Geometric Representation Learning

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28 papers in the last four weeks, up 155% on the four weeks before. 0.3% of all new papers.

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Latest papers 188

Jul 17, 2026cs.IR

Dual-Level Atomic and Coordination Geometry Learning for Crystal Property Prediction Using Graph Neural Networks

Accurate prediction of crystal properties remains a key challenge in computational materials science. While graph neural networks (GNNs) such as CGCNN, MEGNet, ALIGNN, and SchNet have shown strong performance, they primarily represent crystals at the atomic level and implicitly learn local chemical environments through message passing. However, many material properties are governed by coordination polyhedra, the fundamental structural units formed by atoms and their neighboring atoms. To address this limitation, we propose the Coordination Polyhedron Graph Network (CPGN), a multi-scale GNN that jointly learns atomic, bond, and coordination-polyhedron representations. CPGN constructs three coupled graphs: an atom graph encoding elemental and bonding information, a line graph capturing angular interactions, and a coordination polyhedron graph describing Voronoi-derived local environments through corner-, edge-, and face-sharing relationships. Physically meaningful geometric descriptors are incorporated for each polyhedron, while an interleaved message-passing mechanism with bidirectional cross-attention enables effective information exchange across structural levels. Extensive evaluations on the Materials Project, JARVIS-DFT, and QM9 benchmark datasets demonstrate that CPGN outperforms existing state-of-the-art GNN models. It achieves a formation-energy MAE of 0.060 eV/atom and a band-gap MAE of 0.292 eV on the Materials Project, while providing competitive multi-property prediction on JARVIS-DFT and superior HOMO prediction on QM9. The results highlight that explicit modeling of coordination polyhedra improves crystal representation learning and enables accurate, physically interpretable prediction of material properties.

Sanjay Chakraborty

Department of Computer and Information Science (IDA), REAL, AIICS, Linköping University, Linköping, Sweden

Graph Neural Networks Materials Property Prediction Geometric Representation Learning Computational Materials Science

Jul 17, 2026cs.CV

E3DGS: Unified Geometric-Photometric Equivariance for 3D Gaussian Splatting via Color-as-Geometry Embedding

3D Gaussian Splatting (3DGS) captures scenes by coupling explicit geometry (position, covariance) with view-dependent photometry (Spherical Harmonics). However, building

\mathrm{SE}(3)

-equivariant architectures on these primitives presents a fundamental representation bottleneck. Color has been treated as a signal rather than a geometric entity, making it nontrivial to unify symmetry across geometry and appearance as the camera frame changes. While translations are handled by relative coordinates, rotations act heterogeneously across attributes:

μ\mapsto Rμ

,

Σ\mapsto RΣR^\top

, and

f_\ell\mapsto D^\ell(R)f_\ell

. This mismatch complicates strict equivariance, leading existing methods to either discard or flatten SH coefficients, thereby breaking symmetry. We propose a unified solution rooted in representation theory: for SH degrees

\ell\le2

, photometry is algebraically isomorphic to a rank-2 geometric tensor. We prove that the Wigner-

D

action on these SH coefficients can be exactly reformulated as the conjugation action on

3\times3

matrices. Leveraging this, we introduce the Unified Matrix Embedding, a lifting that maps all Gaussian attributes into a unified carrier space,

\mathfrak{gl}(3)

. Building on the "Color-as-Geometry" formulation, we present E3DGS, a rigid-body (

\mathrm{SE}(3)

) equivariant architecture that processes 3D Gaussians without Clebsch-Gordan tensor products. Evaluations on object vision and action-conditioned Gaussian world modeling demonstrate that our unified approach yields strong robustness under camera-frame changes and improved data efficiency.

Chankyo Kim, Maani Ghaffari

University of Michigan, Ann Arbor, MI, USA

3D Gaussian Splatting Equivariant Neural Networks Geometric Representation Learning 3D Scene Representation

Jul 16, 2026cs.CV

SUFLECA: Scaling Up Feature Learning for CAD-to-image Alignment

CAD-to-image alignment aims to estimate an object's 9D pose (rotation, translation, and anisotropic scale) from a single RGB image, enabling applications in robotics and augmented reality. Recent zero-shot methods use visual foundation models to match image regions to CAD models, yet typically their correspondences are appearance-driven and degrade under occlusion or sim-to-real domain shift. To address these limitations, we introduce SUFLECA (Scaling Up Feature LEarning for CAD Alignment), a weakly-supervised framework for zero-shot CAD alignment with two key contributions. First, SUFLECA scales up geometry-grounded feature learning from pretrained visual representations through Normalized Object Coordinates (NOCs) supervision on 674K images spanning 12 real and synthetic datasets, learning compact geometry-aware features that generalize across domains. Second, we propose a geometrically consistent matching algorithm that establishes reliable one-to-one CAD-to-image correspondences. Together, these contributions enable accurate, sub-second alignment per object instance without iterative pose refinement. On ScanNet25k, SUFLECA achieves 33.4%/42.3% category/instance accuracy, outperforming, with a smaller computational footprint, the strongest zero-shot baseline by 10.3/12.2 percentage points and, for the first time on this benchmark, even surpassing fully supervised methods. Code is available at: https://github.com/snt-arg/SUFLECA

Saad Ejaz, Miguel Fernandez-Cortizas, Javier Civera +2

Automation and Robotics Research Group, Interdisciplinary Centre for Security, Reliability, and Trust (SnT), University of Luxembourg, Luxembourg · I3A, Universidad de Zaragoza

Object Pose Estimation Geometric Representation Learning Cross-Modal Feature Matching

Jul 15, 2026cs.CV

SeeSE3: Emergence of 3D Space in Vision Features

In this paper, we ask whether vision foundation models construct representations that reflect the intrinsic properties of 3D Euclidean space. Unlike previous works that probe 3D awareness of vision features by regressing image-centric quantities such as depth or normals, we investigate the relation between the structure of the space of visual features and the group of Euclidean transformations

SE(3)

. We propose a set of probes to evaluate this relation from both topological and geometric perspectives: a mutual neighborhood metric that measures the alignment between feature neighborhoods and spatial topology, and a Poincaré Adapter to test the linear accessibility of the geometry of camera motion from latent displacements in static scenes. We show that self-supervised vision models, which, in principle, have not been trained with direct 3D supervision or active agency, possess latent subspaces that are remarkably strongly correlated with three-dimensional Euclidean space, when probed correctly. Building on this insight we propose a new class of "Latent-Space Navigation" techniques that perform visual odometry and localization purely in the latent space, bypassing the need for explicit 3D reconstruction.

Caroline Chen, Sayna Ebrahimi, Fedor Kitashov +4

Google DeepMind

Geometric Representation Learning Visual Odometry 3D Representation Learning Topological Representation Learning

Jul 14, 2026cs.LG

Contrastive-Collapsed Loss for Flexible and Geometrically Optimal Embeddings and Faster Convergence

In this work, we introduce CoCo, a loss function aimed at learning normalized and well-structured representations. The proposed loss encourages intra-class collapse and inter-class contrast while preserving sufficient flexibility for neural networks to approximate geometrically optimal embeddings with large angular separation between classes. We provide a theoretical analysis positioning CoCo with respect to related objectives such as dot regression and cross-entropy, showing that the new proposed loss benefits from closer initialization to the optimal configuration, more informative gradients, and stronger incentives for class-wise representation collapse. Extensive experiments on diverse tabular datasets from the OpenML-CC18 benchmark show that CoCo achieves competitive performance with state-of-the-art methods, including kernel SVM, Random Forest, dot regression, and cross-entropy-based neural networks. In addition, both theoretical arguments and empirical analyses demonstrate that the proposal promotes tighter class clustering and faster convergence. These results highlight CoCo loss as an effective objective for learning discriminative representations while maintaining competitive predictive performance.

Blanca Cano-Camarero, Ángela Fernández-Pascual, José R. Dorronsoro

1Departamento de Ingeniería Informática, Universidad Autónoma de Madrid, Calle Francisco Tomás y Valiente, 11, 28049, Madrid, España

Geometric Representation Learning Neural Collapse Supervised Contrastive Learning

Jul 13, 2026cs.LG

IG-GAN: A Generative Adversarial Network for Aerodynamic Data Generation Based on Intrinsic Geometry

Existing generative models learn data distributions in flat Euclidean space. However, most data in our real world are manifolds embedded in high dimensional Euclidean space. Therefore, we propose an intrinsic-geometry-based generative adversarial network (IG-GAN) for data generation in the field of aerodynamics. The generator of the IG-GAN represents aerodynamic data as a piecewise smooth manifold constructed by Bézier surfaces, and the generator tries to learn the coefficients of each Bézier surface to further combine multiple Bézier surfaces into a smooth manifold automatically. The discriminator in the IG-GAN is a radial-basis-function based discriminator (RBF-D). Experimental results show that IG-GAN achieves lower predicted Mean Squared Errors (MSEs) than those of three baselines. Specifically, on the Burgers' equation dataset, IG-GAN reduces the predicted MSE of velocity u by 97.41% compared with state of the art SSL-Transformer. Additionally, on the ONERA M6 aircraft dataset, IG-GAN reduces the overall MSE of nine aerodynamic coefficients by 82.95% compared with SSL-Transformer.

Ying Yan, Liwei Hu, Xiaoming Zhang

School of Information Science and Engineering, Hebei University of Science and Technology, Shijiazhuang, 050091, China

Synthetic Data Generation Geometric Representation Learning Generative Adversarial Networks Computational Fluid Dynamics

Jul 12, 2026cs.CV

TriCons-Pose: Triangle-Invariant Geometric Consistency Learning for Category-Level Object Pose Estimation

Category-level object pose estimation is a crucial yet challenging task in both academia and industry, and has achieved remarkable success by leveraging keypoint-based correspondence paradigms. However, most existing methods increasingly rely on stronger feature learning while overlooking whether the established correspondences are geometrically stable across diverse perturbations. This often results in fragile pose recovery under intra-class shape variations and occlusions. To tackle this challenge, we develop a novel Triangle-Invariant Geometric Consistency Learning for Category-Level Object Pose Estimation (TriCons-Pose) to anchor stable keypoints and aggregate pose-invariant cues, yielding reliable canonical mapping and accurate pose estimation. Specifically, a Structure-Consistent Keypoint Detector (SCKD) is designed to identify robust keypoints by enforcing cross-view structural consistency via normalized pairwise distance matching. Moreover, we propose a Pose-Invariant Geometric Aggregator (PIGA) to augment keypoint representations by injecting triangle-based pose-invariant descriptors into a local-to-global attention mechanism. The proposed framework is optimized using standard objective functions while incorporating an additional geometry consistency loss. Extensive experiments on REAL275, CAMERA25, and HouseCat6D datasets demonstrate the effectiveness of the proposed approach.

