night, we are 10times more accurate than previous SCR methods with similar map sizes and require at least 5times smaller map sizes than any other SCR method while still delivering superior accuracy. Code will be available at: https://github.com/cvg/scrstudio .

Physical trajectories contain more than snapshots of a system: they also reveal how its states evolve under governing conditions. However, representation learning for parametric partial differential equations (PDEs) has largely relied on reconstruction-based objectives that emphasize recovering observed physical fields. In this paper, we investigate predictive representation pretraining as an alternative to reconstruction-based learning. We find that predictive representations preserve rich physical information, yet this advantage alone does not ensure accurate field evolution. Based on these observations, we introduce PDE-JEPA for parametric PDE dynamics. Specifically, we first train an encoder using a masked-latent prediction to capture the underlying regularities of PDE dynamics. To explicitly adapt the pretrained representation toward a more dynamics-aligned state space, we then introduce a geometry projector that aligns latent trajectory geometry with the evolution geometry of physical fields. Finally, building on this geometry-aligned latent space, we further develop a physics-structured latent predictor that decomposes the dynamics into parameter-independent evolution and parameter-dependent response components. Extensive experiments on nine widely used PDE benchmarks demonstrate that our framework outperforms existing state-of-the-art methods by an average of 33.4\% in-distribution, while achieving an average improvement of 51.4\% when extrapolating to unseen governing parameters. The project page is available https://tanpig-x.github.io/PDE-JEPA/{here}.

We present 3DRegNet, a novel deep learning architecture for the registration of 3D scans. Given a set of 3D point correspondences, we build a deep neural network to address the following two challenges: (i) classification of the point correspondences into inliers/outliers, and (ii) regression of the motion parameters that align the scans into a common reference frame. With regard to regression, we present two alternative approaches: (i) a Deep Neural Network (DNN) registration and (ii) a Procrustes approach using SVD to estimate the transformation. Our correspondence-based approach achieves a higher speedup compared to competing baselines. We further propose the use of a refinement network, which consists of a smaller 3DRegNet as a refinement to improve the accuracy of the registration. Extensive experiments on two challenging datasets demonstrate that we outperform other methods and achieve state-of-the-art results. The code is available.

We introduce Talk2Move, a reinforcement learning (RL) based diffusion framework for text-instructed spatial transformation of objects within scenes. Spatially manipulating objects in a scene through natural language poses a challenge for multimodal generation systems. While existing text-based manipulation methods can adjust appearance or style, they struggle to perform object-level geometric transformations-such as translating, rotating, or resizing objects-due to scarce paired supervision and pixel-level optimization limits. Talk2Move employs Group Relative Policy Optimization (GRPO) to explore geometric actions through diverse rollouts generated from input images and lightweight textual variations, removing the need for costly paired data. A spatial reward guided model aligns geometric transformations with linguistic description, while off-policy step evaluation and active step sampling improve learning efficiency by focusing on informative transformation stages. Furthermore, we design object-centric spatial rewards that evaluate displacement, rotation, and scaling behaviors directly, enabling interpretable and coherent transformations. Experiments on curated benchmarks demonstrate that Talk2Move achieves precise, consistent, and semantically faithful object transformations, outperforming existing text-guided editing approaches in both spatial accuracy and scene coherence.

Visual retrieval systems face significant challenges when updating models with improved representations due to misalignment between the old and new representations. The costly and resource-intensive backfilling process involves recalculating feature vectors for images in the gallery set whenever a new model is introduced. To address this, prior research has explored backward-compatible training methods that enable direct comparisons between new and old representations without backfilling. Despite these advancements, achieving a balance between backward compatibility and the performance of independently trained models remains an open problem. In this paper, we address it by expanding the representation space with additional dimensions and learning an orthogonal transformation to achieve compatibility with old models and, at the same time, integrate new information. This transformation preserves the original feature space's geometry, ensuring that our model aligns with previous versions while also learning new data. Our Orthogonal Compatible Aligned (OCA) approach eliminates the need for re-indexing during model updates and ensures that features can be compared directly across different model updates without additional mapping functions. Experimental results on CIFAR-100 and ImageNet-1k demonstrate that our method not only maintains compatibility with previous models but also achieves state-of-the-art accuracy, outperforming several existing methods.

