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

May 27, 2026cs.CV

General Covariant Action Modeling: Constructing Generalized Manifolds via Spatio-Temporal Decoupling

Achieving robust generalization from limited data is a central challenge in embodied intelligence. Prevailing methods fail by regressing absolute coordinates, which violates the principle of general covariance. Fundamentally, this conflates the intrinsic task geometry with rigid execution patterns, binding policies to specific motion styles and fixed speeds. To resolve this, we propose the Generalized Action Manifold (GAM) framework that enforces general covariance through structural disentanglement. Specifically, GAM realizes the manifold by enforcing invariance across two orthogonal dimensions: (1) Temporal Invariance, utilizing an Arc-Length Parameterizer to orthogonalize the spatial path geometry from temporal dynamics, ensuring robustness to velocity variations; (2) Geometric Invariance, where a Schema-Affine-Factorization mechanism maps trajectories to canonical ``world lines'' in a pose-normalized coordinate frame. This distinguishes invariant geometric schemas from affine modulations, ensuring spatial generalizability. By integrating GAM within a structured Vision-Language-Action (VLA) architecture, we enable sparse demonstrations to densely populate a continuous, valid action manifold. Empirical results demonstrate that GAM enables superior transfer and robustness capabilities, outperforming geometry-agnostic baselines.

Huaihai Lyu, Chaofan Chen, Mingyu Cao +2

1MAIS, Institute of Automation, Chinese Academy of Sciences · 2Beijing Academy of Artificial Intelligence.

Geometric Representation Learning Vision-Language-Action Models Robot Policy Generalization Invariant Representation Learning

May 26, 2026cs.LG

SPHERE-JEPA: Spherical Prediction with Homogeneous Embeddings

A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both k-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for k-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic k-NN neighborhoods, severely biasing the estimator. To correct this, we introduce SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{é}r-Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6%, while consistently matching or outperforming LeJEPA on standard benchmarks-including a +1.8% linear probing gain on ImageNet-1K (ViT-B/14).

L{é}o Nicollier, Max Dunitz, Marc Pic +3

CB,ATT · ATT · CB,IFUMI +1

Self-Supervised Learning Joint-Embedding Predictive Architecture Hyperspherical Representation Learning Geometric Representation Learning Manifold Learning

May 25, 2026cs.LG

Metric-Aware PCA as a Linear Instance of Geometric Deep Learning

Geometric deep learning organises neural architectures around the symmetries of their data domain, with the choice of symmetry group serving as a geometric prior that determines what representations can be learned. Metric-Aware Principal Component Analysis (MAPCA) parameterises principal component analysis by a positive-definite metric matrix, with a canonical subfamily interpolating between standard PCA and output whitening and a diagonal-metric point recovering Invariant PCA (IPCA). This paper positions MAPCA within the geometric deep learning framework. The metric is read as the geometric prior; the orthogonal group preserving it is the symmetry group it induces; MAPCA solutions are equivariant under this group with the resulting spectrum invariant; and MAPCA's defining constraint is the linear analogue of the Schur-type weight constraints used in equivariant networks. Across six axes - domain, symmetry group, equivariance, invariance, architectural primitive, and geometric prior - we construct a precise dictionary between MAPCA and geometric deep learning. The technical anchor is a uniqueness theorem characterising IPCA as the unique linear data-derived metric in the MAPCA family that is equivariant under arbitrary diagonal rescaling and projects onto the fixed-point set of the action, equivalent under normalisation to the variance-maximisation criterion in its precise form. The paper closes with three bridges: kernel PCA as the nonlinear extension, spectral graph methods as MAPCA on graphs, and a deep MAPCA construction extending the positioning into deep equivariant networks

Michael Leznik

Principal Component Analysis Geometric Representation Learning

May 23, 2026cs.CV

Understanding the Impact of Geometric Foundation Models on Vision-Language-Action Models

Recent work explores new opportunities at the intersection of vision-language-action models (VLAs) and geometric foundation models (GFMs) for 3D reconstruction, such as VGGT. While the resulting geometric VLAs often show improved performance, it remains unclear (i) if modern VLAs already have sufficient geometric understanding to start with, (ii) what is the best architecture to inject geometric understanding into a VLA, and (iii) what is the effect of other design choices that affect geometric VLAs. In this paper we provide a rigorous experimental analysis to shed light on these questions, for a specific choice of VLA (GR00T-N1.5) and GFM (VGGT). Our first contribution is to formalize prior work's intuition that current VLAs lack geometric understanding, by providing a rigorous analysis based on linear probing. The analysis quantifies, for the first time, the "geometric gap" between VLAs and GFMs. Our second contribution is to identify and compare different strategies to bridge GFMs with VLAs. We implement three different architectures, which differ in the way they inject geometry in the VLA, while keeping low-level implementation details as similar as possible, to ensure a fair comparison. Finally, we analyze the impact of non-architectural choices (e.g., training data, number of cameras, reconstruction quality) on the performance of the geometric VLAs.

