Geometric Representation Learning

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The field studies how to represent data whose structure is not well described by ordinary flat vectors, including point clouds, graphs, molecules, shapes, and signals. Approaches include rotation- and group-equivariant networks, learning directly on Riemannian, spherical, or hyperbolic spaces, and using geodesic distances, curvature, or spectral structure to guide training. These methods aim to preserve meaningful geometry so models can generalize across changes in viewpoint, sampling, or domain, with applications from 3D perception and robotics to biology and medical imaging. A central challenge is choosing geometry that fits the task: explicit geometric constraints can improve robustness and data efficiency, but may add computational cost or hinder downstream learning when the chosen structure is mismatched.

Latest papers

259 papers32 papers in the last four weeks, up 7% on the four weeks before. 0.2% of all new papers.

Apr 18, 2026

CCAR: Intrinsic Robustness as an Emergent Geometric Property

Standard supervised learning optimizes for predictive accuracy but remains agnostic to the internal geometry of learned features, often yielding representations that are entangled and brittle. We propose Class-Conditional Activation Regularization (CCAR) to explicitly engineer the feature space, imposing a block-diagonal structure via a soft inductive bias. By shaping the latent representation to confine class energy to orthogonal subspaces, we create an intrinsic geometric scaffold that naturally filters noise and adversarial perturbations. We provide theoretical analysis linking this structural constraint to the maximization of the Fisher Discriminant Ratio, establishing a formal connection between geometric disentanglement and algorithmic stability. Empirically, this approach demonstrates that robustness is an emergent property of a well-engineered feature space, significantly outperforming baselines on label noise and input corruption benchmarks.

Akash Samanta, Manish Pratap Singh, Debasis Chaudhuri

Techno India University, Salt Lake, Kolkata, India · DRDO Young Scientist Laboratory - CT, Chennai, India

Disentangled Representation Learning Representation Learning Algorithmic Stability Neural Network Robustness Adversarial Robustness Orthogonal Representation Learning+3

Apr 16, 2026

STEP-Parts: Geometric Partitioning of Boundary Representations for Large-Scale CAD Processing

Many CAD learning pipelines discretize Boundary Representations (B-Reps) into triangle meshes, discarding analytic surface structure and topological adjacency and thereby weakening consistent instance-level analysis. We present STEP-Parts, a deterministic CAD-to-supervision toolchain that extracts geometric instance partitions directly from raw STEP B-Reps and transfers them to tessellated carriers through retained source-face correspondence, yielding instance labels and metadata for downstream learning and evaluation. The construction merges adjacent B-Rep faces only when they share the same analytic primitive type and satisfy a near-tangent continuity criterion. On ABC, same-primitive dihedral angles are strongly bimodal, yielding a threshold-insensitive low-angle regime for part extraction. Because the partition is defined on intrinsic B-Rep topology rather than on a particular triangulation, the resulting boundaries remain stable under changes in tessellation. Applied to the DeepCAD subset of ABC, the pipeline processes approximately 180{,}000 models in under six hours on a consumer CPU. We release code and precomputed labels, and show that STEP-Parts serves both as a tessellation-robust geometric reference and as a useful supervision source in two downstream probes: an implicit reconstruction--segmentation network and a dataset-level point-based backbone.

Shen Fan, Mikołaj Kida, Przemyslaw Musialski

New Jersey Institute of Technology · Warsaw University of Technology

Geometric Representation Learning 3D Instance Segmentation

Apr 13, 2026

Fairness is Not Flat: Geometric Phase Transitions Against Shortcut Learning

Deep Neural Networks are highly susceptible to shortcut learning, frequently memorizing low-dimensional spurious correlations instead of underlying causal mechanisms. This phenomenon not only degrades out-of-distribution robustness but also induces severe demographic biases in sensitive applications. In this paper, we propose a geometric a priori methodology to mitigate shortcut learning. By deploying a zero-hidden-layer (

N=1

) Topological Auditor, we mathematically isolate features that monopolize the gradient without human intervention. We empirically demonstrate a Capacity Phase Transition: once linear shortcuts are pruned, networks are forced to utilize higher geometric capacity (

N \geq 16

) to curve the decision boundary and learn ethical representations. Our approach outperforms L1 Regularization -- which collapses into demographic bias -- and operates at a fraction of the computational cost of post-hoc methods like Just Train Twice (JTT), successfully reducing counterfactual gender vulnerability from 21.18% to 7.66%.

Nicolas Rodriguez-Alvarez

Instituto de Educacion Secundaria Parquesol, Valladolid, Spain · Instituto de Educación Secundaria Parquesol Valladolid, Castile and León, Spain

Shortcut Learning Algorithmic Fairness Algorithmic Bias Geometric Representation Learning

Apr 13, 2026

GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural Kähler Potentials

We present GlobalCY, a JAX-based framework for globally defined and symmetry-aware neural Kähler-potential models on projective hypersurface Calabi--Yau geometries. The central problem is that local-input neural Kähler-potential models can train successfully while still failing the geometry-sensitive diagnostics that matter in hard quartic regimes, especially near singular and near-singular members of the Cefalú family. To study this, we compare three model families -- a local-input baseline, a globally defined invariant model, and a symmetry-aware global model -- on the hard Cefalú cases

λ=0.75

and

λ=1.0

using a fixed multi-seed protocol and a geometry-aware diagnostic suite. In this benchmark, the globally defined invariant model is the strongest overall family, outperforming the local baseline on the two clearest geometric comparison metrics, negative-eigenvalue frequency and projective-invariance drift, in both cases. The gains are strongest at

λ=0.75

, while

λ=1.0

remains more difficult. The current symmetry-aware model improves projective-invariance drift relative to the local baseline, but does not yet surpass the plain global invariant model. These results show that global invariant structure is a meaningful architectural constraint for learned Kähler-potential modeling in hard quartic Calabi--Yau settings.