Zuzhi Yang, Shuai Wang, Mounir Kaaniche +4

School of Cyberspace, Hangzhou Dianzi University, Hangzhou 310018, China · Universit´e Sorbonne Paris Nord, L2TI, UR 3043, F-93430, Villetaneuse, France · Universit´e Paris-Saclay, Centrale-Sup´elec, CVN, 91190 Gif-sur-Yvette, France +2

Object Pose Estimation Geometric Representation Learning Keypoint Detection

Jul 8, 2026cs.CV

Learning to Unify Deformable Shape and Texture Representations for Cardiac Video Classification

Deformable shape representations have proven to be robust complements to texture features in cardiac image classification, offering geometric priors that are invariant to imaging artifacts and intensity variations. However, existing deep networks perform simple concatenation to combine these distinct feature representations, which neither fully exploits their complementary nature nor learns cross-modal feature dependencies. Furthermore, this results in uniform attention across all timepoints; hence ignoring the varying diagnostic importance across the cardiac phases. In this paper, we propose a novel cardiac video classification model that, for the first time, learns temporal features in an integrated space of deformable shape and image texture representations. In particular, we design a bi-directional cross-attention in the latent space to fuse latent deformable shape and image features, allowing each modality to adaptively weight the other based on spatio-temporal correspondence. In contrast to current methods that apply uniform weighting across all the cardiac phases, our approach learns to dynamically adjust the contributions of shape and texture representations, derived from images, over time. We demonstrate state-of-the-art classification performance on a cine cardiac magnetic resonance (CMR) video dataset, achieving improved interpretability from attention mechanisms that identify diagnostically critical cardiac phases and modality contributions.

Tonmoy Hossain, Miaomiao Zhang

Department of Computer Science, University of Virginia, Virginia, USA · Department of Electrical and Computer Engineering, University of Virginia, USA

Geometric Representation Learning Medical Image Classification Cross-Modal Attention Multimodal Fusion Cardiac MRI

Jul 8, 2026cs.CL

Riemannian Geometry for Pre-trained Language Model Embeddings

Understanding the geometric structure of pre-trained language model embeddings matters for interpretability and safety. We ask whether sentence-level classification signal lives in the Riemannian geometry of contextual token embeddings, and probe it by extracting per-token pullback metrics from a learned encoder's analytical Jacobian and aggregating them with the Fréchet mean on the symmetric positive definite (SPD) manifold; we call this procedure Riemannian Mean Pooling (RMP). Across three datasets with non-trivial linguistic structure (CoLA, CREAK, RTE), RMP outperforms Euclidean mean pooling, while on FEVER-Symmetric, a benchmark constructed to remove annotation-driven lexical artifacts, the method correctly stays at chance. Ablations show that a randomly initialised encoder combined with Fréchet aggregation already beats Euclidean pooling on two of the three signal-bearing datasets, localising the source of the gain to the geometric aggregation rather than to learned manifold structure; the trained encoder contributes additional signal specifically on CREAK, the most knowledge-heavy of the three signal-bearing datasets.

Szczepan Konior, Alexandre Quemy, Przemysław Klocek +2

IBM Automation and AI, Krakow, Poland · Hother, Krakow, Poland · University of Warsaw +1

Sentence Embeddings Geometric Representation Learning Riemannian Geometry Representation Geometry in Language Models

Jul 6, 2026cs.RO

Learning 4D Geometric Priors for Inference-Efficient World Action Models

World Action Models (WAMs) have shown strong potential for robotic manipulation by jointly modeling visual future dynamics and executable action sequences. However, existing video-action co-training methods primarily optimize appearance-oriented video latents, which may insufficiently capture the temporally evolving geometry required for precise manipulation. We propose MECo-WAM, a Multi-Expert Co-Training World Action Model that injects action-relevant 4D geometric priors into video-action representations while preserving the original lightweight inference graph. During training, MECo-WAM combines video and action experts with a lightweight 4D expert supervised by relational targets from a frozen VGGT encoder. Asymmetric expert visibility prevents non-causal shortcuts from auxiliary geometry to action generation. To transfer geometric knowledge into the deployed video-action pathway, we introduce decayed 4D read-mask attention, which provides restricted current-frame geometric guidance early in training and progressively removes this dependency. We further propose action-aware temporal geometric distillation, which aligns within-frame geometric relations and their temporal evolution while emphasizing visual regions most relevant to robot actions. At deployment, all auxiliary 4D components are removed. Experiments on LIBERO (98.2%), RoboTwin 2.0 (92.6%), and challenging real-world manipulation tasks show that MECo-WAM improves manipulation performance without increasing inference cost.

Jianjun Zhang, Jian Zhu, Taiyi Su +4

Tongji University · AIRC, Midea Group

World Models for Robotics Action-Conditioned World Models Geometric Representation Learning Robotic Manipulation World Action Models

Jul 6, 2026cs.RO

Geometry-Aware Motion Latents for Learning Robust Manipulation Policies

Learning motion latents for robotic manipulation heavily relies on extracting motion patterns from visual sequences, yet effective action abstractions require understanding three-dimensional geometric transformations. Here, we introduce GeoMoLa (Geometry-Aware Motion Latents), which learns discrete motion latent codes by predicting how point clouds evolve during manipulation rather than reconstructing visual observations. This four-dimensional objective -- spatial geometry changing through time -- forces latent representations to encode actual physical motion rather than appearance patterns. GeoMoLa achieves state-of-the-art performance using only single-view RGB-D input, while existing methods require multi-view reconstruction, succeeding across diverse manipulation benchmarks. Our ablations reveal that geometric prediction is the key to driving performance, quantitatively validating that manipulation depends on spatial understanding. Furthermore, the learned codes exhibit effective motion abstraction: applying them to novel scenes produces physically consistent transformations regardless of visual context. Our real-world experiments also confirm this robustness capability, achieving robust manipulation with minimal demonstrations in cluttered environments where geometric reasoning determines success. Thus, we demonstrate that effective motion latents for robot control can better emerge from understanding motion through its three-dimensional effects rather than pixel-level patterns.

Yunchao Zhang, Yijia Weng, Ruizhe Liu +3

Department of Computer Science, The University of Hong Kong, Hong Kong, China · Department of Computer Science, Stanford University, Stanford, California, USA · Department of Data Science and AI, Monash University, Melbourne, Australia

Latent Action Learning Geometric Representation Learning Robotic Manipulation Dynamic Robotic Manipulation Motion Representation Learning

Jul 5, 2026cs.LG

Geometry of Ordinal Representations in Language Models

Recent work showed that language models represent character counts on curved 1D manifolds, with attention heads performing geometric transformations to enable computation. We test whether this generalizes across four ordinal tasks (bracket depth, indentation, table position, numeric magnitude) in Gemma-2-2B, Gemma-2-9B, and Qwen3-4B. We find that 1D manifolds with place-cell feature tiling emerge for tasks where the ordinal variable is locally computable from token identity, while tasks requiring cross-position integration or semantic extraction produce higher-dimensional or incoherent representations. Geometric computation is architecture-dependent: Qwen3-4B shows substantially stronger twisting than Gemma models for indentation, and its twisters preserve ordinal order, unlike its numeric twisters. Activation patching confirms that the identified manifold subspaces concentrate task-relevant information, with manifold-direction ablation causing dramatically larger probe accuracy drops than random-direction controls.

Saksham Bassi, Sharvi Tomar

*Equal contribution

Geometric Representation Learning Representation Geometry in Language Models

Jul 5, 2026cs.LG

Geometry-Aware Infrastructure-Anchored Denoiser for UWB Sensing and Work-Zone Reconstruction

Accurate work-zone geometry perception is critical for intelligent transportation systems, and ultra-wideband sensing offers a low-cost approach for infrastructure-aided reconstruction. However, outdoor UWB ranging is often degraded by non-line-of-sight propagation, burst noise, and long-tail errors, which can distort downstream spatial reconstruction. We present GAIA, a geometry-aware, infrastructure-anchored learning framework that couples temporal range modeling with latent anchor-layout estimation and deterministic distance projection. GAIA preserves range denoising as the supervised task while orienting the learned distances toward boundary-consistent reconstruction. We evaluate GAIA on a real-world outdoor UWB dataset with synchronized UWB, GNSS, and IMU measurements, and further test robustness using a real-data-calibrated stress-test simulator. GAIA achieves the lowest overall range MSE and highest polygon IoU among evaluated filtering-based and learning-based baselines, reducing MSE by 18.4% and improving polygon IoU by 15.5% over PoseMLP. These results show that geometry-aware range denoising provides an effective path toward spatially coherent work-zone reconstruction.

Weizhe Tang, Jiaxi Liu, Junwei you +5

Department of Civil & Environmental Engineering, University of Wisconsin-Madison, Madison, 53706, Wisconsin, United States · Department of Civil and Architectural Engineering and Construction Management, University of Wyoming, Laramie, 82071, Wyoming, United States · School of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

3D Reconstruction Geometric Representation Learning Wireless Localization

Jul 2, 2026cs.CV

GeoMix: Descriptor-Free Visual Localization via Global Context and Multi-Detector Training

Descriptor-free visual localization eliminates high-dimensional descriptor storage, preserves scene privacy, and simplifies map maintenance, yet its accuracy still lags far behind descriptor-based pipelines. We identify this gap to insufficient geometric discriminability in geometry-only matching. Without visual appearance, current methods underutilize local geometry cues, lack the global context among keypoints, and overfit to a single keypoint detector. We further observe that descriptor-free matching naturally enables multi-detector training, as heterogeneous keypoints can be optimized in a shared geometry-only space without aligning descriptor spaces. Building on these insights, we propose GeoMix, a descriptor-free 2D-3D matching framework that strengthens geometric discriminability at three levels. Locally, directional and distance-aware embeddings enrich neighborhood aggregation with fine-grained spatial structure. Globally, learnable context nodes aggregate and redistribute scene-wide information via cross-attention to resolve ambiguities beyond local receptive fields. At the training level, Mix-Training exploits this detector-agnostic geometry space to learn representations across multiple keypoint detectors. Extensive experiments on MegaDepth, Cambridge Landmarks, 7Scenes, and Aachen Day-Night show that GeoMix sets a new state of the art among descriptor-free methods, reducing 75th-percentile rotation error by 89% and translation error by up to 90% over the previous best, while generalizing zero-shot to unseen detectors and narrowing the gap to descriptor-based pipelines. Code is available at

\href{https://github.com/YejunZhang/Geomix}{\text{this links}}

.

Yejun Zhang, Xinjue Wang, Zihan Wang +2

Aalto University, Espoo, Finland · Tampere University, Tampere, Finland · University of Oulu, Oulu, Finland

Visual Place Recognition Camera Pose Estimation Geometric Representation Learning Robot Localization

Jul 2, 2026cs.CV

Understanding Geometric Representations in Self-Supervised Vision Transformers via Subspace Intervention

We introduce a controlled subspace intervention framework to investigate how self-supervised Vision Transformers (ViTs) encode dense geometric information. While linear probing is widely used to assess geometric representations, it treats features as a black box, failing to disentangle the underlying topology. To address this issue, we decompose the weights of converged linear probes to isolate the low-rank subspaces containing explicit geometric signals using Singular Value Decomposition (SVD). Our perspective yields three key insights: (1) Pre-training objectives determine how features are encoded. DINOv2 aligns spatial features for efficient linear extraction, while Masked Autoencoders (MAE) tend to disperse these signals, requiring a broader spatial context. (2) Explicit geometric representations are highly compressible, suggesting dense predictive heads could potentially be constrained to low-rank subspaces with minimal performance loss. (3) The layer-wise task affinity suggests that geometric precision peaks at intermediate layers before yielding to semantic abstraction in the final layers. By connecting internal encoding mechanics with downstream performance, these findings provide a basis for effective feature selection and lightweight decoder design. The source code is available at https://github.com/Zhou-Weichen/Geosubprobe.