Yurou Yang, Muyuan Lin, Roberto Martin-Martin +4

Amazon Personal Robotics Group · University of Texas at Austin · Massachusetts Institute of Technology

3D Reconstruction Geometric Representation Learning Vision-Language-Action Models VLMs for Robotics

May 23, 2026cs.LG

Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds

The eigendecomposition of the Laplace--Beltrami Operator (LBO) is fundamental to geometric analysis, yet computing its low-frequency eigenmodes remains a significant bottleneck due to the high cost of iterative solvers on large-scale data. To amortize this cost, we introduce the Neural Eigenspace Operator (NEO), a feed-forward framework designed to predict the spectrum directly from point clouds. Crucially, NEO circumvents the ill-posed nature of standard eigenvector regression, which suffers from intrinsic sign flips and rotation ambiguities, by learning the stable, invariant low-frequency subspace instead. Specifically, the network predicts a redundant set of basis functions whose span robustly covers the target eigenspace, allowing for the recovery of accurate eigenpairs via a lightweight Rayleigh--Ritz refinement. To handle irregular sampling, we propose a mass-aware neural operator that incorporates per-point area weights into attention-based aggregation, improving robustness to non-uniform densities and enabling zero-shot generalization across resolutions. Our approach achieves near-linear runtime scaling and substantial wall-clock speedups over iterative solvers at comparable accuracy, and exhibits strong zero-shot transfer to high-resolution point clouds. The resulting eigenpairs support standard spectral geometry tasks, while the raw basis functions provide effective point-wise features for downstream learning. Code: https://github.com/Adversarr/NEO.

Zherui Yang, Tao Du, Ligang Liu

University of Science and Technology of China, China · Tsinghua University, China · Shanghai Qi Zhi Institute, China +1

Point Cloud Learning Geometric Representation Learning Manifold Learning Spectral Methods Neural Operators

May 22, 2026cs.CV

GIBLy: Improving 3D Semantic Segmentation through an Architecture-Agnostic Lightweight Geometric Inductive Bias Layer

In 3D scene understanding, deep learning models rely on large models and extensive training to capture basic geometric structures that are present in the 3D data. However, existing methods lack explicit mechanisms to incorporate geometric information, such as learnable primitive shapes, often necessitating large models and more training data which in turn increases cost and can limit generalization. We introduce GIBLy, a lightweight geometric inductive bias layer that integrates learnable geometric priors into 3D segmentation pipelines. GIBLy enhances existing architectures -- whether MLP-based, convolution-based, or transformer-based -- by providing features aligned with simple geometric shapes (and thus human-interpretable) that improve segmentation performance with minimal computational overhead. We validate our approach across multiple 3D semantic segmentation benchmarks, demonstrating consistent performance gains, including up to +11.5% mIoU on TS40K with PTV3, while adding only 58K extra parameters. Our results highlight the benefit of explicitly encoding geometric structure to support accurate and efficient 3D scene understanding, with a lightweight add-on layer

Diogo Lavado, Alessandra Micheletti, Clàudia Soares

NOVA School of Science and Technology, Portugal · Universit`a degli Studi di Milano, Italy

3D Semantic Segmentation Geometric Representation Learning 3D Scene Understanding

May 22, 2026cs.LG

VACE: Learning Geometrically Structured Representations for Time Series Anomaly Detection

Anomaly detection in multivariate time series is a critical task across a wide range of real-world applications, where abnormal behaviour is rare, labels are unavailable, and the cost of a miss is high. The central challenge is learning a characterisation of normality precise enough to flag deviations. Representation self-supervised learning, typically through contrastive approaches, addresses this by embedding temporal patches into a latent space where normality occupies a well-defined region, with anomalies detected by geometric deviation. However, contrastive approaches shape this space indirectly through pair-sampling heuristics, providing no explicit control over the geometric structure that distance-based scoring requires. This means how tightly normal representations are grouped, and whether distances are directionally meaningful. We present VACE (Velocity-Aligned Channel Embeddings), a self-supervised anomaly detection method that represents normality as a compact, directionally coherent region in the embedding space. To this end, VACE trains a channel-aware encoder through a velocity-consistency objective, with no negatives and no synthetic anomalies, so that normal trajectories are locally smooth and aligned. At test time, a Mahalanobis positional score and a velocity-bank directional score are combined multiplicatively, flagging points that are simultaneously off-distribution and dynamically atypical. Despite its simplicity, VACE achieves state-of-the-art performance on TSB-AD-M under rigorous evaluation, significantly outperforming more complex methods trained on substantially larger budgets.

Alberto D. Cencillo, Leonardo Concepción, Isaac Triguero +1

1Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI) · Department of Computer Science and Artificial Intelligence (DECSAI), University of Granada, Granada, 18071, Spain

Self-Supervised Time Series Representation Learning Geometric Representation Learning Unsupervised Anomaly Detection Time-Series Anomaly Detection

May 22, 2026cs.LG

Commutator-Induced Uncertainty in VAEs

Variational autoencoders (VAEs) often struggle to represent non-commutative structure in learned latent spaces. Symmetry-aware VAEs commonly address this issue by enforcing commutativity through algebraic regularization, which is appropriate for commutative transformation groups but can suppress meaningful non-commutative structure when it is intrinsic to the data. We argue that non-commutativity should instead be explicitly diagnosed and reflected in reconstruction behavior. We introduce a Lie Group VAE framework that combines geometric and algebraic perspectives on uncertainty while separating discrete generative factors from continuous geometric transformations. In a first phase, the model is trained without structural constraints while algebraic non-commutativity is measured through finite Baker-Campbell-Hausdorff deviations and decoder order sensitivity is measured through reconstruction order-swap tests. These diagnostics reveal a scale mismatch between latent non-commutativity and reconstruction behavior under unconstrained training. In a second phase, we introduce a deformation-stability constraint with a data-driven calibration constant that aligns decoder sensitivity with algebraic non-commutativity. We evaluate the framework on dSprites, 3DShapes, 3DCars, and CelebA against generic and symmetry-aware baselines, including beta-VAE, CLG-VAE, and CFASL. Across synthetic benchmarks, the method improves reconstruction quality and yields decoder-level behavior more consistent with latent non-commutative structure. Qualitative analyses show clearer order-dependent latent compositions and more stable reconstructions. On CelebA, the model yields more faithful reconstructions and factor-specific latent traversals than CFASL, while also exhibiting meaningful order-dependent interactions between learned latent directions.