Abdul Rahman

Geometric Representation Learning Invariant Representation Learning Riemannian Geometry

Apr 8, 2026

Training-free Spatially Grounded Geometric Shape Encoding (Technical Report)

Positional encoding has become the de facto standard for grounding deep neural networks on discrete point-wise positions, and it has achieved remarkable success in tasks where the input can be represented as a one-dimensional sequence. However, extending this concept to 2D spatial geometric shapes demands carefully designed encoding strategies that account not only for shape geometry and pose, but also for compatibility with neural network learning. In this work, we address these challenges by introducing a training-free, general-purpose encoding strategy, dubbed XShapeEnc, that encodes an arbitrary spatially grounded 2D geometric shape into a compact representation exhibiting five favorable properties, including invertibility, adaptivity, and frequency richness. Specifically, a 2D spatially grounded geometric shape is decomposed into its normalized geometry within the unit disk and its pose vector, where the pose is further transformed into a harmonic pose field that also lies within the unit disk. A set of orthogonal Zernike bases is constructed to encode shape geometry and pose either independently or jointly, followed by a frequency-propagation operation to introduce high-frequency content into the encoding. We demonstrate the theoretical validity, efficiency, discriminability, and applicability of XShapeEnc via extensive analysis and experiments across a wide range of shape-aware tasks and our self-curated XShapeCorpus. We envision XShapeEnc as a foundational tool for research that goes beyond one-dimensional sequential data toward frontier 2D spatial intelligence.

Yuhang He

Microsoft Research

Positional Encoding Geometric Representation Learning

Apr 8, 2026

LiftFormer: Lifting and Frame Theory Based Monocular Depth Estimation Using Depth and Edge Oriented Subspace Representation

Monocular depth estimation (MDE) has attracted increasing interest in the past few years, owing to its important role in 3D vision. MDE is the estimation of a depth map from a monocular image/video to represent the 3D structure of a scene, which is a highly ill-posed problem. To solve this problem, in this paper, we propose a LiftFormer based on lifting theory topology, for constructing an intermediate subspace that bridges the image color features and depth values, and a subspace that enhances the depth prediction around edges. MDE is formulated by transforming the depth value prediction problem into depth-oriented geometric representation (DGR) subspace feature representation, thus bridging the learning from color values to geometric depth values. A DGR subspace is constructed based on frame theory by using linearly dependent vectors in accordance with depth bins to provide a redundant and robust representation. The image spatial features are transformed into the DGR subspace, where these features correspond directly to the depth values. Moreover, considering that edges usually present sharp changes in a depth map and tend to be erroneously predicted, an edge-aware representation (ER) subspace is constructed, where depth features are transformed and further used to enhance the local features around edges. The experimental results demonstrate that our LiftFormer achieves state-of-the-art performance on widely used datasets, and an ablation study validates the effectiveness of both proposed lifting modules in our LiftFormer.

Shuai Li, Huibin Bai, Yanbo Gao +5

School of Control Science and Engineering, Shandong University, and Key Laboratory of Machine Intelligence and System Control, Ministry of Education, Jinan 250100, China. · School of Software, Shandong University, Jinan 250100, China · Shandong University-WeiHai Research Institute of Industrial Technology, Weihai 264209, China +2

Depth Estimation Monocular Depth Estimation Geometric Representation Learning

Apr 7, 2026

Let Geometry GUIDE: Layer-wise Unrolling of Geometric Priors in Multimodal LLMs

Multimodal Large Language Models (MLLMs) have achieved remarkable progress in 2D visual tasks but still struggle to understand physical space in real-world visual streams. Recently, feed-forward geometric foundation models that implicitly extract geometric priors from visual inputs have offered a promising direction. However, many existing geometry-aware MLLMs extract features from a single deep encoder layer and perform one-shot fusion at the language-model input, potentially overlooking complementary information across geometric encoder layers, limiting access to fine-grained geometric cues, and hindering progressive cross-modal adaptation. To address these limitations, we propose GUIDE (Geometric Unrolling Inside MLLM Early-layers), a progressive framework for integrating geometric priors. GUIDE samples and aligns multi-level features from the geometric encoder and sequentially injects them into the early layers of the MLLM. This design enables the MLLM to continuously access multi-granularity geometric cues and progressively integrate them with visual representations during layer-wise semantic abstraction. GUIDE further introduces a dual context-aware gating mechanism at both the token and layer levels, selectively regulating geometric information to reduce redundant injection and interference with pretrained representations. Extensive experiments on spatial reasoning and 3D scene understanding benchmarks, including VSI-Bench, ScanRefer, and Scan2Cap, validate the effectiveness and cross-task applicability of GUIDE. Our 5B and 9B models achieve average scores of 71.5 and 72.1 on VSI-Bench, respectively.