Weichen Zhou, Yawen Zou, Chunzhi Gu +3

University of Toyama, Toyama, Japan · University of Fukui, Fukui, Japan · Chukyo University, Aichi, Japan +1

Neural Representation Geometry Linear Probing Vision Transformer Geometric Representation Learning Self-Supervised Visual Representation Learning

Jul 2, 2026cs.LG

Finite-Lag Operator Geometry of Recurrent Representations

Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs

(X_t,X_{t+Δ})

. The primitive is the conditional transport law

Q_Δ(dy\mid x)

, estimated by a dense Gaussian source-smoothing operator. From this directed finite-lag law we derive a source-centered transport tensor

G_Δ

, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation

W_Δ^ρ

, which summarizes directed lagged flow. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite-lag separation result showing that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry. A linear-Gaussian closed form calibrates the quantities in terms of the update

A_Δ

, source covariance, and innovation covariance. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions. In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent.

Kanishka Reddy

Department of Applied Mathematics University of Washington Seattle, WA 98105

Neural Representation Geometry Recurrent Neural Networks Geometric Representation Learning

Jul 1, 2026cs.LG

I\textsuperscript{2}RiMA: Spectral Riemannian Representation with Temporal Attention for Mental Stress Detection based on EEG Signals

Cross-subject EEG stress detection remains challenging because discriminative stress-related patterns are both subject-dependent and frequency-specific. Conventional Riemannian methods model spatial covariance mainly in the time domain, overlooking neural oscillations that are critical for high-level cognitive state decoding, while standard temporal tokenization often fragments inter-slice temporal coherence. To address these limitations, we propose \method{}, an Intra-Inter Riemannian Manifold Attention Network for EEG-based stress detection. \method{} constructs spatial covariance matrices independently at each frequency point and maps them to the SPD tangent space, preserving channel-wise geometry together with frequency-specific discriminative cues. It further introduces frequency cluster aggregation to select informative spectral components and reduce redundancy by forming compact, data-driven frequency clusters aligned with EEG rhythms. Finally, an intra-inter slice attention module adaptively integrates local slice-level spectral dynamics and global temporal context across EEG sequences. Experiments on three datasets show that \method{} consistently outperforms five state-of-the-art baselines, achieving up to 82.78% balanced accuracy while remaining efficient with only 1.60M parameters and 31.95M FLOPs.

Cheng He, Kunyu Peng, Shangen Han +3

Laboratory of Neural Computing and Intelligent Perception, College of Information Engineering, Capital Normal University, Beijing, China · Karlsruhe Institute of Technology, Germany · School of Psychology, Capital Normal University, Beijing, China

Cross-Subject EEG Decoding Electroencephalography Frequency-Domain Feature Learning Geometric Representation Learning

Jul 1, 2026cs.CV

HieDG: A Hierarchical Discrete Geometry-Guided Framework for Multi-Animal Tracking

Multi-animal tracking (MAT) is critical for wildlife monitoring and behavioral analysis, yet remains challenging due to uniform appearance, high density, and irregular motion. Existing methods typically follow heuristic- or query-based paradigms: the former relies on handcrafted geometric associations without end-to-end optimization, whereas the latter enables joint optimization but relies heavily on appearance embeddings. In such conditions, continuous geometric embeddings can be unstable, as small coordinate perturbations may disproportionately alter cross-frame attention weights, degrading identity association performance. To address this limitation, we propose HieDG, a Hierarchical Discrete Geometry-guided tracking framework that reformulates geometric dynamics as structured discrete representations within a query-based tracker. Instead of directly using raw geometric signals, HieDG employs a two-stage residual codebook to discretize position, scale, and velocity cues, transforming unstable continuous geometry into structured, stable discrete tokens. These tokens are aligned with visual embeddings and integrated into the tracking queries to enhance identity consistency. Extensive experiments on animal-specific benchmarks (AnimalTrack, BFT, and BuckTales) demonstrate state-of-the-art association performance with significant improvements in HOTA, AssA, and IDF1. Additional evaluations on generic multi-object tracking benchmarks, including DanceTrack and SportsMOT, show competitive performance, indicating the broader applicability of discretized geometric modeling beyond animal-specific scenarios.

Chenxun Deng, Zhongde Zhang, Ye Yuan +7

State Key Laboratory of Brain Cognition and Brain-inspired Intelligence Technology, Institute of Automation, Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences · School of Technology, Beijing Forestry University +6

Geometric Representation Learning Wildlife Monitoring Multi-Object Tracking

Jun 30, 2026cs.CV

Deep Spectral Models for Robust Dental Shape Generation

Accurate modeling of dental crown morphology is fundamental for diagnosis, orthodontic planning, and computer-aided restoration design. However, datasets suitable for training such models are typically limited in size. We present ToothForge, a deep spectral generative framework that models dental crown geometries from compact, intrinsic representations. By operating in the spectral domain, ToothForge learns a latent manifold of 3D tooth shapes through synchronized spectral embeddings, ensuring consistent modeling across samples with varying connectivity. Spectral synchronization mitigates the instability of Laplace-Beltrami eigenbases and enables efficient learning in a low-dimensional space. The framework is thoroughly evaluated through robustness analysis, ablation studies, and benchmarking against PCA-based statistical shape models and point-based generative frameworks. Results show that synchronized spectral modeling achieves reconstruction and generative performance comparable to or exceeding spatial approaches, while maintaining compactness and geometric interpretability. Together, the compact synchronized coefficients and low-dimensional learning space make the framework particularly suitable for limited datasets, as often encountered in dental and medical domains, and applicable in real-world scenarios where guaranteeing consistent connectivity across shapes from various clinics is unrealistic.

Tibor Kubík, François Guibault, Michal Španěl +1

Polytechnique Montr´eal, Montr´eal, Canada · Brno University of Technology, Brno, Czech Republic

Geometric Representation Learning Generative Modeling 3D Shape Generation

Jun 29, 2026cs.RO

Self-supervised Geometry Reasoning for LiDAR Simultaneous Localization and Mapping

LiDAR simultaneous localization and mapping (SLAM) relies on local geometric quantities such as covariances, correspondences, and surface structures. However, most existing pipelines rely on hand-crafted estimates of local geometry and use them as fixed inputs to LiDAR SLAM, which can make the estimated local geometry noisy and unstable in sparse regions of a point cloud or when using low-resolution LiDAR. To address this issue, this paper introduces a self-supervised framework that learns an explicit symbolic representation of local geometry and uses it to improve LiDAR SLAM recursively. Specifically, each point is represented as a Gaussian distribution, allowing local geometry to be described by a covariance. Without dense geometry labels or ground-truth poses, the framework learns by maximizing the likelihood of local geometry, with self-supervision derived from consistency relations over symbolic geometric representations, including predicted covariances, correspondences, and trajectory from SLAM. The learned geometry is then fed back into LiDAR SLAM, forming a reciprocal loop in which improved geometry enhances localization and mapping, and improved localization provides cleaner supervision for subsequent geometry reasoning. This framework is backend-agnostic and can be plugged into existing LiDAR SLAM pipelines without architectural changes. Experiments on KITTI under varying LiDAR resolutions show that the proposed method improves both odometry and global registration.

Jiwoo Kim, Jinwoo Lee, Woojae Shin +2

Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea · Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu, Republic of Korea

Simultaneous Localization and Mapping Self-Supervised Learning LiDAR Odometry Geometric Representation Learning LiDAR Localization

Jun 26, 2026cs.LG

From Dataset Spectral Geometry to Network Weights: A Geometry-Aware Initialization for Sigmoidal MLPs in Image Classification

Classical universal approximation theorems (UAT) establish the expressive power of sigmoidal multilayer perceptrons, but they do not specify how the weights should be initialized. We study a supervised, data-dependent, geometry-aware initialization for one-hidden-layer sigmoidal MLPs that compiles labeled class geometry into network weights. The construction starts from the idea that sigmoid units can act as smooth half-space gates. For each class, we center the training samples at their mean, apply SVD to estimate principal directions and spectral scales, select retained directions by an energy threshold, and represent each retained direction by a pair of sigmoid slab gates. These class-specific gates are then concatenated into a shared hidden layer initialized directly from the training set. We also formulate a SVD-Mahalanobis subspace classifier as a non-neural geometric reference, which tests whether the estimated spectral class geometry is already discriminative before being embedded into the MLP. Experiments on MNIST, Fashion-MNIST, and CIFAR-10 show that the proposed initializer produces a substantially more informative zero-epoch state than a matched task-agnostic Xavier reference, while full training reaches comparable final accuracy. Frozen-hidden experiments and neutral-head ablations further show that the class-wise SVD gates remain useful fixed features, even after the initially aligned output layer is replaced by a Xavier-initialized one.

Yi-Shan Chu

Institute of Statistical Science, Academia Sinica, Taipei, Taiwan (R.O.C.)

Multilayer Perceptrons Neural Network Initialization Geometric Representation Learning Image Classification

Jun 23, 2026cs.RO

GRAFT: Graph-Based Affordance Transfer via Part Correspondence

Generalizing robotic manipulation to unseen objects remains challenging, as learning-based approaches require many demonstrations and fail in few-shot settings. Prior work transfers affordances through semantic retrieval, but semantics alone neglect geometric similarity, which is critical for manipulation. We propose GRAFT, a geometry-aware correspondence framework for zero-shot manipulation transfer using only one demonstration per object. Objects are represented as part-based graphs, where part-level descriptors support global instance retrieval and part correspondence, and vertex-level descriptors enable fine-grained contact point matching. For an unseen object, our method first retrieves the most functionally and geometrically similar instance from the demonstration buffer with aligned functional parts, and finally propagates the contact points through point-wise correspondence.

Mengying Lin, Utkarsh Mishra, Ajay Mandlekar +1

Georgia Institute of Technology · NVIDIA

Zero-Shot Robotic Manipulation Generalization in Robotic Manipulation Geometric Representation Learning Robotic Manipulation Robot Learning

Jun 23, 2026cs.CV

VistaRef: Boosting Visual Spatial Orientation Awareness for Pointing-to-Object Detection

Grounding deictic gestures in natural images is fundamental to AR and human-robot collaboration, providing a basis for seamless spatial interaction. While Transformer-based visual models have achieved significant progress in general object detection, their global attention mechanisms often neglect micro-geometric relationships, degrading orientation accuracy. In pointing tasks, this deficiency manifests as an inability to accurately capture the pointing ray implied by finger poses, which results in pointing drift and localization ambiguity when dealing with distant or densely packed objects. To address this, we propose VistaRef, a framework designed to explicitly enhance spatial orientation awareness. First, we develop the Local Hand Entity Modeling (LHEM) module, which incorporates hand-pose embeddings to strengthen the model's capability to capture subtle finger deviations. Second, drawing inspiration from multi-view geometry, we construct the Geometric Ray Modeling (GRM) module to transform implicit orientation information into explicit spatial geometric features, guiding feature aggregation and deep fusion via attention mechanisms. Furthermore, we introduce a novel Orientation-Consistent Alignment Loss (OCAL) to synergistically supervise hand presence and pointing consistency, ensuring that all architectural improvements collectively serve the core objective of spatial localization. Experimental results demonstrate that VistaRef significantly outperforms the baseline, achieving a 14-point absolute gain in grounding accuracy. Qualitative analysis further confirms that VistaRef effectively models the geometric correlation from hand to target, bridging the spatial perception gap inherent in traditional Transformers for complex scenarios. Code: https://github.com/lingli1724/VistaRef.