Tahereh Dehdarirad, Michael Felsberg, Gabriel Eilertsen +1

Computer Vision and Learning Systems (CVL), Linköping University, Sweden · Department of Science and Technology, Linköping University, Sweden

Variational Autoencoders Disentangled Representation Learning Representation Learning Geometric Representation Learning

May 22, 2026cs.LG

Emergent Semantic Representations in World Models through Physical Interaction without Linguistic Supervision

What does a world model learn from physical exploration, without any linguistic supervision? We argue the answer is organized by a single principle: the geometric structure of the physical world. Training a VAE-based world model on random embodied exploration, we find that its latent space develops spatial semantic structure that mirrors physical geometry -- direction accuracy 0.677+-0.029 versus 0.547 for a randomly initialized encoder, and position RSA 0.192+-0.047 versus 0.029 for random encoders (6.6x improvement), showing that training induces genuine structural organization beyond CNN inductive bias. Across 20 temporal checkpoints, prediction performance and semantic alignment co-improve (Spearman r=-0.61, p=0.004), consistent with the shared-driver account. We confirm this through a double knockout: standard KL regularization (beta=0.1) forces the encoder away from geometric structure, and both prediction performance and semantic alignment collapse simultaneously to near-chance by step 50,000 -- exactly as the shared-driver account predicts. Reducing beta to 0.001 restores geometric access and recovers both capabilities together. These findings establish physical world geometry as the organizing principle of world model representations, with direct implications for the design of semantically grounded embodied agents.

Jiayi Fang

1Shanghai University of Finance and Economics

Representation Learning World Model Learning Geometric Representation Learning Latent World Models Embodied World Models

May 21, 2026cs.LG

Uncovering the Latent Potential of Deep Intermediate Representations

Foundational Models pretrained on huge amount of data learn representations that evolve across depth, forming a hierarchy of embeddings with distinct semantic content and geometric structure. Contrary to the widespread practice of using only the final layer or shallow mixtures, we show that task-relevant information is distributed non-monotonically across layers and cannot be recovered by naïve aggregation. Through a geometric and empirical study across multiple modalities, we show that effective transfer depends on identifying which layers encode task-discriminative structure and how their embeddings are geometrically organized. We introduce Layer-wise Optimal Embedding Selection (LOES), a constructive spectral method that identifies task-discriminative subspaces by minimizing residual error under orthogonality and isotropy constraints. To align fine-tuning with this selection principle, we further propose Geometric Regularization Loss (GeoReg), which enforces a simplicial structure on class manifolds and stabilizes representation geometry during fine-tuning. Across a wide range of architectures, depths, modalities, and data regimes, LOES consistently outperforms standard baselines, with gains that grow as model depth increases. Beyond accuracy, our method reveals how semantic factors are distributed across layers, thereby enabling cross-lingual and cross-modal interpretability analyses. Together, our results provide strong evidence that layerwise embedding geometry is not incidental but central to how deep models represent and transfer knowledge.

Arnesh Batra, Arush Gumber, Aniket Khandelwal +2

SBILab, Indraprastha Institute of Information Technology Delhi, Delhi, India.

Neural Representation Geometry Hierarchical Representation Learning Geometric Representation Learning Transfer Learning

May 21, 2026cs.LG

Disentanglement Beyond Generative Models with Riemannian ICA

There is a gap between the theoretical foundations of disentanglement and the practice of modern representation learning. Existing theoretical frameworks, particularly Independent Component Analysis (ICA) and its nonlinear variants, assume a generative model with statistically independent latent variables underlying the data so that disentanglement amounts to identifying the latents that could have generated the data. This generative framework is interpretable and theoretically justified, but its strong assumptions make it difficult to apply to modern representation learning. Modern pretrained encoders often learn features that exhibit disentangled properties without making generative assumptions, yet there is no general theory for interpreting these features as independent factors of variation. We take a step toward such a theory by introducing Riemannian ICA (RICA), which replaces ICA's global generative model with local geometric structure. RICA is founded on the observation that in ICA, the factors of variation underlying a data point can be understood through radial curves emanating from the point that map to axis-aligned lines in the latent space. We formalize this perspective using Riemannian geometry and introduce our theory in a way that is consistent with the existing generative approach. Our main contribution is the disentanglement tensor, which encodes a second-order notion of disentanglement that we call pointwise disentanglement. This tensor depends on the Hessian of the data log likelihood as well as the Ricci curvature induced by the model. In a controlled source recovery setting with known ground-truth sources, RICA recovers sources across several manifolds, while the success of ICA baselines depends on the coordinates used to represent the observations. Our work provides a theoretical basis for studying local disentanglement without assuming a global generative model.

Edmond Cunningham

University of Massachusetts Amherst

Disentangled Representation Learning Independent Component Analysis Geometric Representation Learning Riemannian Geometry

May 21, 2026cs.CV

GA-VLN: Geometry-Aware BEV Representation for Efficient Vision-Language Navigation

Despite significant progress in Vision-Language Navigation (VLN), existing approaches still rely on dense RGB videos that produce excessive patch tokens and lack explicit spatial structure, resulting in substantial computational overhead and limited spatial reasoning. To address these issues, we introduce the Geometry-Aware BEV (GA-BEV) - a compact, 3D-grounded feature representation that integrates both explicit and implicit geometric cues into multimodal large language model (MLLM) - based navigation systems. We construct BEV spatial maps from RGB-D inputs by projecting visual features into 3D space and aggregating them into an agent-centric layout that preserves geometric consistency while reducing token redundancy. To further enrich geometric understanding, we incorporate features from a pretrained 3D foundation model into the BEV space, injecting structural priors learned from large-scale 3D reconstruction tasks. Together, these complementary cues - explicit depth-based projection and implicit learned priors - yield compact yet spatially expressive representations that substantially improve navigation efficiency and performance. Experiments show that our method achieves state-of-the-art results using only navigation data, without DAgger augmentation or mixed VQA training, demonstrating the robustness and data efficiency of the proposed GA-VLN framework.