Chongyu Wang, Ting Huang, Chunyu Sun +3

Xi’an Jiaotong University · Peking University · Institute of Automation, Chinese Academy of Sciences +1

Multimodal Large Language Models 3D VLMs Spatial Reasoning 3D Scene Understanding Geometric Representation Learning

Apr 6, 2026

MAVEN: A Mesh-Aware Volumetric Encoding Network for Simulating 3D Flexible Deformation

Deep learning-based approaches, particularly graph neural networks (GNNs), have gained prominence in simulating flexible deformations and contacts of solids, due to their ability to handle unstructured physical fields and nonlinear regression on graph structures. However, existing GNNs commonly represent meshes with graphs built solely from vertices and edges. These approaches tend to overlook higher-dimensional spatial features, e.g., 2D facets and 3D cells, from the original geometry. As a result, it is challenging to accurately capture boundary representations and volumetric characteristics, though this information is critically important for modeling contact interactions and internal physical quantity propagation, particularly under sparse mesh discretization. In this paper, we introduce MAVEN, a mesh-aware volumetric encoding network for simulating 3D flexible deformation, which explicitly models geometric mesh elements of higher dimension to achieve a more accurate and natural physical simulation. MAVEN establishes learnable mappings among 3D cells, 2D facets, and vertices, enabling flexible mutual transformations. Explicit geometric features are incorporated into the model to alleviate the burden of implicitly learning geometric patterns. Experimental results show that MAVEN consistently achieves state-of-the-art performance across established datasets and a novel metal stretch-bending task featuring large deformations and prolonged contacts.

Zhe Feng, Shilong Tao, Haonan Sun +3

Academy for Advanced Interdisciplinary Studies, Peking University · School of Computer Science, Peking University · School of Electrical and Computer Science, University of Southampton

Deformable Object Simulation 3D Mesh Deformation Graph Neural Networks Geometric Representation Learning

Apr 5, 2026

The Geometric Alignment Tax: Tokenization vs. Continuous Geometry in Scientific Foundation Models

Foundation models for biology and physics optimize predictive accuracy, but their internal representations systematically fail to preserve the continuous geometry of the systems they model. We identify the root cause: the Geometric Alignment Tax, an intrinsic cost of forcing continuous manifolds through discrete categorical bottlenecks. Controlled ablations on synthetic dynamical systems demonstrate that replacing cross-entropy with a continuous head on an identical encoder reduces geometric distortion by up to 8.5x, while learned codebooks exhibit a non-monotonic double bind where finer quantization worsens geometry despite improving reconstruction. Under continuous objectives, three architectures differ by 1.3x; under discrete tokenization, they diverge by 3,000x. Evaluating 14 biological foundation models with rate-distortion theory and MINE, we identify three failure regimes: Local-Global Decoupling, Representational Compression, and Geometric Vacuity. A controlled experiment confirms that Evo 2's reverse-complement robustness on real DNA reflects conserved sequence composition, not learned symmetry. No model achieves simultaneously low distortion, high mutual information, and global coherence.

Prashant C. Raju

Neural Representation Geometry Geometric Representation Learning

Apr 2, 2026

Quantum-Inspired Geometric Classification with Correlation Group Structures and VQC Decision Modeling

We propose a geometry-driven quantum-inspired classification framework that integrates Correlation Group Structures (CGR), compact SWAP-test-based overlap estimation, and selective variational quantum decision modelling. Rather than directly approximating class posteriors, the method adopts a geometry-first paradigm in which samples are evaluated relative to class medoids using overlap-derived Euclidean-like and angular similarity channels. CGR organizes features into anchor-centered correlation neighbourhoods, generating nonlinear, correlation-weighted representations that enhance robustness in heterogeneous tabular spaces. These geometric signals are fused through a non-probabilistic margin-based fusion score, serving as a lightweight and data-efficient primary classifier for small-to-moderate datasets. On Heart Disease, Breast Cancer, and Wine Quality datasets, the fusion-score classifier achieves 0.8478, 0.8881, and 0.9556 test accuracy respectively, with macro-F1 scores of 0.8463, 0.8703, and 0.9522, demonstrating competitive and stable performance relative to classical baselines. For large-scale and highly imbalanced regimes, we construct compact Delta-distance contrastive features and train a variational quantum classifier (VQC) as a nonlinear refinement layer. On the Credit Card Fraud dataset (0.17% prevalence), the Delta + VQC pipeline achieves approximately 0.85 minority recall at an alert rate of approximately 1.31%, with ROC-AUC 0.9249 and PR-AUC 0.3251 under full-dataset evaluation. These results highlight the importance of operating-point-aware assessment in rare-event detection and demonstrate that the proposed hybrid geometric-variational framework provides interpretable, scalable, and regime-adaptive classification across heterogeneous data settings.

Nishikanta Mohanty, Arya Ansuman Priyadarshi, Bikash K. Behera +1

Centre for Quantum Software and Information, University of Technology Sydney, 15 Broadway, Ultimo, Sydney, 2007, NSW, Australia. · Bikash’s Quantum (OPC) Pvt. Ltd., Balindi, Mohanpur, 741246, WB, India. · Universit`a degli Studi di Cagliari, Via Is Mirrions, Cagliari, 09123, Italy.

Tabular Classification Hybrid Quantum-Classical ML Geometric Representation Learning Class-Imbalanced Learning

Mar 30, 2026

LA-Sign: Looped Transformers with Geometry-aware Alignment for Skeleton-based Sign Language Recognition

Skeleton-based isolated sign language recognition (ISLR) demands fine-grained understanding of articulated motion across multiple spatial scales, from subtle finger movements to global body dynamics. Existing approaches typically rely on deep feed-forward architectures, which increase model capacity but lack mechanisms for recurrent refinement and structured representation. We propose LA-Sign, a looped transformer framework with geometry-aware alignment for ISLR. Instead of stacking deeper layers, LA-Sign derives its depth from recurrence, repeatedly revisiting latent representations to progressively refine motion understanding under shared parameters. To further regularise this refinement process, we present a geometry-aware contrastive objective that projects skeletal and textual features into an adaptive hyperbolic space, encouraging multi-scale semantic organisation. We study three looping designs and multiple geometric manifolds, demonstrating that encoder-decoder looping combined with adaptive Poincare alignment yields the strongest performance. Extensive experiments on WLASL and MSASL benchmarks show that LA-Sign achieves state-of-the-art results while using fewer unique layers, highlighting the effectiveness of recurrent latent refinement and geometry-aware representation learning for sign language recognition.