Ling Li, Zhizhen Cai, Xinkun Wu +4

Tsinghua University Beijing Shi, China · Beijing University of Posts and Telecommunications Beijing Shi, China · Tsinghua Shenzhen International Graduate School Shenzhen, Guangdong, China +1

Visual Spatial Reasoning Detection Transformer Geometric Representation Learning Object Detection

Jun 23, 2026cs.CV

REDI-Match: Rotation-Equivariant Distillation for Efficient and Robust Dense Matching

Vision Foundation Models (VFMs) have significantly advanced dense feature matching, yet severe in-plane rotation remains a critical challenge. Existing solutions face a fundamental dilemma: data-driven methods require inefficient parameter scaling to implicitly learn rotations, whereas strictly equivariant networks lack the semantic capacity of modern VFMs. Consequently, current frameworks typically freeze VFMs and shift the entire burden of rotation generalization to the downstream decoder. To break this architectural bottleneck, we propose REDI-Match, an efficient framework driven by a novel Rotation-Equivariant Distillation (REDI) paradigm. Instead of relying on rotation data augmentation to establish rotational correspondences, REDI distills the non-equivariant semantic representations of a VFM into a lightweight, strictly rotation-equivariant encoder, leveraging an equivariant geometric architecture to constrain robust high-dimensional semantics. To fully exploit these features, we equip the decoder with an entropy-driven spatial alignment module. By evaluating discrete rotation hypotheses, this mechanism explicitly locks onto the canonical coordinate system, eliminating global ambiguity before continuous refinement. Extensive experiments demonstrate that REDI-Match establishes a new state-of-the-art (SOTA) across multiple benchmarks. Notably, it achieves a 13.89% absolute pose accuracy improvement on the highly challenging SatAst dataset while operating 1.9x faster than the current SOTA (RoMa v2), enabling real-time inference (~41 FPS) on a single RTX 4090 GPU. Code: https://github.com/YinjiGe/REDI-Match.

Yinji Ge, Guixu Zheng, Wulong Guo +5

1Tsinghua University · 2Southern University of Science and Technology · 3Beihang University +1

Vision Foundation Model Adaptation Geometric Representation Learning Knowledge Distillation

Jun 21, 2026cs.CV

MAPS: Multi-Anchor Projection Similarity for Joint Vision-Language Geo-Localization

Humans localize places by integrating perceptual cues from vision with semantic reasoning from language, forming a scene understanding that is both intuitive and structured. Although existing geo-localization models have made substantial progress in cross-view and cross-modal settings, they are largely built upon point-to-point alignment, which is insufficient for joint vision-language queries. In such queries, visual and textual cues do not simply act as independent references, but jointly define a semantic subspace for locating the target. In this paper, we formulate vision-language geo-localization (VLGL) with joint image-text queries as a multi-anchor geometric alignment problem and propose a unified framework for this setting. To realize this formulation, we propose Multi-Anchor Projection Similarity (MAPS), a new metric which constructs an anchor plane from visual and textual query features in a high-dimensional space and measures similarity by the projection length of the target feature onto this plane. Unlike cosine similarity which evaluates isolated pairwise relations, MAPS captures the geometric consistency between the target feature and the joint query subspace, providing a more discriminative ranking criterion during retrieval. To make the learned representation consistent with this geometry, we further introduce a MAPS-based contrastive loss that drives target features toward the corresponding anchor plane. The proposed framework, similarity metric, and training objective jointly yield state-of-the-art performance in VLGL.

Yutong Hu, Siyuan Tan, Shaocheng Yan +3

School of Remote Sensing and Information Engineering, Wuhan University, Wuhan 430079, China

Cross-View Geo-Localization Geometric Representation Learning Vision-Language Retrieval Multimodal Contrastive Learning

Jun 17, 2026cs.RO

Space Is Intelligence: Neural Semigroup Superposition for Riemannian Metric Generation

Traditional approaches place intelligence in the agent, whether as a learned policy or a search procedure. We instead place intelligence in the space itself: a scene induces a Riemannian metric on the configuration manifold, and action reduces to following the geodesics of that metric rather than invoking a separate planner or collision checker. A single Encoder-Router network realizes this idea through three complementary parameter groups -- frame parameters that orient the generators, modulation parameters that govern their spatial propagation, and basic coefficients that determine their strength. These groups combine through a shared semigroup-superposition mechanism to produce a single Riemannian metric field, yielding a compact architecture whose geometry scales naturally with scene complexity. Trained on a single two-obstacle scene, the model demonstrates robust zero-shot generalization across unseen obstacle configurations, with orders-of-magnitude separation between collision-free and obstacle-penetrating path costs.

Chenghao Xu

School of Artificial Intelligence and Robotics and National Engineering Research Center of Robot Visual Perception and Control Technology, Hunan University, Changsha, Hunan, China

Geometric Representation Learning Metric Learning Motion Planning Robot Motion Planning

Jun 17, 2026cs.CV

Deep Image Prototype Learning with Geometric Heat-Kernel Priors

Learning unsupervised representations of medical imaging cohorts can reveal anatomically meaningful prototypes without expert labels, which are often noisy and fail to capture true pathological heterogeneity. However, existing deep latent-variable models estimate Gaussian mixture priors via Euclidean averaging, producing prototypes that drift off the curved data manifold and degenerate as the number of sub-populations grows. We propose a manifold-anchored variational framework built on a geometry-aware Expectation-Maximization (EM) algorithm, whose M-step selects each sub-population prototype as the graph medoid with the highest diffusion centrality on a heat-kernel-weighted latent graph, ensuring that every prototype remains on-manifold. A Dirichlet energy regularizer enforces geometric smoothness of the latent space, and a per-sub-population uncertainty score enables label-free quality assessment. The manifold-anchored EM is a general-purpose geometric tool that extends standard EM and applies readily to other latent-variable models beyond this setting. On cardiac scar and brain MRI benchmarks, our framework attains the highest accuracy among all compared methods, produces the sharpest prototypes reported to date, and remains stable at large sub-population counts where all baselines degenerate. Code and implementation details are available at https://github.com/jr-xing/On-Manifold-Variational-Learning-with-Heat-Kernel-Priors.

Jiarui Xing, Tal Zeevi, Nian Wu +1

Yale School of Medicine, New Haven CT, 06510, USA · University of Virginia, Charlottesville, VA, 22903, USA · Harvard Medical School, Boston MA, 02115, USA

Medical Image Analysis Prototype Learning Unsupervised Representation Learning Geometric Representation Learning Medical Image Representation Learning

Jun 17, 2026cs.LG

What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry

We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation

π

of a Lie group

G

on a space

V

and a learned function

f\colon V \to \mathbb{R}

, we define two objects measuring the symmetry invisible to

f

. The null fiber at a point

x \in V

is the set

N_G(f,x) = \{g \in G : f(π(g^{-1}) \cdot x) = f(x)\}

of group elements whose inverse action on

x

is undetectable by

f

. When

N_G(f,x)

is independent of

x

, it coincides with the stabilizer

\mathrm{Stab}_G(f)

, the largest subgroup of

G

under which

f

is invariant. For smooth maps to

\mathbb{R}

, the preimage theorem guarantees that null fibers have dimension at least

\dim G - 1

at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of

f

. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under

\mathrm{SO}(3)

and spherical image classification under the Möbius group

\mathrm{PSL}(2, \mathbb{C})

. The framework applies uniformly to classical neural networks and variational quantum circuits.

Zachary P. Bradshaw

QodeX Quantum, Inc. Chicago, IL

Geometric Representation Learning Lie Group Methods Privacy-Preserving ML Invariant Representation Learning

Jun 16, 2026stat.ML

Compact Geometric Representations of Hierarchies

Computing geometric representations of data is a cornerstone of modern machine learning, typically achieved by training dual encoders which map queries and documents into a shared embedding space. Recent work of You et al. [NeurIPS '25] has extended this approach to hierarchical retrieval, where relevance is determined by the ancestor-descendant relationships in a Directed Acyclic Graph (DAG). While previous work has shown that valid embeddings exist when the number of descendants is small, these bounds degrade significantly for deep hierarchies, requiring dimensions as large as the total number of nodes. In this paper, we investigate compact reachability embeddings for more general graph classes and provide theoretical guarantees for representing hierarchies using embeddings whose dimension depends on structural graph parameters. We prove that for any directed tree, there exists a reachability embedding in constant dimension 3, independent of the tree's size or depth. We generalize this result to graphs characterized by treewidth

t

, constructing embeddings of dimension

O(t \log n)

, where

n

is the number of nodes. Complementing these upper bounds, we provide matching or near-matching lower bounds, showing that dimension

Ω(n)

is necessary for general DAGs and

Ω(t/\log(n/t))

is required for graphs of treewidth

t

. We also obtain upper and lower bounds parameterized by the number of cross-edges in the DAG. We additionally show that our embeddings can be constructed on real world datasets, and that they give much smaller dimensions in high recall regimes compared to prior embeddings with theoretical guarantees.

Prashant Gokhale, Piotr Indyk, Yuhao Liu +3

UW-Madison · MIT

Hierarchical Representation Learning Geometric Representation Learning Graph-Based Retrieval Graph Embedding

Jun 15, 2026math.ST

Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis

A central objective of machine learning is to identify structure and patterns in data. Advances in data acquisition have increasingly produced datasets whose observations possess rich geometric form, giving rise to shape spaces that encode variability in object geometry. Such datasets arise across a wide range of disciplines, including biology, medicine, anthropology, and computer vision, where subtle geometric differences often carry important scientific information. Traditional machine learning methods, however, are frequently ill-equipped to account for the nonlinear geometric structure underlying these data. This survey synthesizes a rapidly growing body of work on shape space analysis, which provides a mathematical and computational framework for the study of geometric data. Drawing on ideas from differential geometry, statistics, and machine learning, we organize the literature around a common analytical pipeline: shape representation and parameterization, the rigorous construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. We discuss how these tools enable the characterization of shape variability, the comparison of geometric objects, and the analysis of structural trajectories across populations and time. To illustrate the breadth of the field, we highlight applications spanning multiple scales of biological organization, including studies of subcellular morphology and primate tooth evolution. Across these and many other domains, researchers face common challenges arising from complex, nonlinear, and often unaligned geometric variation. The review concludes by identifying key theoretical and computational challenges, as well as emerging opportunities driven by increasingly large and diverse geometric datasets.