Jiahao Yang, Zihan Wang, Xiangyang Li +4

State Key Laboratory of AI Safety, Institute of Computing Technology, Chinese Academy of Sciences, Beijing · University of Chinese Academy of Sciences, Beijing · School of Computing, National University of Singapore +2

Efficient VLM Inference Geometric Representation Learning Bird's-Eye-View Representation Vision-Language Navigation 3D Representation Learning

May 20, 2026cs.CV

UniT: Unified Geometry Learning with Group Autoregressive Transformer

Recent feed-forward models have significantly advanced geometry perception for inferring dense 3D structure from sensor observations. However, its essential capabilities remain fragmented across multiple incompatible paradigms, including online perception, offline reconstruction, multi-modal integration, long-horizon scalability, and metric-scale estimation. We present UniT, a unified model built upon a novel Group Autoregressive Transformer, which reformulates these seemingly disparate capabilities within a single framework. The key idea is to treat groups of sensor observations as the basic autoregressive units and predict the corresponding point maps in an anchor-free and scale-adaptive manner. More specifically, diverse view configurations in both online and offline settings are naturally unified within a single group autoregression process. By varying the group size, online mode operates over multiple autoregressive steps with single-frame groups, whereas offline mode aggregates a multi-frame group in a single forward pass. Meanwhile, a queue-style KV caching mechanism ensures bounded autoregressive memory over long horizons. This is enabled by reducing long-range dependencies on early frames through anchor-free relational modeling, thereby allowing outdated memory to be discarded on the fly. To improve metric-scale generalization across scenes, a scale-adaptive geometry loss is further introduced within this framework. It couples relative geometric constraints with a partial absolute scale term, implicitly regularizing global scale and inducing a progressive transition from scale-invariant geometry to metric-scale solutions. Together with a dedicated modal attention module for integrating auxiliary modalities, UniT achieves state-of-the-art performance in unified geometry perception, as validated on ten benchmarks spanning seven representative tasks.

Haotian Wang, Yusong Huang, Zhaonian Kuang +4

Intelligent Transportation Thrust of the Systems Hub, The Hong Kong University of Science and Technology (GZ), Guangzhou, P.R.China. · The National Key Laboratory of Human-Machine Hybrid Augmented Intelligence, Xi’an Jiaotong University, Xi’an, P.R.China. · Applied Science, Amazon.com, Inc., USA.

Transformer Depth Estimation Multi-View 3D Reconstruction Geometric Representation Learning Feed-Forward 3D Reconstruction

May 20, 2026cs.LG

Learning fMRI activations dictionaries across individual geometries via optimal transport

Dictionary learning is a powerful tool for creating interpretable representations. When applied to functional magnetic resonance imaging (fMRI) data, the resulting patterns of brain activity can be used for various downstream tasks, such as brain state classification or population-level analysis. However, a major challenge is the variability in brain geometry across individuals. This is usually addressed by projecting each individual brain geometry onto a common template, which removes subject-specific information. In this work, we introduce a novel approach to dictionary learning on fMRI data that explicitly accounts for this variability. We use the optimal transport-based Fused Gromov-Wasserstein (FGW) distance to compare graphs with different geometries and features. To address the challenge of computing multiple FGW distances for large graphs such as those arising from fMRI data, we rely on amortized optimization to learn a neural network that predicts an approximation of the optimal transport plans, which substantially reduces the computational cost. Additionally, we learn dictionary atoms that depend on the FGW trade-off parameter, which controls the balance between feature alignment and structural consistency. Numerical experiments on the HCP dataset demonstrate that the proposed approach captures different levels of geometric variability in the data and provides representations that preserve essential information.

Sonia Mazelet, Rémi Flamary, Bertrand Thirion

CMAP, Ecole Polytechnique · Palaiseau, France · Mind, Inria-Saclay

Learning to Optimize Geometric Representation Learning Dictionary Learning Neuroimaging Optimal Transport

May 20, 2026cs.CV

GAMR: Geometric-Aware Manifold Regularization with Virtual Outlier Synthesis for Learning with Noisy Labels

Deep neural networks (DNNs) experience significant performance degradation when processing noisy labels, primarily due to overfitting on mislabeled data. Current mainstream approaches attempt to mitigate this issue by passively filtering clean samples during training. However, simple sample filtering within feature spaces degraded by noise struggles to distinguish between challenging samples and noisy samples, creating a bottleneck for model performance. We highlight for the first time the fundamental importance of actively reshaping feature space geometry for learning from noisy data. We propose a novel Geometry-aware Manifold Regularization Paradigm whose core idea is to explicitly construct energy barriers between data manifolds by actively synthesizing virtual outlier samples. By imposing geometric constraints that promote intra-class compactness and inter-class separation, this approach enhances the discriminability between hard and noisy samples, leading to the learning of more robust representations. Our regularization mechanism exhibits high universality, with effectiveness independent of any prior assumptions about noise patterns. It can be integrated as a standalone mechanism into existing sample selection frameworks, providing stronger robustness against diverse noisy environments. Experiments demonstrate that our paradigm achieves performance surpassing current state-of-the-art (SOTA) methods on multiple benchmarks, including CIFAR-10, with particularly pronounced advantages under more challenging asymmetric noise conditions. Furthermore, this paradigm significantly enhances the model's capability in Out-of-Distribution (OOD) detection, ensuring superior reliability and safety for deployment in open-world scenarios.