Muxin Pu, Mei Kuan Lim, Chun Yong Chong +1

School of Information Technology, Monash University · S-Lab, Nanyang Technological University

Sign Language Recognition Recurrent Transformers Geometric Representation Learning Hyperbolic Representation Learning

Mar 27, 2026

Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow

We introduce the Geometric Evolution Graph Convolutional Network (GEGCN), a novel framework that enhances graph representation learning through explicit modeling of geometric evolution on graph structures. Specifically, GEGCN leverages a Long Short-Term Memory (LSTM) network to capture the dynamic structural sequence generated by discrete Ricci flow, and infuses the learned dynamic representations into a graph convolutional network. Extensive experiments demonstrate that GEGCN achieves excellent performance on classification tasks across various benchmark datasets, including homophilic/heterophilic graphs, filtered graphs, and large-scale graphs.

Jicheng Ma, Yunyan Yang, Juan Zhao +1

School of Mathematics, Renmin University of China, Beijing, China. · School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems of MOE, Beijing Normal University, Beijing, China.

Graph Representation Learning Graph Neural Networks Geometric Representation Learning

Mar 26, 2026

Geo $^\textbf{2}$ : Geometry-Guided Cross-view Geo-Localization and Image Synthesis

Cross-view geo-spatial learning consists of two important tasks: Cross-View Geo-Localization (CVGL) and Cross-View Image Synthesis (CVIS), both of which rely on establishing geometric correspondences between ground and aerial views. Recent Geometric Foundation Models (GFMs) have demonstrated strong capabilities in extracting generalizable 3D geometric features from images, but their potential in cross-view geo-spatial tasks remains underexplored. In this work, we present Geo^2, a unified framework that leverages Geometric priors from GFMs (e.g., VGGT) to jointly perform geo-spatial tasks, CVGL and bidirectional CVIS. Despite the 3D reconstruction ability of GFMs, directly applying them to CVGL and CVIS remains challenging due to the large viewpoint gap between ground and aerial imagery. We propose GeoMap, which embeds ground and aerial features into a shared 3D-aware latent space, effectively reducing cross-view discrepancies for localization. This shared latent space naturally bridges cross-view image synthesis in both directions. To exploit this, we propose GeoFlow, a flow-matching model conditioned on geometry-aware latent embeddings. We further introduce a consistency loss to enforce latent alignment between the two synthesis directions, ensuring bidirectional coherence. Extensive experiments on standard benchmarks, including CVUSA, CVACT, and VIGOR, demonstrate that Geo^2 achieves state-of-the-art performance in both localization and synthesis, highlighting the effectiveness of 3D geometric priors for cross-view geo-spatial learning.

Yancheng Zhang, Xiaohan Zhang, Guangyu Sun +3

Institute of Artificial Intelligence, University of Central Florida · University of Vermont

Cross-View Geo-Localization Geospatial Representation Learning Image Generation Geometric Representation Learning

Mar 24, 2026

Patchwork: A compact representation for 3D polygonal shapes

We introduce Patchwork, a new general-purpose shape representation capable of modeling 2D and 3D geometry with a small number of parameters. Patchwork is grounded in a rigorous mathematical framework, providing provable complexity bounds and the ability to approximate arbitrary shapes with arbitrary precision in any dimension. We propose an efficient gradient-based optimization scheme to fit Patchwork representations to 2D and 3D data, along with a novel regularization loss that progressively prunes redundant elements, yielding high compactness after convergence. Our approach offers fast fitting performance, a fraction of the required parameters compared to existing alternatives, and native support for inside-outside classification, making it a versatile and compact representation for geometric learning and reconstruction tasks, with future potential for 3D generation. Our implementation is available at: https://github.com/Ankbzpx/patchwork-experiment.

Ruichen Zheng, Biao Zhang, Michael Birsak +2

King Abdullah University of Science and Technology, Thuwal, 23955, Saudi Arabia

Geometric Representation Learning 3D Representation Learning 3D Reconstruction

Mar 23, 2026

Riemannian Geometry Speaks Louder Than Words: From Graph Foundation Model to Next-Generation Graph Intelligence

Graphs provide a natural description of the complex relationships among objects, and play a pivotal role in communications, transportation, social computing, the life sciences, etc. Currently, there is strong agreement that Graph Foundation Models (GFMs) are essential for advancing graph learning, yet considerable disagreement persists on how to build a powerful, general-purpose GFM analogous to Large Language Models (LLMs). Graph Neural Networks (GNNs) exhibit limitations in memory retention and principled interpretability when confronted with multi-domain pretraining and adaptation. The challenge of graph serialization hinders the direct application of LLMs, as the words struggle to capture the structural complexity and diversity inherent in graphs. In contrast, Riemannian geometry offers an elegant mathematical framework for modeling structures, while remaining compatible with graph semantic learning, even with LLMs. In this paper, we argue that, for graphs, Riemannian geometry speaks louder than words, and lay out the foundational principles for GFM. Reimagining with Riemannian geometry, we introduce a blue sky idea-Riemannian Foundation Model (RFM)-that opens a new pathway for capturing complex structural patterns and uncovering cross-domain generalities. RFM emphasizes intrinsic graph geometry and embodies endogenous capacities for structural inference and generation, moving beyond mere representation-space switching. Accordingly, we outline a progressive agenda that begins with universal structural understanding through intrinsic geometry, and then rebuilds LLM with a Riemannian engine for general-purpose graph modeling and beyond. Thus, RFM enables a paradigm shift from designing graph models to solving graph-structured applications with RFM agents, unlocking the next-generation graph intelligence.