Single-image 3D object generation can now produce high-fidelity assets, yet accurately placing them into a coherent scene layout remains an open challenge. A central difficulty lies in how object layout is represented. Holistic methods absorb placement into a scene-level generation process, sacrificing object-level detail. Compositional methods preserve object fidelity by decoupling geometry from layout, but typically parameterize layout as sparse, unbounded pose variables that are difficult to learn and generalize poorly under scarce scene-level supervision.We present Mira-Scene, a compositional 3D scene reconstruction framework that replaces sparse pose regression with dense, bounded correspondence recovery. At its core is the Canonical Coordinate Map (CCM), a pixel-aligned field that maps each visible object pixel to a surface coordinate in the object's bounded canonical space. When paired with a scene-space Point Cloud Map (PCM) from monocular geometry estimation, CCM induces dense canonical-to-scene correspondences from which object transformations are recovered through robust geometric alignment. Because CCM operates in bounded canonical space, it provides a stable prediction target that can be trained from scalable object-level 3D data without requiring scene-level layout annotations. Mira-Scene further introduces a multimodal diffusion transformer that jointly generates object geometry and CCMs, using modality-specific expert streams with shared attention and positional encoding to promote geometry-layout consistency. Experiments on indoor, outdoor, synthetic, and in-the-wild scenes show that Mira-Scene substantially outperforms strong baselines in layout accuracy, achieving relative gains of 39.8% in 3D-IoU and 16.5% in 2D-IoU over SAM3D, using limited open-source training data.

Modeling spatiotemporal coupling is a key challenge in building physical intelligence across scales, from microscopic to macroscopic. Existing models capture such structure broadly through physics-motivated dynamical formulations or learning-motivated architectures. The former provide stronger priors but may constrain flexibility, whereas the latter are more flexible but leave the spatiotemporal coupling largely implicit. We therefore seek an approach that combines flexible learning with an explicit geometric bias for jointly modeling time and space. To this end, we propose Minkowski Positional Encoding (MinkowskiPE), which uses joint temporal and spatial coordinates to parameterize Lorentz transformations applied to query and key features. With MinkowskiPE, the query-key attention score depends on position only through the relative spacetime displacement between the two tokens and is therefore invariant to global translation of the coordinates. This paradigm retains the standard dot-product attention interface and remains compatible with efficient attention implementations. We evaluate MinkowskiPE on microscopic molecular dynamics and macroscopic video prediction tasks, achieving the best results on all nine multi-trajectory molecular evaluations and reducing KTH video-prediction MSE by 9.9% relative to the best baseline while using roughly one-tenth as many parameters.

Recently, methods leveraging diffusion model priors to assist monocular geometric estimation (e.g., depth and normal) have gained significant attention due to their strong generalization ability. However, most existing works focus on estimating geometric properties within the camera coordinate system of individual video frames, neglecting the inherent ability of diffusion models to determine inter-frame correspondence. In this work, we demonstrate that, through appropriate design and fine-tuning, the intrinsic consistency of video generation models can be effectively harnessed for consistent geometric estimation. Specifically, we 1) select geometric attributes in the global coordinate system that share the same correspondence with video frames as the prediction targets, 2) introduce a novel and efficient conditioning method by reusing positional encodings, and 3) enhance performance through joint training on multiple geometric attributes that share the same correspondence. Our results achieve superior performance in predicting global geometric attributes in videos and can be directly applied to reconstruction tasks. Even when trained solely on static video data, our approach exhibits the potential to generalize to dynamic video scenes.

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.