Ningkang Peng, Jingyang Mao, Xiaoqian Peng +4

Nanjing Normal University Nanjing, China · Nanjing University of Chinese Medicine Nanjing, China

Geometric Representation Learning Manifold Learning Learning with Noisy Labels

May 19, 2026cs.LG

Beyond Isotropy in JEPAs: Hamiltonian Geometry and Symplectic Prediction

JEPAs often regularize one-view embeddings toward an isotropic Gaussian, implicitly baking Euclidean symmetry into the representation. We show that this is not merely a benign default. For a known structured downstream geometry

H\succ0

, the minimax and maximum-entropy covariance under a Hamiltonian energy budget is

(c/d)H^{-1}

, and Euclidean isotropy incurs a closed-form price of isotropy. More importantly, when the downstream geometry is unknown, no geometry-independent fixed marginal target is canonical: every fixed covariance shape can be maximally misaligned for some structured geometry. We further show that even oracle one-view marginals do not identify the JEPA view-to-view predictive coupling. These results suggest that the structural bias in JEPAs should enter the cross-view coupling rather than a fixed encoder marginal. We instantiate this principle with \textbf{HamJEPA}, which encodes each view as a phase-space state

(q,p)

and predicts view-to-view transitions with a learned Hamiltonian leapfrog map, while non-isotropic scale and spectral floors prevent collapse. In a deliberately headless token protocol, HamJEPA improves over SIGReg on CIFAR-100 by

+4.89

kNN@20 and

+3.52

linear-probe points at 30 epochs, and by

+6.45

kNN@20 and

+10.64

linear-probe points at 80 epochs, while a matched MLP predictor ablation shows that the symplectic coupling is the ingredient driving the neighborhood-geometry gain. On ImageNet-100, HamJEPA-

q

improves by

+4.82

kNN@20 and

+7.52

linear-probe points at 45 epochs.

Robert Jenkinson Alvarez

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

May 19, 2026cs.CV

GeoMamba: A Geometry-driven MambaVision Framework and Dataset for Fine-grained Optical-SAR Object Retrieval

Multi-source remote sensing enables complementary observation of ground objects, while cross-modal fine-grained object retrieval remains challenging, especially under unaligned optical and SAR conditions. Unlike conventional retrieval settings that rely on paired or spatially aligned samples, practical optical-SAR retrieval is affected by substantial modality discrepancy, speckle noise, and structural inconsistency, which limit robust cross-modal representation learning. To address this problem, we propose GeoMamba, a geometry-driven framework tailored for optical-SAR fine-grained retrieval. Specifically, GeoMamba introduces a Geometric Feature Injection (GFI) module that enhances cross-modal feature interaction and incorporates structural priors, thereby improving the robustness of SAR representations and promoting geometry-consistent feature learning. In addition, a Geometric Consistency Constraint (GCC) module, together with a Deep Supervision (DS) strategy, imposes hierarchical geometric constraints using classical operators, which helps preserve informative object structures during representation learning. We further construct a new dataset, FGOS-as, containing 11 aerospace and maritime categories for evaluating unaligned cross-modal fine-grained object retrieval in realistic remote sensing scenarios. Extensive experiments on FGOS-as demonstrate that GeoMamba outperforms existing methods, achieving 63.3% mAP and 77.0% Rank-1 accuracy in all-to-all retrieval setting.

Tiantong Fang, Xiuwei Wang, Jing Xiao +3

School of Artificial Intelligence, Wuhan University · School of Artificial Intelligence and Information Engineering, Zhejiang University of Science & Technology · Hangzhou Institute of Technology, Xidian University +1

Cross-Modal Representation Learning Geometric Representation Learning Cross-Modal Retrieval

May 18, 2026cs.LG

Riemannian Networks over Full-Rank Correlation Matrices

Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.

Ziheng Chen, Xiaojun Wu, Bernhard Schölkopf +1

Department of Information Engineering and Computer Science, University of Trento, Trento, Italy · Max Planck Institute for Intelligent Systems, Tübingen, Germany · School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, China

Geometric Representation Learning Manifold Learning Riemannian Geometry

May 18, 2026cs.AI

Learning to Solve Compositional Geometry Routing Problems

We study the Compositional Geometry Routing Problem (CGRP), a unified superclass of traditional routing problems that covers point-only, line-only, area-only, and arbitrary hybrid task geometries, providing a broad abstraction for real-world routing scenarios. Beyond standard point-based routing, CGRP with non-point tasks can be inherently asymmetric, tightly coupled travel routes with the intrinsic path, and enlarges the action space with numerous feasible yet often irrelevant options, thereby posing significant challenges for both representation learning and decision-making. To address these challenges, we propose DiCon, a differential attention-assisted solver with contrastive learning, as a plug-and-play framework that tackles the problem from two complementary angles. First, we introduce a differential attention mechanism that actively suppresses the probability mass on less competitive candidate actions. Second, we design a double-level contrastive learning objective to promote robust global instance representations and regularize geometry-aware task representations. Extensive experiments demonstrate that DiCon achieves strong performance, broad versatility, and superior generalization across diverse CGRP instances with different compositions.