Philip S. Yu, Li Sun

University of Illinois Chicago, Chicago, USA · Beijing University of Posts and Telecommunications, Beijing, China

Graph Foundation Models Graph Representation Learning LLMs for Graph Learning Geometric Representation Learning

Mar 20, 2026

Continual Learning as Shared-Manifold Continuation Under Compatible Shift

Continual learning methods usually preserve old behavior by regularizing parameters, matching old outputs, or replaying previous examples. These strategies can reduce forgetting, but they do not directly specify how the latent representation should evolve. We study a narrower geometric alternative for the regime where old and new data should remain on the same latent support: continual learning as continuation of a shared manifold. We instantiate this view within Support-Preserving Manifold Assimilation (SPMA) and evaluate a geometry-preserving variant, SPMA-OG, that combines sparse replay, output distillation, relational geometry preservation, local smoothing, and chart-assignment regularization on old anchors. On representative compatible-shift CIFAR10 and Tiny-ImageNet runs, SPMA-OG improves over sparse replay baselines in old-task retention and representation-preservation metrics while remaining competitive on new-task accuracy. On a controlled synthetic atlas-manifold benchmark, it achieves near-perfect anchor-geometry preservation while also improving new-task accuracy over replay. These results provide evidence that geometry-aware anchor regularization is a useful inductive bias when continual learning should preserve a shared latent support rather than create a new one.

Henry J. Kobs

Continual Learning Geometric Representation Learning Manifold Learning

Mar 20, 2026

Cov2Pose: Leveraging Spatial Covariance for Direct Manifold-aware 6-DoF Object Pose Estimation

In this paper, we address the problem of 6-DoF object pose estimation from a single RGB image. Indirect methods that typically predict intermediate 2D keypoints, followed by a Perspective-n-Point solver, have shown great performance. Direct approaches, which regress the pose in an end-to-end manner, are usually computationally more efficient but less accurate. However, direct pose regression heads rely on globally pooled features, ignoring spatial second-order statistics despite their informativeness in pose prediction. They also predict, in most cases, discontinuous pose representations that lack robustness. Herein, we therefore propose a covariance-pooled representation that encodes convolutional feature distributions as a symmetric positive definite (SPD) matrix. Moreover, we propose a novel pose encoding in the form of an SPD matrix via its Cholesky decomposition. Pose is then regressed in an end-to-end manner with a manifold-aware network head, taking into account the Riemannian geometry of SPD matrices. Experiments and ablations consistently demonstrate the relevance of second-order pooling and continuous representations for direct pose regression, including under partial occlusion.

Nassim Ali Ousalah, Peyman Rostami, Vincent Gaudillière +4

CVI2, Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg⋆ · Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France† · Infinite Orbits⋊ +1

Riemannian Neural Networks Object Pose Estimation Geometric Representation Learning

Mar 20, 2026

Geometric Mixture-of-Experts with Curvature-Guided Adaptive Routing for Graph Representation Learning

Graph-structured data typically exhibits complex topological heterogeneity, making it difficult to model accurately within a single Riemannian manifold. While emerging mixed-curvature methods attempt to capture such diversity, they often rely on implicit, task-driven routing that lacks fundamental geometric grounding. To address this challenge, we propose a Geometric Mixture-of-Experts framework (GeoMoE) that adaptively fuses node representations across diverse Riemannian spaces to better accommodate multi-scale topological structures. At its core, GeoMoE leverages Ollivier-Ricci Curvature (ORC) as an intrinsic geometric prior to orchestrate the collaboration of specialized experts. Specifically, we design a graph-aware gating network that assigns node-specific fusion weights, regularized by a curvature-guided alignment loss to ensure interpretable and geometry-consistent routing. Additionally, we introduce a curvature-aware contrastive objective that promotes geometric discriminability by constructing positive and negative pairs according to curvature consistency. Extensive experiments on six benchmark datasets demonstrate that GeoMoE outperforms state-of-the-art baselines across diverse graph types.

Haifang Cao, Yu Wang, Timing Li +2

College of Intelligence and Computing, Tianjin University, Tianjin, China

Ollivier-Ricci Curvature Graph Representation Learning Geometric Representation Learning Mixture-of-Experts Models

Mar 17, 2026

Manifold-Matching Autoencoders

We study a simple unsupervised regularization scheme for autoencoders called Manifold-Matching (MMAE): we align the pairwise distances in the latent space to those of the input data space by minimizing mean squared error. Because alignment occurs on pairwise distances rather than coordinates, it can also be extended to a lower-dimensional representation of the data, adding flexibility to the method. We find that this regularization outperforms similar methods on metrics based on preservation of nearest-neighbor distances and persistent homology-based measures. We also observe that MMAE provides a scalable approximation of Multi-Dimensional Scaling (MDS).

