Riemannian Geometry

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Overview

Riemannian geometry extends familiar ideas such as distance, straight lines, and curvature to spaces that may be curved, including covariance matrices, probability distributions, and neural representations. Researchers develop intrinsic tools such as geodesics, curvature, and manifold optimization, alongside computational methods including quotient-manifold algorithms, learned metrics, and geometric flow matching. These methods are used in settings such as EEG covariance classification, graph learning, robot motion planning, and generative modeling, where respecting the data’s geometry may improve analysis or generation. A central challenge is balancing geometric fidelity and theoretical guarantees with scalable computation, while testing when manifold-aware methods truly outperform Euclidean or simpler baselines.

Latest papers

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44 papers2 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

  1. Second-order Sinkhorn Geometry and n\sqrt{n}-Inference

    Oct 27, 2026Tristan Luca Saidi, Florian Gunsilius, Gonzalo MenaOptimal TransportSinkhorn Divergence

    Department of Statistics and Data Science, Carnegie Mellon University · Department of Economics, Emory University

  2. Learning Normal Coordinates on Riemannian Manifolds with Flow Charts

    Oct 20, 2026Edmond CunninghamRiemannian GeometryNormalizing Flows

    University of Massachusetts Amherst

  3. SPD-MetaFormer is what you need for small-data brain decoding

    Oct 7, 2026Zhida Wang, Wei Lyu, Guo Yu +1Small-Data LearningEEG Decoding

  4. Optimization over covariance matrices with a parameterized metric

    Sep 15, 2026Yibang Li, Bamdev Mishra, Pratik Jawanpuria +1Riemannian OptimizationSecond-Order Optimization

    Nanyang Technological University, Singapore · Microsoft India · Indian Institute of Technology Bombay, India

  5. Information-Induced Training Geometry: Exact Reduction, Canonical Completion, and Structured Expressivity

    Sep 14, 2026Zavier LiInformation GeometryRiemannian Optimization

    Xidian University Xi’an, China

  6. Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature

    Aug 23, 2026Hao Li, Yuhan Peng, Junwen DongRiemannian Geometry

    Department of Mathematics, National University of Singapore, Singapore. · School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore. · Chern Institute of Mathematics, Nankai University, Tianjin, China.

  7. The Field Knows: Cross-Dimensional Geometry from Navigation to Black Holes

    Aug 3, 2026Chenghao XuGeometric Representation LearningRiemannian Geometry

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

  8. Sphere Retraction Normalizations

    Aug 2, 2026Jie Zhang, Cheng-Fang Su, Yi-Jui Huang +1Residual LearningRiemannian Geometry

    Department of Computer Science and Information Engineering, National Central University, Taiwan · Department of Applied Mathematics, National Yang Ming Chiao Tung University, Taiwan · Department of Mathematics, National Central University, Taiwan

  9. Landmark shape spaces with induced metrics

    Jul 30, 2026Sarang Joshi, Peter W. Michor, Stefan SommerNon-Rigid Shape CorrespondenceRiemannian Geometry

  10. A Riemannian View on Active Subspaces

    Jul 28, 2026Zachary GreyDimensionality ReductionRiemannian Geometry

  11. Riemannian Geometry for Pre-trained Language Model Embeddings

    Jul 8, 2026Szczepan Konior, Alexandre Quemy, Przemysław Klocek +2Sentence EmbeddingsGeometric Representation Learning

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

  12. On the convergence of graph Laplacians with a symmetric divergence

    Jul 7, 2026Liane XuSinkhorn DivergenceManifold Learning

    Program in Applied and Computational Mathematics, Princeton University

  13. Fisher Width: A Geometric Measure of Complexity on Statistical Manifolds

    Jun 16, 2026Vu Khac KyInformation GeometryRiemannian Geometry

    Department of Mathematics, FPT University, Vietnam

  14. Riemannian Metric Matching for Scalable Geometric Modeling of Distributions

    Jun 12, 2026Jacob Bamberger, Adam Gosztolai, Pierre Vandergheynst +2Geometric Representation LearningMetric Learning

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

  15. Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds

    Jun 11, 2026Srinivas NambirajanMarkov Random FieldsRiemannian Geometry

  16. Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

    Jun 6, 2026Kisung YouRiemannian GeometryOptimal Transport

    Department of Mathematics Baruch College

  17. Geodesic Flow Matching on a Riemannian Degradation Manifold for Blind Image Restoration

    Jun 4, 2026Akshay Janardan Bankar, Ankita Chatterjee, Sayan Banerjee +3Flow MatchingRiemannian Flow Matching