Gary P. T. Choi, Khanh Dao Duc, Shira Faigenbaum-Golovin +6

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR · Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z4, Canada · Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel +6

Statistical Shape Modeling Geometric Representation Learning

Jun 14, 2026cs.RO

GeoTLM: Geometry-aware Tactile-Language Models for Contact Motion Orientation Reasoning of Dynamic Objects

Modern tactile-language models (TLMs) have shown potential for robot learning tasks, such as material and texture recognition. However, for contact-rich scenarios, these TLMs struggle to understand the physical properties of dynamic objects, such as rotation and sliding directions. For instance, our preliminary experiments reveal that popular TLMs, such as Sparsh and AnyTouch2, exhibit weak performance on basic rotation direction reasoning from GelSight Mini tactile data. This surprising gap inspires us to explore a novel research question: Can we inject physically grounded geometric priors into TLMs to enable reliable contact orientation reasoning of dynamic object properties? To this end, we propose GeoTLM, a novel geometric representation-guided TLM for the perception of dynamic contact events. Our key idea is to preserve and structure tactile shear-field geometry before language-level reasoning, rather than forcing low-resolution tactile tokens into fragile closed-form physics operators. To achieve this, we propose a lightweight (only 14k parameters) yet novel Differentiable Geometric Representation (DGR). Specifically, DGR learns a contact-mask-guided representation in the shear field and aggregates it through an antisymmetric seven-region pooling design, motivated by the physical intuition that rotational contact produces antisymmetric deformation patterns. We conduct experiments on two representative tasks: rotation direction and sliding direction reasoning. Extensive experiments show that GeoTLM improves novel-object rotation accuracy by +14.6% and real-sensor sliding accuracy by +16.2% over the same backbone without the geometric encoder. Overall, our work paves a new way for physically grounded tactile-language reasoning, with strong potential for dynamic object understanding and contact-rich robotic manipulation.

Qiutian Li, Zinan Liu, Lin Wang

School of EEE, Nanyang Technological University (NTU), Singapore

Tactile Sensing Contact-Rich Robotic Manipulation Geometric Representation Learning Robotic Tactile Sensing

Jun 13, 2026cs.LG

LieBN: Batch Normalization over Lie Groups

Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold-valued sample distributions. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization (RBN) over Lie groups. Our approach leverages the theoretically convenient left- and right-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices. Notably, among the SPD metrics, we introduce a novel right-invariant metric and extend three existing Lie group structures via matrix power deformation. Extensive experiments on different manifolds validate the effectiveness of our framework. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Ziheng Chen, Yue Song, Rui Wang +2

Department of Information Engineering and Computer Science, University of Trento, Trento, Italy · Computing and Mathematical Sciences, Caltech, CA, USA · School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, China

Geometric Representation Learning Lie Group Methods

Jun 12, 2026cs.LG

Riemannian Metric Matching for Scalable Geometric Modeling of Distributions

High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose Riemannian metric matching: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the carré du champ operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carré du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of

k

-NN-based diffusion geometry estimators, while enabling amortized inference that is up to

400\times

faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.

Jacob Bamberger, Adam Gosztolai, Pierre Vandergheynst +2

Department of Computer Science, University of Oxford, Oxford, UK · Institute of Artificial Intelligence, Medical University of Vienna, Vienna, Austria · EPFL, Lausanne, Switzerland +1

Geometric Representation Learning Metric Learning Riemannian Geometry

Jun 12, 2026cs.LG

Curvature-Guided Geometric Representation for Protein-Ligand Binding Affinity Prediction

Protein-ligand binding affinity (PLA) prediction is critical in drug discovery. Despite the notable advancements in machine learning-based approaches, existing methods struggle to jointly characterize local geometric organization and globally coordinated cross-molecular interactions, limiting their ability to model complex binding mechanisms. Here, we propose RicciBind, a geometric representation framework that integrates curvature-guided hierarchical structure learning with optimal transport (OT)-based cross-domain alignment to model molecular interactions. Specifically, RicciBind leverages Ricci curvature to capture local interaction tightness within molecular structures, enhancing structural awareness and organizing atomic interactions into curvature-aware hierarchical representations. An OT-based cluster matching mechanism then aligns protein and ligand clusters across heterogeneous domains under geometric constraints, enabling globally consistent correspondences and revealing higher-order interaction patterns beyond local neighborhoods. By coupling curvature-guided structure encoding with OT-driven cross-domain alignment, RicciBind effectively models complex interaction semantics and substantially improves both the accuracy and interpretability of binding affinity prediction. Extensive experiments demonstrate that RicciBind achieved superior predictive performance and generalization across PLA benchmarks and virtual screening tasks. Ablation studies further confirmed the essential role of Ricci curvature in enhancing molecular interaction representations.

Shuai Li, Chuan-Xian Ren, Yuhao Li +4

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China · Department of Electrical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

Protein-Ligand Binding Affinity Prediction Hierarchical Representation Learning Geometric Representation Learning Molecular Property Prediction Drug Discovery

Jun 11, 2026cs.LG

The Stable Recovery Manifold: Geometric Principles Governing Recoverability in Continual Learning

Catastrophic forgetting is often viewed as the destruction of previously learned knowledge during sequential learning. Building on the Accessibility Collapse framework, we investigate the geometric structure of recoverability in continual learning. Using Split CIFAR-100 and a sequentially trained ResNet-18, we analyze recoverability, representational drift, and recovery complexity across ten tasks. We introduce Recovery Subspace Dimensionality (k_t), a measure of the minimum number of singular directions required to preserve 90 percent of full probe performance. Contrary to our Recoverability Diffusion hypothesis, recovery dimensionality remains stable throughout training (mean k_t = 8.0) despite substantial representational drift. Principal-angle drift strongly predicts recoverability (r = -0.862), and a simple geometric model explains 82.2 percent of recoverability variance. These findings support the Stable Recovery Manifold hypothesis, suggesting that forgotten knowledge remains compactly decodable despite representational reorganization. The results indicate that catastrophic forgetting is primarily an accessibility and manifold-alignment problem rather than information destruction.

Ayushman Trivedi, Bhavika Melwani

Continual Learning Geometric Representation Learning Representation Drift Catastrophic Forgetting

Jun 11, 2026cs.RO

GeoHAT: Geometry-Adaptive Hybrid Action Transformer for Mobile Manipulation

Whole-body mobile manipulation requires coordinating mobile base and manipulator under shifting viewpoints, posing challenges in geometric perception and action generation. Current policies either rely on 2D features or sparse 3D representations that lack dense spatial structure, and typically encode arm and base within one action vector that ignores their distinct control demands. Moreover, existing dense fusion strategies risk corrupting pretrained representations under noisy depth while incurring heavy computational overhead. We present GeoHAT, an end-to-end diffusion-based framework built on a simple principle: geometry should be injected only where reliable and attended to only where needed. GeoHAT employs a lightweight Fourier spatial encoder that maps dense per-pixel 3D coordinates into geometric tokens without an additional 3D vision backbone. These tokens are then selectively injected into vision foundation model features through per-token gated fusion modulated by depth validity, preserving the semantic prior while enriching spatial understanding. For action generation, a Hybrid Whole-Body Action Decoder decomposes arm and base into distinct subspaces and lets each action modality attend to its task-relevant visual context through sparse cross-attention, while causal temporal modeling captures intra-timestep coordination and inter-timestep dependencies. Experiments on the ManiSkill-HAB simulation benchmark demonstrate that GeoHAT achieves a 79.3% mean success rate, surpassing the strongest baseline by 23.7%. Furthermore, real-world experiments on diverse tasks also confirm consistent improvements over all baselines.

Xiangyu Zhu, Renjun Wu, Luzhou Ge +2

Beijing Institute of Technology

Mobile Manipulation Whole-Body Control Diffusion Policy Geometric Representation Learning Robotic Manipulation

Jun 11, 2026cs.CV

Learning Task-Aware Sampling with Shared Saliency through Density-Equalizing Mappings

In image and surface-based learning tasks, convolutional features are typically extracted using receptive fields that are sampled uniformly across the entire domain. However, informative structures are rarely distributed uniformly in practice and are often concentrated in localized regions. Such phenomena are particularly common in medical imaging, where pathological changes are spatially confined. Consequently, uniform convolution allocates equal computational effort to both informative and uninformative regions, resulting in inefficient feature extraction and suboptimal utilization of model capacity. To address this issue, we propose a framework for task-adaptive sampling that dynamically redistributes computational attention according to the spatial importance of the data. Specifically, we introduce the Density-Equalizing Convolutional Neural Network (DECNN), which employs density-equalizing mappings to guide convolution through a learned density function. The density function encodes the relative importance of different regions and induces a transformation that enlarges informative areas while compressing less relevant ones. As a result, convolutional receptive fields are redistributed non-uniformly over the domain, enabling denser sampling in task-relevant regions. By coupling this importance-driven transformation with convolution, DECNN performs adaptive feature extraction that focuses computational resources on informative structures. This leads to more efficient use of model capacity, yielding a lightweight yet expressive architecture while simultaneously producing an interpretable saliency map. Experiments on image classification and craniofacial surface analysis demonstrate that DECNN achieves competitive or superior performance with fewer parameters, accurately identifies task-relevant regions, and remains robust under complex geometric variations.

Tsz Lok Ip, Han Zhang, Lok Ming Lui

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China. · Department of Mathematics, City University of Hong Kong, Hong Kong, China.

Geometric Representation Learning Convolutional Neural Networks Medical Image Representation Learning

Jun 9, 2026cs.LG

Encoding the Euler Characteristic Transform

The Euler Characteristic Curve (ECC) records the Euler characteristic of a linearly embedded cell complex as a function of filtration height in a given direction, and the Euler Characteristic Transform (ECT) is the injective shape descriptor obtained by collecting ECCs over many directions. How the ECT is encoded for a neural network is itself an inductive bias, conventionally fixed by discretizing each ECC. We introduce a continuous encoding: for each direction and each vertex it records the net Euler-characteristic change attributed to that vertex, producing a per-direction token sequence that a small transformer maps to a feature vector. We separate the resulting pipeline into two stages on orthogonal axes: an ECC encoder that acts within each direction, mapping its curve to a fixed-length vector, and an ECT representation that acts across directions, aggregating the per-direction vectors into one. We study six ECT representation architectures spanning a range of inductive biases, from a structure-agnostic feedforward baseline to convolutional and complex-valued models that preserve equivariance under planar rotations. Across six classification benchmarks covering point clouds, graphs, cubical complexes, and meshes, the continuous encoding improves accuracy on five of six datasets, and control experiments attribute the gain to the tokenization itself rather than to the added transformer capacity. The representation architecture matters less than the encoding, and the payoff from its inductive biases depends on the encoding: a feedforward network performs best under continuous encoding but is less robust under discretization than convolutional architectures.