Laurent Cheret, Vincent Létourneau, Isar Nejadgholi +3

Department of Computer Science, University of Ottawa, Ottawa, Canada · MILA, Montréal, Canada · National Research Council of Canada, Ottawa, Canada

Multidimensional Scaling Autoencoders Latent-Space Regularization Dimensionality Reduction Latent Space Geometry Latent Space Alignment+2

Mar 17, 2026

RecBundle: A Next-Generation Geometric Paradigm for Explainable Recommender Systems

Recommender systems are inherently dynamic feedback loops where prolonged local interactions accumulate into macroscopic structural degradation such as information cocoons. Existing representation learning paradigms are universally constrained by the assumption of a single flat space, forcing topologically grounded user associations and semantically driven historical interactions to be fitted within the same vector space. This excessive coupling of heterogeneous information renders it impossible for researchers to mechanistically distinguish and identify the sources of systemic bias. To overcome this theoretical bottleneck, we introduce Fiber Bundle from modern differential geometry and propose a novel geometric analysis paradigm for recommender systems. This theory naturally decouples the system space into two hierarchical layers: the base manifold formed by user interaction networks, and the fibers attached to individual user nodes that carry their dynamic preferences. Building upon this, we construct RecBundle, a framework oriented toward next-generation recommender systems that formalizes user collaboration as geometric connection and parallel transport on the base manifold, while mapping content evolution to holonomy transformations on fibers. From this foundation, we identify future application directions encompassing quantitative mechanisms for information cocoons and evolutionary bias, geometric meta-theory for adaptive recommendation, and novel inference architectures integrating large language models (LLMs). Empirical analysis on real-world MovieLens and Amazon Beauty datasets validates the effectiveness of this geometric framework.

Hui Wang, Tianzhu Hu, Mingming Li +6

Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China · JD.com, Beijing, China · School of Cyberspace Security, University of Chinese Academy of Sciences, Beijing, China

Recommender Systems Geometric Representation Learning Explainable Recommendation User Preference Modeling

Mar 16, 2026

Parallelised Differentiable Straightest Geodesics for 3D Meshes

Machine learning has been progressively generalised to operate within non-Euclidean domains, but geometrically accurate methods for learning on surfaces are still falling behind. The lack of closed-form Riemannian operators, the non-differentiability of their discrete counterparts, and poor parallelisation capabilities have been the main obstacles to the development of the field on meshes. A principled framework to compute the exponential map on Riemannian surfaces discretised as meshes is straightest geodesics, which also allows to trace geodesics and parallel-transport vectors as a by-product. We provide a parallel GPU implementation and derive two different methods for differentiating through the straightest geodesics, one leveraging an extrinsic proxy function and one based upon a geodesic finite differences scheme. After proving our parallelisation performance and accuracy, we demonstrate how our differentiable exponential map can improve learning and optimisation pipelines on general geometries. In particular, to showcase the versatility of our method, we propose a new geodesic convolutional layer, a new flow matching method for learning on meshes, and a second-order optimiser that we apply to centroidal Voronoi tessellation. Our code, models, and pip-installable library (digeo) are available at: circle-group.github.io/research/DSG.

Hippolyte Verninas, Caner Korkmaz, Stefanos Zafeiriou +2

Imperial College London

Riemannian Geometry Geometric Representation Learning Riemannian Optimization Riemannian Flow Matching

Mar 16, 2026

Riemannian Motion Generation: A Unified Framework for Human Motion Representation and Generation via Riemannian Flow Matching

Human motion generation is often learned in Euclidean spaces, although valid motions follow structured non-Euclidean geometry. We present Riemannian Motion Generation (RMG), a unified framework that represents motion on a product manifold and learns dynamics via Riemannian flow matching. RMG factorizes motion into several manifold factors, yielding a scale-free representation with intrinsic normalization, and uses geodesic interpolation, tangent-space supervision, and manifold-preserving ODE integration for training and sampling. On HumanML3D, RMG achieves state-of-the-art FID in the HumanML3D format (0.043) and ranks first on all reported metrics under the MotionStreamer format. On MotionMillion, it also surpasses strong baselines (FID 5.6, R@1 0.86). Ablations show that the compact

\mathscr{T}+\mathscr{R}

(translation + rotations) representation is the most stable and effective, highlighting geometry-aware modeling as a practical and scalable route to high-fidelity motion generation.

Fangran Miao, Jian Huang, Ting Li

Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University · Department of Applied Mathematics, The Hong Kong Polytechnic University · Department of Data Science and Statistics, Southern University of Science and Technology

Human Motion Generation Motion Representation Learning Riemannian Flow Matching Geometric Representation Learning

Mar 15, 2026

Geometric and Topological Deep Learning for Predicting Thermo-mechanical Performance in Cold Spray Deposition Process Modeling

This study presents a geometric deep learning framework for predicting cold spray particle impact responses using finite element simulation data. A parametric dataset was generated through automated Abaqus simulations spanning a systematic range of particle velocity, particle temperature, and friction coefficient, yielding five output targets including maximum equivalent plastic strain, average contact plastic strain, maximum temperature, maximum von Mises stress, and deformation ratio. Four novel algorithms i.e. a GraphSAGE-style inductive graph neural network, a Chebyshev spectral graph convolution network, a topological data analysis augmented multilayer perceptron, and a geometric attention network were implemented and evaluated. Each input sample was treated as a node in a k-nearest-neighbour feature-space graph, enabling the models to exploit spatial similarity between process conditions during training. Three-dimensional feature space visualisations and two-dimensional contour projections confirmed the highly non-linear and velocity-dominated nature of the input-output relationships. Quantitative evaluation demonstrated that GraphSAGE and GAT consistently achieved R-square values exceeding 0.93 across most targets, with GAT attaining peak performance of R-square equal to 0.97 for maximum plastic strain. ChebSpectral and TDA-MLP performed considerably worse, yielding negative R-square values for several targets. These findings establish spatial graph-based neighbourhood aggregation as a robust and physically interpretable surrogate modelling strategy for cold spray process optimisation.