Mingfeng Fan, Jianan Zhou, Jiaqi Cheng +3

National University of Singapore · Nanyang Technological University · Central South University

Geometric Representation Learning Neural Combinatorial Optimization

May 18, 2026cs.CV

SGSoft: Learning Fused Semantic-Geometric Features for 3D Shape Correspondence via Template-Guided Soft Signals

Learning dense correspondences across deformable 3D shapes remains a long-standing challenge due to structural variability, non-isometric deformation, and inconsistent topology. Existing methods typically trade off generalization, geometric fidelity, and efficiency. We address this by proposing SGSoft, a unified intrinsic pipeline that (i) constructs a geodesic correspondence field on a canonical template, (ii) learns multimodal dense descriptors guided by pretrained semantic priors with this geodesic correspondence field supervision, (iii) retrieves dense correspondences in a single feed-forward pass via nearest-neighbor search in descriptor space. This formulation enables stable and topology-invariant supervision under large pose variation, structural differences, and remeshing. SGSoft achieves state-of-the-art inter-category generalization while offering the best accuracy-efficiency trade-off among prior methods. It also achieves near real-time inference without pre-alignment, pairwise optimization, or post-refinement. Learned descriptors can be transferred effectively to downstream tasks such as semantic segmentation and deformation transfer, establishing a scalable and deployment-ready paradigm for dense 3D correspondence.

Soyeon Yoon, Chang Wook Seo, Hyunjung Shim

1KAIST AI · 2Anigma Technologies

Geometric Representation Learning Non-Rigid Shape Correspondence 3D Representation Learning

May 18, 2026cs.CV

Beyond Euclidean Prototypes: Spectral Disentanglement and Geodesic Matching for Few-Shot Medical Image Segmentation

Few-Shot Medical Image Segmentation (FSMIS) aims to delineate novel anatomical targets from one or a few annotated support images, addressing the annotation scarcity in medical imaging. Notwithstanding recent advancements, current prototype-based methods are bottlenecked by two coupled limitations: 1) cue entanglement, where a single spatial-domain prototype is forced to summarise organ silhouette, parenchymal texture and boundary appearance simultaneously, so any support-query mismatch on one cue propagates indiscriminately to the others; and 2) topology-blind matching, where cosine similarity measures distance in the ambient Euclidean space and ignores the connectivity of the underlying feature manifold, causing fragmented activations inside low-contrast organs and leakage into neighbouring tissues. To this end, we propose Spectral-Geodesic Prototype Network (SGP-Net), built around a Spectral-Geodesic Prototype Module with two coupled components. A Spectral Prototype Bank (SPB) decomposes support and query features into low-, mid- and high-frequency bands via learnable radial Fourier filters, yielding three disentangled prototypes per class that separately encode shape, texture and boundary cues. A Geodesic Matcher (GM) then replaces cosine similarity with a differentiable heat-diffusion approximation of geodesic distance, propagating matching signals along a feature affinity graph so that on-manifold pixels accumulate consistent responses while off-manifold look-alikes are suppressed. Experiments on three public FSMIS benchmarks demonstrate that SGP-Net achieves competitive performance against recent state-of-the-art methods.

Penghao Jia, Zhiyong Huang, Mingyang Hou +4

School of Microelectronics and Communication Engineering, Chongqing University, Chongqing 400044, China · Health Management Institute, The Second Medical Center, and the National Clinical Research Center for Geriatric Diseases, Chinese PLA General Hospital, Beijing 100853, China · Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China

Prototype-Based Learning Image Segmentation Geometric Representation Learning Few-Shot Medical Image Segmentation

May 18, 2026cs.LG

AURORA: Contextual Orthogonalization for Geometric Representation Learning in Healthcare Foundation Models

Recent healthcare foundation models have achieved strong predictive performance through large scale self supervised learning, yet their latent representations frequently entangle physiologic severity, intervention intensity, observational structure, and institutional workflow into shared embedding directions. While effective for downstream prediction, such representations remain semantically opaque and unstable under contextual shift. We introduce AURORA, Adaptive Uncertainty aware Representations through Orthogonalized Relational Alignment, a new framework for healthcare representation learning based on contextual latent geometry. Rather than optimizing a single unified embedding manifold, AURORA decomposes representations into orthogonal semantic subspaces corresponding to distinct contextual factors and learns relational consistency objectives within each subspace. This induces latent spaces that are both semantically disentangled and geometrically interpretable. Across multiple clinical prediction and retrieval tasks, AURORA consistently outperforms reconstruction, contrastive, and self distillation baselines while substantially improving contextual disentanglement, neighborhood purity, and robustness under institutional distribution shift. Our results suggest that latent geometry itself constitutes an important axis of healthcare foundation model design and that explicitly structuring representation space according to contextual semantics provides a complementary direction beyond conventional predictive compression objectives.

Yuanyun Zhang, Shi Li

University of the Chinese Academy of Sciences · Columbia University

Disentangled Representation Learning Representation Learning Healthcare Orthogonal Representation Learning Geometric Representation Learning

May 15, 2026cs.RO

GAP: Geometric Anchor Pre-training for Data-Efficient Visuomotor Learning of Manipulation Tasks

Learning visuomotor policies from scarce expert demonstrations remains a core challenge in robotic manipulation. A primary hurdle lies in distilling high-dimensional RGB representations into control-relevant geometry without overfitting. While using frozen pre-trained Vision Foundation Models (VFMs) improves data efficiency, it also shifts most task adaptation onto a small spatial pooling module, which can latch onto task-irrelevant shortcuts and lose geometric grounding when finetuned with few data samples. More broadly, pre-trained visual representations used for policy learning have been observed to struggle under even minor scene perturbations, highlighting the need for robustness-oriented inductive biases. We propose Geometric Anchor Pre-training (GAP), a simple, action-free warm-up stage that regularizes the spatial adapter before downstream imitation learning. GAP pre-trains the pooling layer on a lightweight simulated proxy task where object masks are available at no cost, encouraging the adapter to produce keypoints that lie on the object, cover its spatial extent, and remain sharp and repeatable over time. This yields stable geometric anchors that provide a reliable coordinate interface for few-shot policy learning, while keeping the VFM frozen. We evaluate GAP on RoboMimic and ManiSkill under severe data scarcity (15-50 demonstrations) and domain shift. A simple adapter regularized with GAP consistently outperforms stronger attention-based poolers and end-to-end fine-tuning, achieving 62% success on RoboMimic Can with 15 demonstrations (+16% over AFA), 63% on the long-horizon high-precision Tool Hang task with 50 demonstrations, and 61% on ManiSkill StackCube with 30 demonstrations (+11% over full fine-tuning). The proxy stage is lightweight and fully decoupled from downstream tasks, making it practical to reuse across environments and manipulation skills.