Positional encoding mechanisms enable Transformers to model sequential structure and long-range dependencies in text. While absolute positional encodings struggle with extrapolation to longer sequences due to fixed positional representations, and relative approaches like Alibi exhibit performance degradation on extremely long contexts, the widely-used Rotary Positional Encoding (RoPE) introduces oscillatory attention patterns that hinder stable long-distance dependency modelling. We address these limitations through a geometric reformulation of positional encoding. Drawing inspiration from Lorentz transformations in hyperbolic geometry, we propose Hyperbolic Rotary Positional Encoding (HoPE), which leverages hyperbolic functions to implement Lorentz rotations on token representations. Theoretical analysis demonstrates that RoPE is a special case of our generalized formulation. HoPE fundamentally resolves RoPE's slation issues by enforcing monotonic decay of attention weights with increasing token distances. Extensive experimental results, including perplexity evaluations under several extended sequence benchmarks, show that HoPE consistently exceeds existing positional encoding methods. These findings underscore HoPE's enhanced capacity for representing and generalizing long-range dependencies. Data and code will be available.

It has been observed that representations learned by distinct neural networks conceal structural similarities when the models are trained under similar inductive biases. From a geometric perspective, identifying the classes of transformations and the related invariances that connect these representations is fundamental to unlocking applications, such as merging, stitching, and reusing different neural modules. However, estimating task-specific transformations a priori can be challenging and expensive due to several factors (e.g., weights initialization, training hyperparameters, or data modality). To this end, we introduce a versatile method to directly incorporate a set of invariances into the representations, constructing a product space of invariant components on top of the latent representations without requiring prior knowledge about the optimal invariance to infuse. We validate our solution on classification and reconstruction tasks, observing consistent latent similarity and downstream performance improvements in a zero-shot stitching setting. The experimental analysis comprises three modalities (vision, text, and graphs), twelve pretrained foundational models, nine benchmarks, and several architectures trained from scratch.

Popular representation learning methods encourage feature invariance under transformations applied at the input. However, in 3D perception tasks like object localization and segmentation, outputs are naturally equivariant to some transformations, such as rotation. Using pre-training loss functions that encourage equivariance of features under certain transformations provides a strong self-supervision signal while also retaining information of geometric relationships between transformed feature representations. This can enable improved performance in downstream tasks that are equivariant to such transformations. In this paper, we propose a spatio-temporal equivariant learning framework by considering both spatial and temporal augmentations jointly. Our experiments show that the best performance arises with a pre-training approach that encourages equivariance to translation, scaling, and flip, rotation and scene flow. For spatial augmentations, we find that depending on the transformation, either a contrastive objective or an equivariance-by-classification objective yields best results. To leverage real-world object deformations and motion, we consider sequential LiDAR scene pairs and develop a novel 3D scene flow-based equivariance objective that leads to improved performance overall. We show our pre-training method for 3D object detection which outperforms existing equivariant and invariant approaches in many settings.

Geometric camera calibration is often required for applications that understand the perspective of the image. We propose perspective fields as a representation that models the local perspective properties of an image. Perspective Fields contain per-pixel information about the camera view, parameterized as an up vector and a latitude value. This representation has a number of advantages as it makes minimal assumptions about the camera model and is invariant or equivariant to common image editing operations like cropping, warping, and rotation. It is also more interpretable and aligned with human perception. We train a neural network to predict Perspective Fields and the predicted Perspective Fields can be converted to calibration parameters easily. We demonstrate the robustness of our approach under various scenarios compared with camera calibration-based methods and show example applications in image compositing.

We propose an approach for generating crochet instructions (patterns) from an input 3D model. We focus on Amigurumi, which are knitted stuffed toys. Given a closed triangle mesh, and a single point specified by the user, we generate crochet instructions, which when knitted and stuffed result in a toy similar to the input geometry. Our approach relies on constructing the geometry and connectivity of a Crochet Graph, which is then translated into a crochet pattern. We segment the shape automatically into chrochetable components, which are connected using the join-as-you-go method, requiring no additional sewing. We demonstrate that our method is applicable to a large variety of shapes and geometries, and yields easily crochetable patterns.