Nello Blaser, Odin Hoff Gardaa, Lars M. Salbu +2

Department of Informatics, University of Bergen, Norway · Department of Informatics, University of Fribourg, Switzerland

Representation Learning Geometric Representation Learning Topological Representation Learning Topological Data Analysis

Jun 9, 2026cs.LG

Geometry-Aware Reinforcement Learning for 2D Irregular Nesting

Traditional heuristic solvers for the 2D irregular nesting problem share a fundamental limitation: they are blind to polygon geometry, relying on guided brute-force to navigate the continuous placement space with minimal geometrical guidance. In this paper, we argue that Reinforcement Learning is uniquely positioned to overcome this bottleneck. By pairing an optimization policy with a geometry-aware neural encoder, an agent can automatically discover rich geometric priors directly from data, utilizing these learned intuitions to strategically guide exploration. To realize this, we introduce the Polygons Transformer (PoT), a novel architecture that encodes 2D continuous vector geometries while allowing cross-polygons attention. We couple this novel architecture with a Combinatorial Optimization Reinforcement Learning (CORL) training framework to find optimal solutions. To support this paradigm, we release an open-source training dataset derived from complex geographic contours alongside a dedicated evaluation benchmark. Our empirical validation demonstrates that our trained agent achieves area utilization performance highly competitive with Sparrow, the state-of-the-art heuristic solver, proving that reinforcement learning can successfully discover and exploit geometric awareness for precise spatial tasks.

Auguste Lehuger, Guillaume Henon-Just

Valeo Brain

Reinforcement Learning Geometric Representation Learning RL for Combinatorial Optimization

Jun 7, 2026cs.RO

GEAR-VLA: Learning Geometry-Aware Action Representations for Generalizable Robotic Manipulation

Vision-Language-Action (VLA) models achieve strong benchmark performance but still struggle in real-world deployment with unseen objects, background shifts, and different robot embodiments. We argue that this stems from the lack of a unified geometry-aware manipulation representation, leaving existing VLAs vulnerable to low-level trajectory supervision, misaligned 3D features, and embodiment differences. To address this, we propose GEAR-VLA, a VLA framework for learning unified geometry-aware action representations for generalizable robotic manipulation. GEAR-VLA adopts coarse-to-fine action learning, where multi-source embodied pretraining equips the VLM with embodied reasoning and discrete action understanding before latent action tokens connect action semantics to a gradient-decoupled DiT continuous action expert. It further performs semantic-aligned 3D integration by aligning a trainable 3D spatial backbone with the VLA representation while freezing the original VLM-aligned visual pathway. To share this representation across robots, GEAR-VLA uses embodiment canonicalization, where embodiment-aware states and embodiment-invariant actions confine robot differences to the low-level interface. Extensive simulation and real-world experiments demonstrate strong generalization: GEAR-VLA achieves state-of-the-art performance on LIBERO, zero-shot LIBERO-Plus, and RoboTwin 2.0, reaches 85.9% success on AgileX and 81.0% on the pretraining-unseen LDT-01 embodiment, and obtains 90.1% success on a 6,360-trial universal grasping benchmark with 212 unseen objects. Code and models will be released at https://github.com/babynabeauty/GEAR-VLA.

Yuan Zhang, Shiqi Zhang, Yedong Shen +11

Anhui University · University of Science and Technology of China · iFLYTEK

Generalization in Robotic Manipulation Cross-Embodiment Robot Learning Geometric Representation Learning Robotic Manipulation Vision-Language-Action Models Robot Learning

Jun 6, 2026cs.RO

Revisiting Articulated Parts Perception in Robot Manipulation

We are surrounded by various objects with movable, articulated parts, e.g., box, handle, door. An accurate and generalizable perception of articulated parts is essential to enhance robotic manipulation capabilities. Building on this need, recent efforts in articulated parts perception have followed two main directions: One line of work uses pose-based representation, which requires high manual cost; in parallel, affordance-based methods extract future object motion from point tracking without additional manual efforts, but suffer from low-quality data. In this paper, we propose a new representation of articulated parts, Geometric Primary Structure (GPS), an abstraction of the part geometry structure to balance scalability and quality. For efficient and scalable data collection, GPS is integrated with a portable Virtual Reality (VR) device and requires only one minute to annotate one object sequence. This direct human annotation provides higher quality than the estimated affordance. With this efficient VR-GPS system, we collect 41K frames for 234 objects across six part classes, and train a generalizable GPS model with a single RGB-D object image as input. For object manipulation, we deploy a heuristic policy based on GPS prediction. Without any in-domain fine-tuning, our method achieves an 73% success rate, covering 270 initial states for 9 objects. Our code, data and reusable tool are available at https://enlighten0707.github.io/gps.

Xiaoqian Wu, Yejie Guo, Xiaoyang Chen +3

Shanghai Jiao Tong University

Geometric Representation Learning Articulated Object Manipulation Robotic Perception

Jun 4, 2026cs.LG

The Post-GCN Decade Revisited: Curvature-Stratified Evaluation of Relational Learning

Current evaluation practices in relational learning rely heavily on flat leaderboards that average performance across heterogeneous datasets, implicitly assuming a uniform underlying structure. We show that this assumption introduces systematic bias: it obscures geometry-dependent performance variations and can lead to misleading conclusions about model generalization. In this work, we identify intrinsic geometry as a key latent factor governing model effectiveness. We demonstrate that conventional aggregated metrics mask critical performance trade-offs that only become visible when datasets are stratified by their geometric properties. To address this issue, we introduce a curvature-stratified evaluation framework that partitions datasets into positive, negative, and near-zero curvature regimes. Our benchmark evaluates 18 representative models including Graph Convolutional Networks (GCNs), Graph Foundation Models (GFMs), and tabular learning methods across 14 datasets. We find that model rankings are highly stable within each curvature regime but shift significantly across regimes, indicating that performance is fundamentally geometry-dependent rather than universally transferable. Notably, we identify regimes where GFMs offer diminishing returns compared to geometry-aligned GNNs. Based on these findings, we propose a geometry-aware evaluation protocol that yields more reliable and interpretable comparisons than standard aggregated benchmarks. We release all code, curvature-stratified dataset splits, and evaluation tools to support reproducible and rigorous assessment of future relational learning methods. Code and datasets are provided in our project homepage: https://sirbabbage.github.io/CurvBench_HOME/.

Shuo Wang, Xiangyu Wang, Quanxin Wang +9

University of Electronic Science and Technology of China · Tsinghua University · Western University +1

Graph Neural Networks Graph Representation Learning Geometric Representation Learning Graph Foundation Models

Jun 4, 2026cs.SD

Geometric Second-Order Feature Correlation Learning for Self-Supervised Speech Emotion Recognition

Self-supervised learning (SSL) yields powerful, context-rich representations for speech emotion recognition (SER), yet aggregating these representations into holistic descriptors remains a bottleneck. Conventional first-order aggregation implicitly assumes feature independence, which overlooks the latent Riemannian geometry and discards higher-order relationships essential to the representational power of the backbone. To address this problem, this paper proposes a novel Second-Order Correlation (SOC) layer. Instead of treating features in isolation, SOC models feature correlations as covariance descriptors to capture synergistic co-occurrence patterns, which serve as discriminative signatures for robust emotion recognition. By mapping these descriptors from the Riemannian manifold to a Euclidean tangent space through Log-Euclidean mapping (LEM), the proposed method preserves geometric integrity while enabling direct linear discriminative learning. Extensive experiments on the ESD and RAVDESS datasets demonstrate that SOC recovers discriminative information lost in first-order pooling and effectively aggregates high-dimensional SSL features.

Shuanglin Li, Ruxiao Qian, Siyang Song

Xiangjiang Laboratory, Changsha, China · University of Exeter, Exeter, UK

Geometric Representation Learning Speech Emotion Recognition

Jun 4, 2026cs.CV

Learning Geometric Representations from Videos for Spatial Intelligent Multimodal Large Language Models

Multimodal Large Language Models (MLLMs) excel at 2D semantic understanding but lack intrinsic 3D awareness, resulting in representations that fail to maintain geometric and spatial consistency across video frames. Given the scarcity of large-scale 3D data, we present GeoVR, a novel framework that learns geometric representations using purely 2D video sequences. This approach effectively restructures the semantic latent space within MLLMs to unlock spatial intelligence. Rather than employing superficial feature mixing, GeoVR reshapes the internal representations of the MLLM by distilling geometry knowledge from pre-trained 3D foundation models. This is accomplished through a multi-objective learning strategy driven by four complementary geometric targets: (1) estimating inter-frame camera poses to embed varying viewpoint dynamics, (2) regressing dense depth maps to anchor physical distances, (3) predicting a metric scale factor for real-world calibration, and (4) distilling multi-scale 3D features to align the intermediate feature space. Guided by these explicit physical and geometric constraints, the model's internal representations naturally develop strong 3D awareness. Extensive experiments on spatial reasoning benchmarks demonstrate that GeoVR achieves state-of-the-art performance, establishing a new paradigm for endowing foundation models with spatial intelligence.

Haibo Wang, Lifu Huang

University of California, Davis

3D Spatial Reasoning Multimodal Large Language Models Geometric Representation Learning Video Representation Learning 3D Representation Learning

Jun 3, 2026cs.CV

SFMambaNet: Spectral-Frequency Enhanced Selective State Space Model for Correspondence Pruning

Correspondence pruning aims to identify inliers from an initial set of correspondences. Most existing Graph Neural Network (GNN)-based methods rely on geometric features mapped from coarse Euclidean coordinates, which struggle to capture the subtle geometric consistencies presented by inliers. While Mamba-based methods possess global receptive fields and long sequence modeling capabilities, they tend to accumulate substantial inconsistent features within the hidden state space, making it difficult to distinguish inliers from outliers. In this paper, we integrate frequency domain perception into this task for the first time and propose SFMambaNet, a novel Spectral-Frequency enhanced Mamba-based two-view correspondence pruning network. Our method is collaboratively composed of two components: First, we design a Local Spectral-Geometric Attention (LSGA) block. LSGA incorporates spectral positional encoding into local graph interactions and introduces multi-scale Mamba processing to enhance the capture of subtle geometric consistencies and improve local feature discriminability. Building upon this, we design a Spectral-Integrated Global Mamba (SIGM) block. SIGM embeds a frequency gating mechanism within the state space, utilizing the frequency information provided by LSGA to explicitly suppress high-frequency noise accumulation within hidden states and mitigate the propagation of inconsistent features. This enhances inlier-outlier separability and achieves robust global context modeling capabilities with nearly linear complexity. Extensive experiments demonstrate that SFMambaNet outperforms current state-of-the-art methods on several challenging tasks. The code is available at https://github.com/Kirito14IT/SFMambaNet.

Zhihua Wang, Yanping Li, Yizhang Liu

School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China · Institute of Artificial Intelligence, Shanghai Jiao Tong University, Shanghai, China · College of Computer and Data Science/College of Software, Fuzhou University, Fuzhou 350108, China

Selective SSMs Geometric Representation Learning

Jun 3, 2026cs.LG

From Symbolic to Geometric: Enabling Spatial Reasoning in Large Language Models

Recent large language models (LLMs) often appear to exhibit spatial reasoning ability; however, this capability is largely \emph{symbolic}, arising from pattern matching over spatial language rather than true \emph{geometric} reasoning over space. Because LLMs operate on discrete tokens, they lack native support for continuous spatial representations, explicit geometric computation, and structured spatial operators. To address this limitation, we introduce the \emph{Spatial Language Model (SLM)}, the first multimodal LLM that treats location information as a first-class modality and enables geometric spatial reasoning within the model's inference process. SLM directly operates on learned spatial representations rather than textual descriptions of spatial relations. To support effective training, we construct a \emph{Spatial Instruction Dataset} that aligns spatial representations, atomic geometric operations, and natural language instructions. We further propose a new benchmark named \emph{SpatialEval}, which is designed to evaluate spatial reasoning across attributes, distance, topology, and relative-position tasks. Extensive experiments show that SLM significantly outperforms existing LLM-based approaches that rely on symbolic reasoning via prompt engineering or textual abstraction, demonstrating the benefits of integrating geometric spatial representations for robust spatial reasoning. Our instruction dataset, evaluation benchmark, model training codes, and models' checkpoints can be found at: \hyperlink{https://github.com/chuchen2017/SLM}{https://github.com/chuchen2017/SLM}.