Davide Buoso, Andrea Protopapa, Stefano Di Carlo +2

Department of Control and Computer Engineering, Polytechnic University of Turin, Italy

Vision Foundation Model Adaptation Visuomotor Policy Learning Geometric Representation Learning Robot Imitation Learning Robot Learning

May 15, 2026cs.LG

Neural Point-Forms

Point cloud learning often rests on the premise that observed samples are noisy traces of an underlying geometric object, such as a manifold embedded in a high-dimensional feature space. Yet much of this geometry is not captured directly by coordinates, pairwise distances, or learned graph neighborhoods alone. In the smooth setting, differential forms are devices to encode higher order tangency information. In this work, we introduce a new family of principled learnable geometric features for point clouds called neural point-forms (NPFs). In the absence of a natural tangency structure, we instead use Laplacian-based techniques from Diffusion Geometry to build a discrete model for comparing differential forms on point clouds via inner products. In the continuum, submanifolds of a shared ambient feature space are represented as comparison matrices, whose entries describe how pairs of feature forms interact with extrinsic tangency information. We make this intuition precise by proving the long-run consistency of comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This yields a compact, efficient and permutation-invariant neural layer whose output is a learned form-comparison matrix. Across synthetic and biologically relevant experiments, we show that NPFs provide a competitive, and interpretable representation, with the strongest benefits appearing when labels depend on sampling density, manifold-like structure, or response-relevant population geometry.

Bruno Trentini, Jacob Hume, Vincenzo Antonio Isoldi +3

University of Oxford Oxford, UK · School of Engineering and Applied Sciences Harvard University Cambridge, MA, US · Max Planck Institute of Molecular Cell Biology and Genetics Dresden, Germany

Point Cloud Learning Geometric Representation Learning Manifold Learning

May 13, 2026cs.LG

Chem-GMNet: A Sphere-Native Geometric Transformer for Molecular Property Prediction

Modern SMILES-based chemical language models obtain strong MoleculeNet performance by treating SMILES as generic text and compensating with multi-million-molecule self-supervised pretraining. We ask: when a domain carries structural priors as rich as chemistry's, does it warrant a domain-native transformer rather than a generic one rescued by scale? We answer affirmatively with \textbf{GM-Net} (Geometric Measure Network), a transformer family in which every module is replaced by a sphere-native counterpart, and instantiate it as \textbf{Chem-GMNet}. Three blocks follow: SH-Embedding (tokens as learnable directions on

S^{k-1}

lifted through a Gegenbauer feature map); DualSKA (a per-head fusion of a linear-time gated Sphere-Flow recurrence whose persistent state we prove is the truncated multipole expansion of the input distribution, and a softmax Sphere-Kernel branch over the same Schoenberg-valid kernel); and SH-FFN (sphere projection

\to

Gegenbauer lift

\to

moment readout). On canonical DeepChem scaffold splits, against same-shape ChemBERTa-2 baselines under the chemberta3-faithful protocol: (i) random-initialised, Chem-GMNet wins on 7 of 10 MoleculeNet endpoints at

\sim\!35\%

fewer parameters; (ii) pretrained on the same 10M-SMILES ZINC corpus as ChemBERTa-2 MLM-10M, it matches or beats the public release on 6 of 8 shared endpoints (5/7 excluding a known ClinTox release anomaly). A

(k,L)

ablation shows that increasing the sphere dimension from

k\!=\!8

to

k\!=\!10

at fixed

L\!=\!3

lowers ESOL RMSE to

0.938

at scratch, beating pretrained ChemBERTa-2 MLM-10M on this endpoint without any pretraining at all.

Deepak Warrier, Raja Sekhar Pappala

MSTACK AI

Transformer Geometric Representation Learning Molecular Property Prediction Molecular Representation Learning

May 12, 2026cs.CE

A nonlinear extension of parametric model embedding for dimensionality reduction in parametric shape design

Dimensionality reduction is essential in simulation-based shape design, where high-dimensional parameterizations hinder optimization, surrogate modeling, and systematic design-space exploration. Parametric Model Embedding (PME) addresses this issue by constructing reduced variables from geometric information while preserving an explicit backmapping to the original design parameters. However, PME is intrinsically linear and may become inefficient when the sampled design space is governed by nonlinear geometric variability. This paper introduces a nonlinear extension of PME, denoted NLPME. The proposed framework preserves the defining principle of PME -- geometry-driven latent variables and parameter-mediated reconstruction -- while replacing the linear reduced subspace with a nonlinear latent representation. Geometry is not reconstructed directly from the latent variables; instead, the latent representation is decoded into admissible design parameters, and the corresponding geometry is recovered through a forward parametric map. The method is assessed on a bio-inspired autonomous underwater glider with a 32-dimensional parametric shape description and a CAD-based geometry-generation process. NLPME reaches a 5% reconstruction-error threshold with

N=5

latent variables, compared with

N=8

for linear PME, and a 1% threshold with

N=9

, compared with

N=15

for PME. Comparison with a deep autoencoder shows that most of the nonlinear compression gain can be retained while preserving an explicit backmapping to the original design variables. The results establish NLPME as a compact, admissible, and engineering-compatible nonlinear reduced representation for parametric shape design spaces.

