Riemannian Optimization

Ask about this topic

Overview

Riemannian optimization studies how to minimize objectives when parameters lie on curved spaces such as spheres, Stiefel manifolds, and spaces of low-rank matrices. Common methods project gradients onto tangent spaces and take feasible steps with retractions or geodesics; natural-gradient, mirror-descent, and second-order methods adapt updates to the chosen metric. These techniques are used for orthogonality-constrained learning, low-rank model adaptation, covariance and probability optimization, tensor networks, and robotics. A central tension is balancing geometric fidelity and convergence guarantees against the cost of computing geometric updates, while empirical gains over well-tuned Euclidean optimizers are not consistent across settings.

Latest papers

CardsList

76 papers8 papers in the last four weeks, down 47% on the four weeks before. 0.0% of all new papers.

  1. Corrected Block Sparse Spectral Optimization: Exact Models, Finite Identification, and Local Dynamics

    Oct 19, 2026Omid Shams SolariSpectral OptimizationStructured Sparsity

    sAIm Labs

  2. A Riemannian Geometry for Low-rank Adaptation

    Oct 6, 2026Shoichiro Takeda, Shin'ya Yamaguchi, Satoshi Suzuki +1Riemannian OptimizationLow-Rank Adaptation

    NTT, Inc.

  3. How Inefficient Is Natural Gradient Descent? From Exact Optimality to Θ( \sqrt{ \log d } ) Divergence

    Oct 5, 2026Guni Sharon, Alan KuhnleInformation GeometryRiemannian Optimization

    Department of Computer Science and Engineering, Texas A&M University, College Station, TX 77843, USA.

  4. The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective

    Oct 5, 2026Xingrun Li, Sho Kuno, Yusuke Mukuta +2Riemannian OptimizationImplicit Bias

    The University of Tokyo · RIKEN

  5. Hyperbolic Graph Representation Learning: Embed in One Metric, Optimize with Another

    Oct 5, 2026Federico Larroca, Paola Bermolen, Marcelo Fiori +1Riemannian OptimizationHierarchical Representation Learning

    Facultad de Ingenier´ıa Universidad de la Rep´ublica Uruguay

  6. Hamiltonian Metric Learning and Energy-Based Training: A Dissipative Geometric Framework for Optimization

    Oct 4, 2026Sparsho Chakraborty, Mohammad Alamgir, Nishanth M +3Neural Network OptimizationRiemannian Optimization

    Department of Physics, School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Odisha, India · Department of Electronics and Computer Science, Narula Institute of Technology, West Bengal, India · Department of Computer Science and Engineering, School of Electrical and Computer Sciences, Indian Institute of Technology Bhubaneswar, Odisha, India +1

  7. Riemannian Difference-of-Convex Optimization for K-Means Clustering

    Sep 28, 2026Meng Xu, Bo Jiang, Hanfu Zhang +2Riemannian OptimizationClustering

    AMSS, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, Beijing, China · Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, China · Ministry of Education Key Laboratory of Mathematics and Information Networks, School of Mathematical Sciences, Beijing University of Posts and Telecommunications, Beijing, China +1

  8. Rotated Manifold Optimization for Low-Rank Adaptation

    Sep 28, 2026Yuhui Ding, Javier Zazo, James HensmanRiemannian OptimizationLow-Rank Adaptation

    ETH Zurich · Microsoft Research Cambridge

  9. Retraction-Based Gradient Projection Algorithms on Manifolds

    Sep 25, 2026Conglong Xu, Hao WuGradient ProjectionRiemannian Optimization

    Department of Mathematics, The George Washington University, Phillips Hall, Room 739, 801 22nd Street, N.W., Washington, DC 20052, USA.

  10. Riemannian Structure and Optimization for a Class of Low-Parametric Orthogonal Matrices

    Sep 23, 2026Ali Aliev, Maxim RakhubaRiemannian OptimizationMatrix Optimization

    HSE University

  11. Guaranteed Low-Rank Tensor Recovery from Modewise Measurements via Normalized Block-Weighted Riemannian Gradient Descent

    Sep 21, 2026Yushi Zhou, Feng ZhangRiemannian OptimizationTensor Decomposition

    School of Mathematics and Statistics, Southwest University, Chongqing, China.

  12. Locally Private Inference for Riemannian Stochastic Optimization

    Sep 18, 2026Xiaotian Chang, Yangdi Jiang, Qirui HuRiemannian OptimizationConfidence Region Estimation

    School of Physical and Mathematical Sciences, Nanyang Technological University · School of Statistics and Data Science, Shanghai University of Finance and Economics.