    Samsung Research Institute, Bangalore, India

  18. Knowledge Manifold: A Riemannian Geometric Framework for Semantic Mapping and Geodesic Analysis of Scientific Literature

    Jun 4, 2026Tomonaga Okabe, Kazuhiko KomatsuKnowledge RepresentationManifold Learning

    Department of Aerospace Engineering, Tohoku University, 6-6-01, Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8579, Japan · Research Center for Green X-Tech, Tohoku University, 6-6-11, Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8579, Japan

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

    Jun 2, 2026Li Sun, Zhenhao Huang, Yiding Wang +3GNN ExpressivityGeometric Representation Learning

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

  20. Direct Informed Sampling on Riemannian Manifolds via Loewner Order Lower Bounds

    Jun 1, 2026Phone Thiha Kyaw, Jonathan KellyMotion PlanningRiemannian Geometry

    Space and Terrestrial Autonomous Robotic Systems (STARS) Laboratory, University of Toronto Institute for Aerospace Studies (UTIAS), Toronto, Ontario M3H 5T6, Canada

  21. Riemannian Gradient Descent for Low-Rank Architectures

    Jun 1, 2026Nicholas KnightRiemannian OptimizationLow-Rank Approximation

    NVIDIA

  22. Riemannian Optimization for Hadamard Products of Low-Rank Matrices

    May 31, 2026Pratik Jawanpuria, Ankish Chandresh, Bamdev MishraRiemannian OptimizationLow-Rank Approximation

    Centre for Machine Intelligence and Data Science, Indian Institute of Technology Bombay, India · Microsoft India

  23. Riemannian-Manifold Steering: Geometry-Aware Generative Autoencoders for Label-Free Steering

    May 24, 2026Narmeen Oozeer, Shivam Raval, Philip Quirke +4Language Model SteeringRepresentation Steering

    Martian · Harvard University · Thoughtworks +1

  24. Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions

    May 22, 2026Willem Diepeveen, Deanna NeedellManifold LearningDimensionality Reduction

    Department of Mathematics University of California, Los Angeles Los Angeles, CA 90095, USA

  25. Disentanglement Beyond Generative Models with Riemannian ICA

    May 21, 2026Edmond CunninghamDisentangled Representation LearningIndependent Component Analysis

    University of Massachusetts Amherst

  26. Riemannian geometry meets fMRI: the advantages of modeling correlation manifolds and eigenvector subspaces

    May 21, 2026Mario Severino, Manuela Moretto, Robert A. McCutcheon +1Functional ConnectivityNeuroimaging

    Department of Information Engineering, University of Padova, Padova, Italy · Department of Neuroimaging, Institute of Psychiatry, Psychology and Neuroscience (IoPPN), King’s College London, London, United Kingdom · Department of Psychiatry, University of Oxford, Warneford Hospital, Warneford Ln, Headington, Oxford OX3 7JX, United Kingdom +2

  27. Riemannian Networks over Full-Rank Correlation Matrices

    May 18, 2026Ziheng Chen, Xiaojun Wu, Bernhard Schölkopf +1Riemannian Neural NetworksGeometric Representation Learning

    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

  28. Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learning

    May 6, 2026Tushar Das, Subrata Dutta, Sarmistha Neogy +1Geometric Representation LearningMetric Learning

    National Institute of Technology Jamshedpur · Jadavpur University · Machine Vision and Intelligence Lab National Institute of Technology Jamshedpur

  29. Entropic Riemannian Neural Optimal Transport

    May 5, 2026Alessandro Micheli, Silvia Sapora, Anthea Monod +1Riemannian GeometryOptimal Transport

    Imperial College London London, UK · University of Oxford Oxford, UK · Statens Serum Institut Copenhagen, Denmark +1

  30. Flow Matching on Symmetric Spaces

    May 5, 2026Francesco Ruscelli, Ferdinando Zanchetta, Rita FioresiFlow MatchingRiemannian Flow Matching

    Institute for Mathematics University of Heidelberg 69120 Heidelberg, Germany · Department of Pharmacy and Biotechnologies University of Bologna Bologna, Italy