Andrea Serani, Giorgio Palma, Matteo Diez

National Research Council-Institute of Marine Engineering, Rome, Italy

Geometric Representation Learning Dimensionality Reduction

May 12, 2026cs.CV

WorldComp2D: Spatio-semantic Representations of Object Identity and Location from Local Views

Learning latent representations that capture both semantic and spatial information is central to efficient spatio-semantic reasoning. However, many existing approaches rely on implicit latent structures combined with dense feature maps or task-specific heads, limiting computational efficiency and flexibility. We propose WorldComp2D, a novel lightweight representation learning framework that explicitly structures latent space geometry according to object identity and spatial proximity using multiscale local receptive fields. This framework consists of (i) a proximity-dependent encoder that maps a given observation into a spatio-semantic latent space and (ii) a localizer that infers the coordinates of objects in the input from the resulting spatio-semantic representation. Using facial landmark localization as a proof-of-concept, we show that, compared to SoTA lightweight models, WorldComp2D reduces the numbers of parameters and FLOPs by up to 4.0X and 2.2X, respectively, while maintaining real-time performance on CPU. These results demonstrate that explicitly structured latent spaces provide an efficient and general foundation for spatio-semantic reasoning. This framework is open-sourced at https://github.com/JinSeongmin/WorldComp2D.

SeongMin Jin, Doo Seok Jeong

1Division of Materials Science and Engineering · Department of Semiconductor Engineering

Representation Learning Visual Representation Learning Geometric Representation Learning

May 11, 2026cs.LG

Steerable Neural ODEs on Homogeneous Spaces

We introduce steerable neural ordinary differential equations on homogeneous spaces

M=G/H

. These models constitute a novel geometric extension of manifold neural ordinary differential equations (NODEs) that transport associated feature vectors transforming under the local symmetry group

H

. We interpret features as sections of associated vector bundles over

M

, and describe their evolution as parallel transport. This results in a coupled system of ODEs consisting of a flow equation on

M

and a steering equation acting on features. We show that steerable NODEs are

G

-equivariant whenever the vector field generating the flow and the connection governing parallel transport are both

G

-invariant. Furthermore, we demonstrate how steerable NODEs incorporate existing NODE models and continuous normalizing flows on Lie groups. Our framework provides the geometric foundation for learning continuous-time equivariant dynamics of general vector-valued features on homogeneous spaces.

Emma Andersdotter, Daniel Persson, Fredrik Ohlsson

Department of Mathematics and Mathematical Statistics Umeå University Umeå SE-901 87, Sweden · Department of Mathematical Sciences2026 Chalmers University of Technology and Gothenburg University Gothenburg SE-412 96, Sweden · Department of Mathematics and Mathematical Statistics 11 Umeå University Umeå SE-901 87, Sweden

Equivariant Neural Networks Geometric Representation Learning Neural ODEs

May 11, 2026cs.RO

Learning Point Cloud Geometry as a Statistical Manifold: Theory and Practice

Point clouds are a fundamental representation for robotic perception tasks such as localization, mapping, and object pose estimation. However, LiDAR-acquired point clouds are inherently sparse and non-uniform, providing incomplete observations of the underlying scene geometry. This makes reliable geometric reasoning challenging and degrades downstream perception performance. Existing approaches attempt to compensate for these limitations by estimating local geometry, but often rely on hand-crafted statistics or end-to-end supervised learning, which can suffer from limited scalability or require large amounts of accurately labeled data. To address these challenges, we explicitly model point cloud geometry under a principled mathematical formulation. We represent local geometry as a statistical manifold induced by a family of Gaussian distributions, where each point is associated with a Gaussian capturing its local geometric structure. Based on this formulation, we introduce Point-to-Ellipsoid (POLI), a deep neural estimator that predicts per-point Gaussian geometry. POLI learns a mapping from point cloud observations to their underlying geometry in a self-supervised manner, removing the need for labeled data while preserving strong geometric inductive biases. The resulting representation integrates seamlessly into existing robotic perception pipelines without architectural modifications. Extensive experiments show that POLI enables accurate and robust geometry estimation and consistently improves performance across diverse robotic perception tasks.

Jinwoo Lee, Jiwoo Kim, 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

Geometric Representation Learning Manifold Learning Point Cloud Understanding Robotic Perception

May 11, 2026cs.SE

MARGIN: Margin-Aware Regularized Geometry for Imbalanced Vulnerability Detection

Software vulnerability detection is critical for ensuring software security and reliability. Despite recent advances in deep learning, real-world vulnerability datasets suffer from two severe challenges: frequency imbalance and difficulty imbalance. We reinterpret these challenges from an embedding geometry perspective, observing that such imbalances induce geometric distortions in hyperspherical representation space. To address this issue, we propose MARGIN, a metric-based framework that learns discriminative vulnerability representations through adaptive margin metric learning and hyperspherical prototype modeling. MARGIN dynamically adjusts geometric regularization according to the distribution structure estimated by the von Mises-Fisher concentration, aligning the probability mass of embedding distributions with their corresponding Voronoi cells, thereby reducing geometric distortion and yielding more stable decision boundaries. Extensive experiments on public vulnerability datasets show that MARGIN consistently outperforms strong baselines, achieving notable improvements in classification and detection, especially on challenging, imbalanced datasets. Further analysis demonstrates that MARGIN produces more structured embedding geometries, improving robustness, interpretability, and generalization.

Yuteng Zhang, Huifang Ma, Jiahui Wei +2

College of Artificial Intelligence and Computer Science, Northwest Normal University, Lanzhou 730070, China

Class-Imbalanced Learning Software Vulnerability Detection Hyperspherical Representation Learning Geometric Representation Learning Metric Learning