Visual navigation using only a single camera and a topological map has recently become an appealing alternative to methods that require additional sensors and 3D maps. This is typically achieved through an "image-relative" approach to estimating control from a given pair of current observation and subgoal image. However, image-level representations of the world have limitations because images are strictly tied to the agent's pose and embodiment. In contrast, objects, being a property of the map, offer an embodiment- and trajectory-invariant world representation. In this work, we present a new paradigm of learning "object-relative" control that exhibits several desirable characteristics: a) new routes can be traversed without strictly requiring to imitate prior experience, b) the control prediction problem can be decoupled from solving the image matching problem, and c) high invariance can be achieved in cross-embodiment deployment for variations across both training-testing and mapping-execution settings. We propose a topometric map representation in the form of a "relative" 3D scene graph, which is used to obtain more informative object-level global path planning costs. We train a local controller, dubbed "ObjectReact", conditioned directly on a high-level "WayObject Costmap" representation that eliminates the need for an explicit RGB input. We demonstrate the advantages of learning object-relative control over its image-relative counterpart across sensor height variations and multiple navigation tasks that challenge the underlying spatial understanding capability, e.g., navigating a map trajectory in the reverse direction. We further show that our sim-only policy is able to generalize well to real-world indoor environments. Code and supplementary material are accessible via project page: https://object-react.github.io/

World building with 3D scene representations is increasingly important for content creation, simulation, and interactive experiences, yet real workflows are inherently iterative: creators must repeatedly extend an existing scene under user control. Motivated by this research gap, we study 3D scene expansion in a user-centric workflow: starting from a real scene captured by multi-view images, we extend its coverage by inserting an additional view synthesized by a generative model. Unlike simple object editing or style transfer in a fixed scene, the inserted view is often 3D-misaligned with the original reconstruction, introducing geometry shifts, hallucinated content, or view-dependent artifacts that break global multi-view consistency. To address the challenge, we propose SceneExpander, which applies test-time adaptation to a parametric feed-forward 3D reconstruction model with two complementary distillation signals: anchor distillation stabilizes the original scene by distilling geometric cues from the captured views, while inserted-view self-distillation preserves observation-supported predictions yet adapts latent geometry and appearance to accommodate the misaligned inserted view. Experiments on ETH scenes and online data demonstrate improved expansion behavior and reconstruction quality under misalignment.

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.

In recent years, huge progress has been made on learning neural implicit representations from multi-view images for 3D reconstruction. As an additional input complementing coordinates, using sinusoidal functions as positional encodings plays a key role in revealing high frequency details with coordinate-based neural networks. However, high frequency positional encodings make the optimization unstable, which results in noisy reconstructions and artifacts in empty space. To resolve this issue in a general sense, we introduce to learn neural implicit representations with quantized coordinates, which reduces the uncertainty and ambiguity in the field during optimization. Instead of continuous coordinates, we discretize continuous coordinates into discrete coordinates using nearest interpolation among quantized coordinates which are obtained by discretizing the field in an extremely high resolution. We use discrete coordinates and their positional encodings to learn implicit functions through volume rendering. This significantly reduces the variations in the sample space, and triggers more multi-view consistency constraints on intersections of rays from different views, which enables to infer implicit function in a more effective way. Our quantized coordinates do not bring any computational burden, and can seamlessly work upon the latest methods. Our evaluations under the widely used benchmarks show our superiority over the state-of-the-art. Our code is available at https://github.com/MachinePerceptionLab/CQ-NIR.

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.

