Riemannian Optimization

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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

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

  1. NS-RGS: Newton-Schulz based Riemannian gradient method for orthogonal group synchronization

    Apr 7, 2026Haiyang Peng, Deren Han, Xin Chen +1Lie Group MethodsRiemannian Optimization

  2. Inversion-Free Natural Gradient Descent on Riemannian Manifolds

    Apr 3, 2026Dario Draca, Takuo Matsubara, Minh-Ngoc TranNatural Gradient MethodsRiemannian Optimization

    School of Mathematics and Statistics, University of Sydney, Australia · University of Sydney Business School, Australia

  3. Random Coordinate Descent on the Wasserstein Space of Probability Measures

    Apr 2, 2026Yewei Xu, Qin LiNonconvex OptimizationRiemannian Optimization

    Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1388, USA

  4. Mirror Descent on Riemannian Manifolds

    Mar 18, 2026Jiaxin Jiang, Lei Shi, Jiyuan TanRiemannian OptimizationStochastic Optimization Convergence

    School of Mathematical Sciences, Fudan University, Shanghai 200433, China · Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China

  5. Parallelised Differentiable Straightest Geodesics for 3D Meshes

    Mar 16, 2026Hippolyte Verninas, Caner Korkmaz, Stefanos Zafeiriou +2Riemannian GeometryGeometric Representation Learning

    Imperial College London

  6. Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization

    Mar 12, 2026Haotong Duan, Zhongming Chen, Ngai WongTensor NetworksRiemannian Optimization

    Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China. · Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong.

  7. Information Theoretic Bayesian Optimization over the Probability Simplex

    Mar 10, 2026Federico Pavesi, Antonio Candelieri, Noémie JaquierBayesian OptimizationConstrained Optimization

    Department of Robotics, Perception and Learning, KTH Royal Institute of Technology, Stockholm, Sweden · Department of Economics, Management and Statistics, University of Milan, Bicocca, Milan, Italy

  8. Riemannian Optimization in Modular Systems

    Mar 4, 2026Christian Pehle, Jean-Jacques SlotineContraction TheoryRiemannian Optimization

    Cold Spring Harbor Laboratory Cold Spring Harbor, NY 11724 · Massachusetts Institute of Technology Cambridge, MA 02139

  9. Geodesic Gradient Descent: A Generic and Learning-rate-free Optimizer on Objective Function-induced Manifolds

    Feb 28, 2026Liwei Hu, Guangyao Li, Wenyong Wang +2Neural Network OptimizationRiemannian Optimization

    the School of Information Science and Engineering, Hebei University of Science and Technology, Shijiazhuang 050091, China · the School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China · the International Institute of Next Generation Internet, Macau University of Science and Technology, Macau 519020

  10. MSINO: Curvature-Aware Sobolev Optimization for Manifold Neural Networks

    Feb 26, 2026Suresan ParethRiemannian Neural NetworksNeural Network Optimization

  11. Smooth Quasar-Convex Optimization with Constraints

    Oct 2, 2025David Martínez-RubioRiemannian OptimizationNonconvex Optimization

    IMDEA Software Institute · Madrid, Spain

  12. Geometrically Principled Randomized Optimization for Efficient LLM Training

    Oct 2, 2025Sahar Rajabi, Nayeema Nonta, Sirisha RambhatlaLanguage Model PretrainingDeep Learning Optimization

    CriticalML Lab, University of Waterloo

  13. Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning

    Jul 29, 2025Marius Willner, Marco Trenti, Dirk LebiedzTensor NetworksRiemannian Optimization

    Chair of Mathematical Data Science, University of Augsburg, Augsburg, Germany · Tensor AI Solutions GmbH, Pfaffenhofen a.d. Roth, Germany · Institute for Numerical Mathematics, University of Ulm, Ulm, Germany

  14. Inexact calculus of variations on the hyperspherical tangent bundle with connections to the attention mechanism

    Jul 21, 2025Andrew GracykRiemannian OptimizationTransformer Attention

  15. Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates

    Mar 31, 2025Flavia Esposito, Andersen AngRiemannian OptimizationConstrained Optimization

    Department of Mathematics, University of Bari Aldo Moro, Bari, 70125, Italy. · School of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, United Kingdom.

  16. Generalizing Adam to Manifolds for Efficiently Training Transformers

    May 26, 2023Benedikt BrantnerStiefel Manifold OptimizationRiemannian Optimization

    Max-Planck-Institut für Plasmaphysik Boltzmannstraße 2, 85748 Garching, Deutschland · Technische Universität München, Zentrum Mathematik Boltzmannstraße 3, 85748 Garching, Deutschland