Chen Chu, Bita Azarijoo, Li Xiong +2

University of Southern California Los Angeles, CA, USA · Emory University Atlanta, GA, USA · Novateur Research Solutions Ashburn, Virginia, USA

Spatial Reasoning Benchmarks Multimodal Large Language Models Geometric Representation Learning Spatial Reasoning

Jun 2, 2026q-bio.QM

EpiFormer: Learning Antigen-Antibody Interactions for Epitope Prediction via Geometric Deep Learning

Antibodies neutralize foreign antigens by binding to specific surface regions called epitopes. Computational epitope prediction is critical for understanding immune recognition and guiding antibody engineering. However, existing methods face three fundamental challenges: antibody-aware models encode each chain independently and combine them only at a late stage, failing to capture co-dependent structural features that define binding interfaces, whereas severe class imbalance and scarcity of known antibody-antigen complexes render standard training objectives ineffective. We propose EpiFormer, a general encoder-decoder framework that addresses these challenges jointly. Our key design principle is interleaved cross-attention within GNN encoding layers, enabling bidirectional antigen-antibody information flow throughout representation learning rather than only at the output. This early-fusion principle is backbone-agnostic, providing consistent gains across GNN architectures from simple GCNs to equivariant models. We further show that sparsity-aware objectives are effective when paired with early-fusion architectures for the epitope prediction task. EpiFormer improves over the previous best method by over 40% in F1 score on standard benchmarks, demonstrating generalizability and cross-dataset transferability. Notably, EpiFormer discovers known biological principles as emergent behaviors of end-to-end training, where the learned cross-attention gates favor antigen-to-antibody information flow, consistent with the asymmetric roles of the two chains at the binding interface, and the model's preference for geometric over evolutionary features aligns with the established finding that epitope residues are not evolutionarily conserved. The source code is available at: https://github.com/mansoor181/epiformer.git

Mansoor Ahmed, Huirong Chai, Haoxin Wang +2

Georgia State University, Atlanta, GA, USA · Georgia Institute of Technology, Atlanta, GA, USA

Graph Representation Learning Cross-Attention Geometric Representation Learning

Jun 2, 2026cs.LG

Are Common Substructures Transferable? Riemannian Graph Foundation Model with Neural Vector Bundles

Foundation models have sparked a revolution via a pretraining-adaptation paradigm, with recent efforts extending this success to graphs. Unlike other modalities, graphs contain rich structural patterns, yet their structural transferability remains poorly understood. Prior studies consider common substructures in the discrete realm, and we are motivated by a fundamental question: Are common substructures transferable? The underlying theory is largely underexplored. In this work, we shift toward learning transferable structures through the lens of functional behavior. Theoretically, we connect transferable substructures to intrinsic geometry of the representation space. However, characterizing such intrinsic geometry has rarely been touched. Grounded in Riemannian geometry, we develop a graph intrinsic geometry learning framework called Neural Vector Bundle, which enables parsing intrinsic geometry with local coordinates. Building on this, we design GAUGE, a pretrainable neural architecture that constructs the vector bundle, flattening geometrically compatible local coordinates, and a new Dirichlet loss, which also measures the transfer effort. We empirically validate its superior expressiveness in challenging tasks including zero-shot link prediction and graph isomorphism.

Li Sun, Zhenhao Huang, Yiding Wang +3

Beijing University of Posts and Telecommunications, Beijing, China · North China Electric Power University, Beijing, China · University of Cambridge, Cambridge, UK +1

GNN Expressivity Geometric Representation Learning Transfer Learning Graph Foundation Models Riemannian Geometry

Jun 2, 2026cs.RO

GeoAlign: Beyond Semantics with State-Guided Spatial Alignment in VLA Models

Current Vision-Language-Action (VLA) models often optimize for semantic grounding, whereas executable manipulation requires geometry-aware spatial alignment. We introduce GeoAlign, a state-guided spatial alignment architecture for VLA policy learning. Offline robot-domain RGB-D supervision post-trains an RGB geometry branch to produce Geometry-Enhanced Post-Trained (GEP) features; the depth head is then discarded, so policy training and rollout use RGB, language, and proprioceptive state. State-generated queries attend to the GEP grid to produce eight compact geometry tokens for action prediction. GeoAlign achieves 99.0% on LIBERO, 85.3% across three SimplerEnv-Fractal task families, and 78.8% on eight real-world ALOHA tasks, with ablations supporting the combined recipe of geometry post-training and state-guided querying on Isaac-GR00T N1.6-3B. Project website: https://chenyizhi123.github.io/geoalign-project/.

Yizhi Chen, Zhanxiang Cao, Xinyi Peng +14

Tongji University · Shanghai Innovation Institute · Shanghai Jiao Tong University +5

Visuomotor Policy Learning Geometric Representation Learning Vision-Language-Action Models

Jun 1, 2026cs.CV

From Extrinsic to Intrinsic: Geodesic-Guided Representation Learning for 3D Geometric Data

Geometric analysis fundamentally distinguishes between \textit{extrinsic} and \textit{intrinsic} perspectives. The dominant paradigm in current 3D representation learning relies on either extrinsic spatial structures or high-level semantics, struggling to capture the essence of shape identity and underlying manifold topology. To bridge this gap, we introduce a novel 3D representation learning paradigm, namely \textbf{PRISM}, for \textbf{P}re-training, which learns isometric embeddings by \textbf{R}ecovering the \textbf{I}ntrinsic \textbf{S}urface geodesic \textbf{M}etric. PRISM incorporates a topology-enforcing objective that explicitly constrains the structure of latent space, alongside a specialized two-stage training recipe mitigating sample imbalance inherent in the distribution of geodesic distances. Experiments demonstrate that our approach shows satisfactory accuracy, robustness, and high efficiency in geodesic distance prediction and achieves superior performance across diverse downstream tasks, including shape recognition, surface parameterization, and non-rigid correspondence. The code will be publicly available at https://github.com/AidenZhao/PRISM.

Yuming Zhao, Junhui Hou, Qijian Zhang +2

Department of Computer Science, City University of Hong Kong, Hong Kong, China · Bambu Lab, Shenzhen, China · Meshy AI, California, USA +1

Geometric Representation Learning Manifold Learning 3D Representation Learning

Jun 1, 2026cs.LG

Planar Symmetric Pattern Generation

Generating objects with specific symmetries is essential in various real-world scenarios. However, adapting existing 2D continuous representations to enforce planar group symmetry remains a challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a symmetrization framework for arbitrary planar groups. Our method transforms any 2D continuous representation into a symmetric one while preserving continuity. We provide the mathematical formulation of this representation, demonstrate its approximation capability for symmetric functions, and detail the construction methodology. We validate our approach through three visual design tasks (pattern design, paper-cutting design and stylized topology design) and one material design task. Experiments confirm that our representation enables effective symmetry control and demonstrate its broader applicability.

Ning Lin, Luxi Chen, Huaguan Chen +4

Gaoling School of Artificial Intelligence, Renmin University of China.

Image Generation Geometric Representation Learning Procedural Content Generation

Jun 1, 2026cs.CV

FlatVPR: Plug-and-play Geo-linear Residual Adapter for Geometric Rectification of Foundation Model Feature Manifolds

This paper proposes ``FlatVPR,'' a novel geometric rectification paradigm that effectively bridges the trade-off between map lightweightness and localization accuracy in visual place recognition (VPR) by enforcing a feature manifold structure where any descriptor between two adjacent anchors

\mathbf{z}_A

and

\mathbf{z}_B

can be accurately reconstructed via linear interpolation

\hat{\mathbf{z}}_{pseudo} = (1-t)\mathbf{z}_A + t\mathbf{z}_B

, where

t \in [0,1]

denotes the relative position. While state-of-the-art foundation models such as DINOv2-ViT-S/14 provide robust semantic features, their latent manifolds exhibit prominent curvature, projecting uniform linear motion in physical space onto highly non-linear trajectories in the feature space, which hinders reliable reconstruction under sparse anchor conditions. To enable the aforementioned interpolation-based reconstruction, we introduce a residual transformation

\hat{\mathbf{z}} = \mathbf{z} + \text{Res}(\mathbf{z})

to the raw foundation features

\mathbf{z}

, where

\text{Res}(\cdot)

represents a learnable adapter. Our method explicitly suppresses manifold curvature using a mathematically grounded Pullback Flatness Loss that minimizes the deviation of intermediate features from the linear segment connecting adjacent anchors, thereby minimizing the intrinsic curvature of the manifold. Through this spatial flattening, map construction is formulated within an Expectation-Maximization (EM) framework, decoupled into a continuous M-step for manifold adaptation and a conceptual E-step for optimal anchor selection guidelines. Experiments on the NCLT dataset demonstrate that the application of our adapter leads to significant performance improvements even under extremely sparse anchor conditions with 100m intervals and extreme seasonal changes.

Rai Hisada, Kanji Tanaka

Fundamental Engineering for Knowledge-Based Society, Graduate School of Engineering, University of Fukui, Japan.

Visual Place Recognition Vision Foundation Model Adaptation Geometric Representation Learning

Jun 1, 2026cs.CV

$\text{VG}^2$ GT: Voxel-Gaussian Splatting Visual Geometry Grounded Transformer

Gaussian splatting has shown strong potential for 3D reconstruction and novel view synthesis. However, most existing methods require accurate camera parameters and per-scene optimization, while feed-forward methods with pixel-aligned Gaussian primitives often suffer from artifacts and non-uniform primitives. In this paper, we propose

\text{VG}^2

GT, a Voxel-Gaussian Splatting Visual Geometry-Grounded Transformer.

\text{VG}^2

GT leverages a frozen pretrained visual foundation model (VFM), incorporates a multi-scale differentiable voxel module to enhance geometric understanding, and directly splits and regresses Gaussian primitive parameters from voxel features. During training, depth maps are supervised through stochastic solid volume rendering, enabling geometrically accurate Gaussian scene reconstruction while keeping the visual foundation model fully frozen. This design enables

\text{VG}^2

GT to be seamlessly plugged into any patch-feature-based VFM, while substantially reducing the required training cost.

\text{VG}^2

GT outperforms current state-of-the-art methods on widely used DTU, Replica, TAT, and ScanNet datasets.