Residual connections are the de facto mechanism for training deep neural networks stably. Geodesic Normalization (GeoNorm) recasts them on a Riemannian manifold, orthogonalizing each layer output against the current hidden state and applying the resulting update through the Riemannian exponential map. Every hidden state thus keeps a constant ell_{2}-norm, confining the residual stream to a hypersphere. The exponential map, however, is only one member of a broad family of retraction maps. We show that on the hypersphere this entire family collapses to a single scalar design choice. What distinguishes one retraction from another is only how the magnitude of an update is converted into a rotation angle within the plane spanned by the hidden state and the update. This view places Euclidean residual connections and GeoNorm in one framework. Instantiating it with the metric projection retraction and the Cayley retraction yields Proj-SpheretNorm and Cay-SpheretNorm, which are exactly norm-preserving yet require only algebraic operations. Both prove to be members of a one-parameter family of angular retractions, p-SpheretNorm, whose rotation angle saturates rather than growing without bound. The two methods above are recovered exactly at p = 1 and p = 2, while the identity map and GeoNorm arise only as limits at either end. On nanoGPT, all three methods outperform existing lightweight deep connection schemes, and the best validation loss is attained at finite p, indicating that the exponential map is not the preferred retraction for spherical residual streams but merely one end of a spectrum.

Self-supervised learning (SSL) methods have achieved remarkable success in learning image representations allowing invariances in them - but therefore discarding transformation information that some computer vision tasks actually require. While recent approaches attempt to address this limitation by learning equivariant features using linear operators in feature space, they impose restrictive assumptions that constrain flexibility and generalization. We introduce a weaker definition for the transformation relation between image and feature space denoted as equivariance-coherence. We propose a novel SSL auxiliary task that learns equivariance-coherent representations through intermediate transformation reconstruction, which can be integrated with existing joint embedding SSL methods. Our key idea is to reconstruct images at intermediate points along transformation paths, e.g. when training on 30-degree rotations, we reconstruct the 10-degree and 20-degree rotation states. Reconstructing intermediate states requires the transformation information used in augmentations, rather than suppressing it, and therefore fosters features containing the augmented transformation information. Our method decomposes feature vectors into invariant and equivariant parts, training them with standard SSL losses and reconstruction losses, respectively. We demonstrate substantial improvements on synthetic equivariance benchmarks while maintaining competitive performance on downstream tasks requiring invariant representations. The approach seamlessly integrates with existing SSL methods (iBOT, DINOv2) and consistently enhances performance across diverse tasks, including segmentation, detection, depth estimation, and video dense prediction. Our framework provides a practical way for augmenting SSL methods with equivariant capabilities while preserving invariant performance.

Robotic manipulation can be formulated as inducing a sequence of spatial displacements: where the space being moved can encompass an object, part of an object, or end effector. In this work, we propose the Transporter Network, a simple model architecture that rearranges deep features to infer spatial displacements from visual input - which can parameterize robot actions. It makes no assumptions of objectness (e.g. canonical poses, models, or keypoints), it exploits spatial symmetries, and is orders of magnitude more sample efficient than our benchmarked alternatives in learning vision-based manipulation tasks: from stacking a pyramid of blocks, to assembling kits with unseen objects; from manipulating deformable ropes, to pushing piles of small objects with closed-loop feedback. Our method can represent complex multi-modal policy distributions and generalizes to multi-step sequential tasks, as well as 6DoF pick-and-place. Experiments on 10 simulated tasks show that it learns faster and generalizes better than a variety of end-to-end baselines, including policies that use ground-truth object poses. We validate our methods with hardware in the real world. Experiment videos and code are available at https://transporternets.github.io

This paper addresses the task of estimating the 6 degrees of freedom pose of a known 3D object from depth information represented by a point cloud. Deep features learned by convolutional neural networks from color information have been the dominant features to be used for inferring object poses, while depth information receives much less attention. However, depth information contains rich geometric information of the object shape, which is important for inferring the object pose. We use depth information represented by point clouds as the input to both deep networks and geometry-based pose refinement and use separate networks for rotation and translation regression. We argue that the axis-angle representation is a suitable rotation representation for deep learning, and use a geodesic loss function for rotation regression. Ablation studies show that these design choices outperform alternatives such as the quaternion representation and L2 loss, or regressing translation and rotation with the same network. Our simple yet effective approach clearly outperforms state-of-the-art methods on the YCB-video dataset. The implementation and trained model are avaliable at: https://github.com/GeeeG/CloudPose.