  13. Stiefel-AdamW: Geometry-Aware AdamW for Linear Factorization Blocks

    Sep 17, 2026Emanuele Zangrando, Marco Sutti, Francesco TudiscoDeep Learning OptimizationRiemannian Optimization

    Gran Sasso Science Institute, L’Aquila, Italy · University of Edinburgh, Edinburgh, UK

  14. Directions That Don't Drift: Stiefel Manifold Routing for Transformer Attention

    Sep 16, 2026Rubén Darío GuerreroStiefel Manifold OptimizationRiemannian Optimization

    NeuroTechNet S.A.S., 1108831, Bogotá, Colombia

  15. 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

  16. Near-Optimal Nonconvex Matrix Completion

    Sep 15, 2026Jian-Feng Cai, Xiliang Lu, Juntao YouRiemannian OptimizationNonconvex Optimization

    Department of Mathematics, Hong Kong University of Science and Technology. · School of Mathematics and Statistics, Hubei Center for Applied Mathematics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China. · School of Artificial Intelligence, Hubei Center for Applied Mathematics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China.

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

    Sep 14, 2026Zavier LiInformation GeometryRiemannian Optimization

    Xidian University Xi’an, China

  18. Riemannian ascent--descent for nonconvex nonconcave minimax landscapes: convergence to basin saddle points and applications to distributionally robust optimization

    Sep 12, 2026Rishabh Dixit, Pranav Upadrashta, Alex CloningerMin-Max OptimizationWasserstein Distributionally Robust Optimization

    Department of Mathematics, UC San Diego · Department of Mathematics and Halıcıoglu Data Science Institute, UC San Diego

  19. Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds

    Sep 12, 2026Zhanyuan Cai, Emre Sahinoglu, Shahin ShahrampourRiemannian OptimizationDecentralized Optimization

    Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115

  20. Riemannian Optimization for Multi-Player Quantum Games on Product Unitary Manifolds

    Sep 7, 2026Alireza Habibi, Setareh MaghsudiMulti-Agent System OptimizationRiemannian Optimization

    Faculty of Electrical Engineering and Information Technology, Ruhr University Bochum, Bochum, Germany

  21. Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size

    Sep 3, 2026A. AfhamRiemannian OptimizationOptimization Convergence Analysis

    Centre for Quantum Technologies, National University of Singapore, Singapore

  22. LoRA-TSD: Tangent-Space Spectral Descent for LoRA via Muon-Style Updates

    Sep 2, 2026Dmitrii Andriianov, Andrey Veprikov, Aleksandr BeznosikovLow-Rank AdaptationParameter-Efficient Fine-Tuning

    BRAIn Lab · Basic Research of Artificial Intelligence Laboratory (BRAIn Lab) · SB AI Lab +1

  23. Stochastic Optimization of Tree Tensor Networks

    Sep 1, 2026Marius Willner, Maximilian Scharf, André Uschmajew +2Tensor NetworksRiemannian Optimization

    Institute of Mathematics, University of Augsburg, 86159 Augsburg, Germany · Centre for Advanced Analytics and Predictive Sciences, University of Augsburg, 86159 Augsburg, Germany · Tensor AI Solutions GmbH, 89284 Pfaffenhofen an der Roth, Germany +1

  24. Denoising as Projection: Constrained Optimization with Gradient-Guided Diffusion

    Aug 30, 2026Runyu Zhang, Jiawei Zhang, Gioele Zardini +2Riemannian OptimizationDiffusion Guidance

    MIT · University of Wisconsin–Madison

  25. Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

    Aug 20, 2026Andrew GracykRiemannian OptimizationGradient Descent

    Department of Mathematics Purdue University West Lafayette, IN 47907, USA

  26. The Advective Fisher-Rao Geometry of Deterministic Measure Transport

    Aug 12, 2026Benjamin Gess, Johannes MüllerInformation GeometryRiemannian Optimization

    Institut für Mathematik, Technische Universität Berlin & Max-Planck-Institut für Mathematik in den Naturwissenschaften · Institut für Mathematik, Technische Universität Berlin

  27. Muon on the Stiefel Manifold Admits an Exact Closed-Form Update

    Aug 6, 2026Mikhail Solonko, Molozhavenko Alexander, Maxim RakhubaStiefel Manifold OptimizationRiemannian Optimization

    HSE University, Moscow, Russia.

  28. Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration

    Aug 3, 2026Shashwat Kumar, Arafat Rahman, Anuj Srivastava +1Riemannian OptimizationTensor Decomposition

    Department of Biomedical Engineering, Johns Hopkins University, USA · Department of Systems and Information Engineering, University of Virginia, USA · Department of Applied Mathematics and Statistics, Johns Hopkins University, USA +1

  29. Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks

    Jul 28, 2026Pengcheng ChengRiemannian OptimizationImplicit Regularization

    School of Mathematics, Jilin University Changchun, 130012, China

  30. Retraction-Free Optimization over the Stiefel Manifold for the LoRA Fine-Tuning

    Jul 28, 2026Yuan Zhang, Jiang Hu, Zhijian Lai +2Stiefel Manifold OptimizationRiemannian Optimization

    Center for Data Science, Peking University, Beijing 100871, China · Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China · Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China +2