Akshansh Mishra

School of Industrial and Information Engineering, Politecnico di Milano, Milan, Italy

Surrogate Modeling Manufacturing Graph Neural Networks Geometric Representation Learning

Mar 13, 2026

SGMatch: Semantic-Guided Non-Rigid Shape Matching with Flow Regularization

Establishing accurate point-to-point correspondences between non-rigid 3D shapes remains a critical challenge, particularly under non-isometric deformations and topological noise. Existing functional map pipelines suffer from ambiguities that geometric descriptors alone cannot resolve, and spatial inconsistencies inherent in the projection of truncated spectral bases to dense pointwise correspondences. In this paper, we introduce SGMatch, a learning-based framework that couples 3D-lifted semantic cues with trajectory-level feature transport regularization. Specifically, we design a Semantic-Guided Local Cross-Attention module that integrates semantic features from vision foundation models into geometric descriptors while preserving local structural continuity. Furthermore, we adapt conditional flow matching as a time-conditioned feature transport regularizer that promotes spatially coherent point-wise recovery. Experimental results on multiple benchmarks demonstrate that SGMatch achieves competitive performance across near-isometric settings and consistent improvements under non-isometric deformations and topological noise.

Tianwei Ye, Xiaoguang Mei, Yifan Xia +3

Wuhan University, Wuhan 430072, China

Functional Maps Geometric Representation Learning Non-Rigid Shape Correspondence 3D Shape Matching

Mar 12, 2026

LatentGeo: Learnable Auxiliary Constructions in Latent Space for Multimodal Geometric Reasoning

Despite recent advances in multimodal reasoning, representing auxiliary geometric constructions remains a fundamental challenge for multimodal large language models (MLLMs). Such constructions are absent from the original diagram and must be introduced before theorems apply. Existing approaches predominantly rely on explicit construction paradigms, including text-based geometric specification, visual-token interleaving during reasoning, and tool-augmented geometric execution. However, these methods either fail to faithfully represent complex spatial relationships, incur representation mismatch between discrete symbols and continuous geometric structures, or rely on external capabilities that hinder end-to-end optimization. To address these limitations, we propose LatentGeo, a framework that learns continuous latent visual representations to internalize auxiliary geometric constructions without pixel-level rendering or external executors. We design a three-stage curriculum that progressively aligns and internalizes these latent representations through auxiliary visual supervision, followed by LaGDPO, a latent-aware reinforcement learning procedure that stabilizes latent representations during policy optimization while improving end-task correctness. To systematically evaluate construction-centric representation quality, we introduce GeoAux, a new benchmark targeting visually dependent geometry problems, and conduct experiments on GeoAux and MathVerse. Results show that LatentGeo achieves substantial gains on geometric reasoning tasks, particularly those requiring auxiliary constructions. Extensive analyses and ablation studies further validate the effectiveness of each component in our framework.

Haiying Xu, Zihan Wang, Song Dai +3

The Hong Kong University of Science and Technology (Guangzhou) · Nankai University · Communication University of China

Multimodal Large Language Models Multimodal Reasoning Latent Visual Reasoning Geometric Representation Learning Mathematical Reasoning Benchmarks RL for Language Model Reasoning

Mar 11, 2026

TractoRC: A Unified Probabilistic Learning Framework for Joint Tractography Registration and Clustering

Diffusion MRI tractography enables in vivo reconstruction of white matter (WM) pathways. Two key tasks in tractography analysis include: 1) tractogram registration that aligns streamlines across individuals, and 2) streamline clustering that groups streamlines into compact fiber bundles. Although both tasks share the goal of capturing geometrically similar structures to characterize consistent WM organization, they are typically performed independently. In this work, we propose TractoRC, a unified probabilistic framework that jointly performs tractogram registration and streamline clustering within a single optimization scheme, enabling the two tasks to leverage complementary information. TractoRC learns a latent embedding space for streamline points, which serves as a shared representation for both tasks. Within this space, both tasks are formulated as probabilistic inference over structural representations: registration learns the distribution of anatomical landmarks as probabilistic keypoints to align tractograms across subjects, and clustering learns streamline structural prototypes that capture geometric similarity to form coherent streamline clusters. To support effective learning of this shared space, we introduce a transformation-equivariant self-supervised strategy to learn geometry-aware and transformation-invariant embeddings. Experiments demonstrate that jointly optimizing registration and clustering significantly improves performance in both tasks over state-of-the-art methods that treat them independently. Code will be made publicly available at https://github.com/yishengpoxiao/TractoRC .

Yijie Li, Xi Zhu, Junyi Wang +3

University of Electronic Science and Technology of China, Chengdu, China · Nanjing University of Science and Technology, Nanjing, China · Harvard Medical School and Brigham and Women’s Hospital, Boston. USA

Clustering Diffusion MRI Geometric Representation Learning

Mar 10, 2026

Geometry-Aware Metric Learning for Cross-Lingual Few-Shot Sign Language Recognition on Static Hand Keypoints

Sign language recognition (SLR) systems typically require large labeled corpora for each language, yet the majority of the world's 300+ sign languages lack sufficient annotated data. Cross-lingual few-shot transfer, pretraining on a data-rich source language and adapting with only a handful of target-language examples, offers a scalable alternative, but conventional coordinate-based keypoint representations are susceptible to domain shift arising from differences in camera viewpoint, hand scale, and recording conditions. This shift is particularly detrimental in the few-shot regime, where class prototypes estimated from only K examples are highly sensitive to extrinsic variance. We propose a geometry-aware metric-learning framework centered on a compact 20-dimensional inter-joint angle descriptor derived from MediaPipe static hand keypoints. These angles are invariant to SO(3) rotation, translation, and isotropic scaling, eliminating the dominant sources of cross-dataset shift and yielding tighter, more stable class prototypes. Evaluated on four fingerspelling alphabets spanning typologically diverse sign languages, ASL, LIBRAS, Arabic Sign Language, and Thai Sign Language, the proposed angle features improve over normalized-coordinate baselines by up to 25 percentage points within-domain and enable frozen cross-lingual transfer that frequently exceeds within-domain accuracy, using a lightweight MLP encoder with about 10^5 parameters. These findings demonstrate that invariant hand-geometry descriptors provide a portable and effective foundation for cross-lingual few-shot SLR in low-resource settings.