Yibin Zhao, Yihan Pan, Jun Nan +3

East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China · Shanghai Open University, 288 Guoshun Road, 200433, Shanghai, China · Shanghai Xiaoyuan Innovation Center, 599 Xingmei Road, 200237, Shanghai, China

3D Gaussian Splatting Novel View Synthesis Geometric Representation Learning Feed-Forward 3D Reconstruction

May 31, 2026cs.LG

UR-JEPA: Uniform Rectifiability as a Regularizer for Joint-Embedding Predictive Architectures

A central difficulty in training Joint-Embedding Predictive Architectures (JEPAs) is preventing representation collapse. LeJEPA addresses this by enforcing an isotropic Gaussian target on the embeddings via Sketched Isotropic Gaussian Regularization (SIGReg). This target is in tension with the manifold hypothesis, which expects embeddings to concentrate on a low-dimensional subset of the ambient space. We propose \emph{UR-JEPA}, which targets a uniformly

n

-rectifiable measure of local tangent dimension

n

at small scales, realized through a Gaussian-kernel smoothed Carleson-type square function

\mathcal{L}^{\text{CGLT}}

, with a complementary Jones

β

-number formulation. On Inet10, UR-JEPA(

\mathcal{L}^{\text{CGLT}}

) attains

0.9141 \pm 0.0014

for a

+0.83

,pp gain over LeJEPA(

\mathcal{L}^{\text{SIGReg}}

) with

\sim 30\%

lower seed standard deviation; on matched-recipe Galaxy10~SDSS, a single-seed ImageNet-

100

run, and a

3

-seed EuroSAT remote-sensing run, the two methods lie in the same peak-accuracy band at convergence, with UR-JEPA retaining its lower-seed-variance signature. On EuroSAT the in-domain pair is competitive at

96.0

to

96.1\%

with large remote-sensing foundation-model transfer at a

25\times

smaller backbone. The distinction is geometric: direct visualization of the projector output distribution shows that on all four datasets UR--JEPA(

\mathcal{L}^{\text{CGLT}}

) produces a global PCA spectrum with a

4

to

5

order-of-magnitude drop at index

\sim 20

to

25

out of

D = 32

, while LeJEPA's spectrum is near-flat (top-to-bottom ratio at most

3.6

). Per-dimension marginals are simultaneously near-Gaussian for both methods (mean Shapiro-Wilk

W \in [0.992, 0.996]

) as a Diaconis-Freedman consequence. At matched accuracy the two regularizers therefore yield structurally distinct projected representations.

Triet M. Le

Joint-Embedding Predictive Architecture Geometric Representation Learning Self-Supervised Visual Representation Learning

May 30, 2026cs.CV

hZACH-ViT: Curved Latent Geometry for Compact Vision Transformers in Low-Data Medical Imaging

Compact Vision Transformers are attractive for medical imaging in low-data and resource-constrained settings, but most existing variants assume that Euclidean latent geometry is sufficient for organizing image representations. We introduce hZACH-ViT, a family of curved-geometry extensions of ZACH-ViT, a compact zero-token Vision Transformer that removes positional embeddings and the class token and relies on global average pooling over patch representations. To isolate the role of geometry, we preserve the verified ZACH-ViT backbone and modify only the final representation space and prototype-based classifier head, enabling a controlled comparison between Euclidean, hyperbolic, and spherical latent geometries. We evaluate Poincaré, Klein, and spherical hZACH-ViT heads on seven MedMNIST datasets under an identical few-shot protocol with 50 samples per class and five random seeds. The completed benchmark contains 770 training runs spanning seven datasets, three non-Euclidean geometries, seven curvature magnitudes, and a Euclidean baseline. Across all seven datasets, the best non-Euclidean hZACH-ViT configuration improves over Euclidean ZACH-ViT, with an average gain of +0.021 in the dataset-specific primary metric and the largest improvement on OCTMNIST (+0.055 MacroF1). Fixed low-curvature configurations retain positive gains on the majority of datasets, and low curvature values (c = 0.1 or 0.2) account for six of the seven dataset-level winners. Rather than identifying a universally optimal manifold, our results establish geometry and curvature as dataset-dependent model-selection variables, with fixed low-curvature analyses confirming that gains persist beyond exhaustive per-dataset tuning.

Athanasios Angelakis

BioML Lab, Research Institute CODE, UniBw, Munich, Germany · Department of Epidemiology and Data Science, Amsterdam UMC, Amsterdam, Netherlands

Geometric Representation Learning Efficient ViTs Medical Image Representation Learning

May 29, 2026cs.LG

Learning Hyperspherical Time-Frequency Representations for Time-Series Out-of-Distribution Detection

Out-of-distribution (OOD) detection for time-series data remains comparatively underexplored compared to vision and language, with a limited principled understanding of how supervised time-series representations can be leveraged for reliable detection under distributional shifts. This work formulates time-series OOD detection as representation learning with hyperspherical embeddings, where class-conditional structure is induced by a von Mises-Fisher (vMF) likelihood-based objective on the unit sphere. The learned representation combines time- and frequency-domain views of the input signal via domain-specific encoders, integrating them into a joint embedding space for OOD detection. Detection uses distance-based scores over the learned embeddings, including k-nearest neighbors (k-NN) and Mahalanobis scores. We evaluate the approach at scale on the complete UCR and UEA time-series archives under a cross-dataset protocol. Empirical results show consistent improvements under both k-NN and Mahalanobis scoring over strong contrastive learning and post-hoc baselines in the same setting. Code is available at https://github.com/tiiuae/hypertf-time-series-ood.

Willian T. Lunardi, Samridha Shrestha, Martin Andreoni

Technology Innovation Institute, Abu Dhabi, United Arab Emirates · Khalifa University, Abu Dhabi, United Arab Emirates

Hyperspherical Representation Learning Distribution Shift Detection Geometric Representation Learning OOD Detection Time Series Representation Learning

May 29, 2026cs.CV

Equivariant Latent Alignment via Flow Matching under Group Symmetries

Geometry-aware generative models and novel view synthesis approaches have shown strong potential in visual fidelity and consistency. In parallel, equivariant representation learning has emerged as a powerful framework for constructing latent spaces where analytically known group transformations could act directly, capturing geometric structure in data and enhancing both interpretability and generalization in novel view synthesis. However, we identify that existing approaches often suffer from latent misalignment, a discrepancy between the intended group action and the actually required transformations in the latent space. Consequently, the learned latents often fail to consistently preserve the equivariant relations imposed by the underlying group symmetry. To address this, we propose Residual Latent Flow, a flow-based framework that corrects the misaligned latents, thereby improving compliance with the underlying equivariance relation. Our comprehensive experiments show that our method significantly reduces latent misalignment and improves novel view synthesis quality, under rotation groups SO(n).

Sunghyun Kim, Jaehoon Hahm, Jeongwoo Shin +1

1Seoul National University, Seoul, Korea · University of Illinois Urbana-Champaign, Illinois, USA

Flow Matching Equivariant Neural Networks Novel View Synthesis Geometric Representation Learning

May 28, 2026cs.CV

Beyond 3D VQAs: Injecting 3D Spatial Priors into Vision-Language Models for Enhanced Geometric Reasoning

Vision-Language Models (VLMs) often struggle with robust 3D spatial reasoning. Prevailing methods that rely on fine-tuning with 3D visual question-answering (VQA) datasets may overfit dataset-specific biases, while integrating specialized 3D visual encoders is often inflexible and cumbersome. In this paper, we argue that genuine spatial understanding should emerge from learning fundamental geometric priors, not only from high-level VQA supervision. We propose GASP (Geometric-Aware Spatial Priors), a framework that injects these priors directly into the LLM's transformer layers. GASP employs a small correspondence head, applied as a deep supervision signal across all layers, and is trained with a dual objective leveraging ground-truth geometry from large-scale video scenes: a contrastive loss on ground-truth point correspondences enforces 2D view-invariance, while a depth consistency supervision resolves 3D geometric ambiguities. Our analysis first provides a diagnostic showing that standard VLMs' internal correspondence matching accuracy is very low (often below 5%). We then demonstrate that our training substantially improves this behavior, boosting peak layer-wise correspondence to over 70% and maintaining over 85% temporal robustness while baselines remain below 5%. These internal improvements translate to significant gains on downstream spatial benchmarks including +18.2% on All-Angles Bench and +29.0% on VSI-Bench, all without training on any 3D VQA data. Our findings indicate that learning from fundamental geometric priors is a promising and generalizable pathway towards VLMs with more reliable 3D spatial reasoning.

Chun-Hsiao Yeh, Shengyi Qian, Manchen Wang +3

FAIR at Meta · UC Berkeley · HKU

Vision-Language Models 3D Spatial Reasoning Geometric Representation Learning 3D VLMs

May 28, 2026cs.LG

A Geometric View of SRC: Learning Representations for Stable Residual Inference

Reconstruction-based inference assigns a class by comparing class-wise reconstruction residuals; Sparse Representation Classification (SRC) is a canonical instance whose reliability depends on the geometry of the learned representation. We adopt a strict training-inference separation: SRC is used only as a fixed test-time rule and is never differentiated, unrolled, or optimized during training. In a span-level idealization based on class-conditional spans and their associated projection residuals, we formalize residual-ordering stability through a residual margin and characterize geometric obstructions -- span overlap, dominance, and near-overlap via small principal angles -- that can collapse this margin in worst-case directions. This span-level theory is primary: it specifies when the idealized residual family is well-separated, and it provides a conditional solver-level interpretation for practical residual approximations (e.g., OMP) insofar as they remain close to the span-level residual ordering. Under explicit coverage and separation assumptions, we derive a quantitative lower bound on the (idealized) residual margin. Guided by these targets, we propose geometry-shaping objectives that promote masked within-class self-expressiveness, discourage cross-class reconstruction pathways and inter-class span alignment, and prevent collapse -- without invoking SRC residuals or predictions during training. Experiments on images (COIL-100), text (TREC), and EEG connectivity evaluate all representations under identical fixed SRC/OMP inference and report residual margins and geometric diagnostics; cross-entropy is included only as a reference geometry under the same evaluation protocol.

Vangelis P. Oikonomou

arXiv:2605.29673v1

Representation Learning Geometric Representation Learning

May 28, 2026cs.CV

Learning Representations from 3D Gaussian Splats

3D Gaussian Splatting (3DGS) is a recent approach for scene rendering. Although primarily designed for view synthesis, its potential for scene understanding tasks remains underexplored. In this work, we conduct a comparative evaluation of various geometric deep learning architectures for the classification of 3D scenes represented using Gaussian Splatting. We benchmark point-based and graph-based models across both traditional point cloud datasets and dedicated Gaussian Splatting datasets. Scenes are embedded into latent representations, which are evaluated through end-to-end classification, linear probing, and clustering analysis. Our study provides insight into the suitability of different geometry-aware architectures and input feature configurations for learning effective 3D Gaussian Splat representations. The results highlight consistent differences between architectural families and reveal the impact of Gaussian-specific attributes on the quality of representation.

Julia Farganus, Krzysztof Żurawicki, Arkadiusz Gaweł +2

Department of Artificial Intelligence, Wrocław University of Science and Technology, Wrocław, Poland

3D Gaussian Splatting Geometric Representation Learning 3D Representation Learning 3D Scene Understanding