Chayanin Chamachot, Kanokphan Lertniponphan

Department of Computer Engineering Faculty of Engineering, Chulalongkorn University

Sign Language Recognition Cross-Lingual Transfer Metric Learning Geometric Representation Learning

Mar 8, 2026

Neural Precoding in Complex Projective Spaces

Deep-learning (DL)-based precoding in multi-user multiple-input single-output (MU-MISO) systems involves training DL models to map features derived from channel coefficients to labels derived from precoding weights. Traditionally, complex-valued channel and precoder coefficients are parameterized using either their real and imaginary components or their amplitude and phase. However, precoding performance depends on magnitudes of inner products between channel and precoding vectors, which are invariant to global phase rotations. Conventional representations fail to exploit this symmetry, leading to inefficient learning and degraded generalization. To address this, we propose a DL framework based on complex projective space (CPS) parameterizations of both the wireless channel and the weighted minimum mean squared error (WMMSE) precoder vectors. By removing the global phase redundancies inherent in conventional representations, the proposed framework enables the DL model to learn geometry-aligned and physically distinct channel-precoder mappings. Two CPS parameterizations based on real-valued embeddings and complex hyperspherical coordinates are investigated and benchmarked against two baseline methods. Simulation results demonstrate substantial improvements in sum-rate performance and generalization, with negligible increase in model complexity.

Zaid Abdullah, Merouane Debbah, Symeon Chatzinotas +1

Interdisciplinary Centre for Security, Reliability, and Trust (SnT) at the University of Luxembourg, Luxembourg. · 6G Research Center, Khalifa University of Science and Technology, Abu Dhabi, UAE.

Wireless Communications Geometric Representation Learning Hyperspherical Representation Learning

Mar 8, 2026

SLNet: A Super-Lightweight Geometry-Adaptive Network for 3D Point Cloud Recognition

We present SLNet, a lightweight backbone for 3D point cloud recognition designed to achieve strong performance without the computational cost of many recent attention, graph, and deep MLP based models. The model is built on two simple ideas: NAPE (Nonparametric Adaptive Point Embedding), which captures spatial structure using a combination of Gaussian RBF and cosine bases with input adaptive bandwidth and blending, and GMU (Geometric Modulation Unit), a per channel affine modulator that adds only 2D learnable parameters. These components are used within a four stage hierarchical encoder with FPS+kNN grouping, nonparametric normalization, and shared residual MLPs. In experiments, SLNet shows that a very small model can still remain highly competitive across several 3D recognition tasks. On ModelNet40, SLNet-S with 0.14M parameters and 0.31 GFLOPs achieves 93.64% overall accuracy, outperforming PointMLP-elite with 5x fewer parameters, while SLNet-M with 0.55M parameters and 1.22 GFLOPs reaches 93.92%, exceeding PointMLP with 24x fewer parameters. On ScanObjectNN, SLNet-M achieves 84.25% overall accuracy within 1.2 percentage points of PointMLP while using 28x fewer parameters. For large scale scene segmentation, SLNet-T extends the backbone with local Point Transformer attention and reaches 58.2% mIoU on S3DIS Area 5 with only 2.5M parameters, more than 17x fewer than Point Transformer V3. We also introduce NetScore+, which extends NetScore by incorporating latency and peak memory so that efficiency can be evaluated in a more deployment oriented way. Across multiple benchmarks and hardware settings, SLNet delivers a strong overall balance between accuracy and efficiency. Code is available at: https://github.com/m-saeid/SLNet.

Mohammad Saeid, Amir Salarpour, Pedram MohajerAnsari +1

Sirjan University of Technology, Sirjan, Iran · Clemson University, Clemson, SC, USA

Point Cloud Learning Point Cloud Classification Geometric Representation Learning 3D Point Cloud Segmentation

Mar 7, 2026

LEPA: Learning Geometric Equivariance in Satellite Remote Sensing Data with a Predictive Architecture

Geospatial foundation models provide precomputed embeddings that serve as compact feature vectors for large-scale satellite remote sensing data. While these embeddings can reduce data-transfer bottlenecks and computational costs, Earth observation (EO) applications can still face geometric mismatches between user-defined areas of interest and the fixed precomputed embedding grid. Standard latent-space interpolation is unreliable in this setting because the embedding manifold is highly non-convex, yielding representations that do not correspond to realistic inputs. We verify this using Prithvi-EO-2.0 to understand the shortcomings of interpolation applied to patch embeddings. As a substitute, we propose a Learned Equivariance-Predicting Architecture (LEPA). Instead of averaging vectors, LEPA conditions a predictor on geometric augmentations to directly predict the transformed embedding. We evaluate LEPA on NASA/USGS Harmonized Landsat-Sentinel (HLS) imagery and ImageNet-1k. Experiments show that standard interpolation achieves a mean reciprocal rank (MRR) below 0.2, whereas LEPA increases MRR to over 0.8, enabling accurate geometric adjustment without re-encoding.

Erik Scheurer, Rocco Sedona, Stefan Kesselheim +1

Jülich Supercomputing Centre (JSC), Forschungszentrum Jülich, 52428 Jülich, Germany · Institute for Visualization and Interactive Systems (VIS), University of Stuttgart, 70569 Stuttgart · School of Engineering and Natural Sciences (SENS), University of Iceland, 102 Reykjavík, Iceland

Equivariant Representation Learning Geometric Representation Learning Remote Sensing Geospatial Foundation Models Predictive Representation Learning