Riemannian Optimization

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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.

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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.

RODR: Riemannian Orthogonally Decoupled Regularization for Disentangled Manifold Representation
Jul 27, 2026Jiayu Zhu, Wenlai ZhaoRiemannian Optimization Point Cloud Denoising
Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, China
Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers
Jul 24, 2026Zhangyong Liang, Huanhuan GaoNeural PDE Solvers Riemannian Optimization
National Center for Applied Mathematics Tianjin University Tianjin, 300072, China · School of Mechanical and Aerospace Engineering Jilin University Changchun, 130025, China
Regularized Optimization on Grassmann Manifold: Theory, Algorithm and Applications
Jul 23, 2026Zhuan Liang, Zheng ZhaiSpectral Regularization Riemannian Optimization
Department of Statistics, Faculty of Arts and Sciences at Beijing Normal University, Zhuhai
Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions
Jul 22, 2026Zhanyuan Cai, Emre Sahinoglu, Shahin ShahrampourOnline Convex Optimization Riemannian Optimization
Department of Mechanical & Industrial Engineering at Northeastern University, Boston, MA 02115, USA
Learning Adaptive Solvers for Distributed Factor Graph Optimization on Matrix Lie Groups
Jul 9, 2026Jaeho Shin, Maani Ghaffari, Yulun TianFactor Graph Optimization Riemannian Optimization
University of Michigan
Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates
Jul 8, 2026Jianing Liu, Dong H. ZhangGauss-Newton Optimization Riemannian Optimization
State Key Laboratory of Chemical Reaction Dynamics and Department of Chemical Physics, University of Science and Technology of China, Hefei, 230026, China · State Key Laboratory of Chemical Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, 116023, China. · University of Chinese Academy of Sciences, Beijing, 100049, China +1
Restricted Dynamic Geometric Complexity: Path-Space Reduction and Möbius--Jacobi Response
Jul 8, 2026Zavier LiRiemannian Optimization Constrained Optimization
Xidian University, Xi’an, China
Optimization Geometrodynamics: Variational Reduction and Interaction Curvature
Jul 7, 2026Zavier LiRiemannian Optimization Second-Order Optimization
Xidian University Xi’an, China
Diffeomorphic Optimization
Jul 1, 2026Ludwig Winkler, Andrew Leaver-Fay, Joseph Kleinhenz +1Riemannian Optimization Protein Design
Microsoft Research AI4Science · Prescient Design, Genentech Roche
Decentralized Pose Graph Riemannian Optimization for Object-based Multi-Robot SLAM
Jun 23, 2026Yixian Zhao, Yan Huang, Yang Xu +2Simultaneous Localization and Mapping Riemannian Optimization
College of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China · Division of Decision and Control Systems, School of EECS, KTH Royal Institute of Technology, Stockholm, SE-100 44, Sweden · Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai, 201210, China
Muown Implicitly Performs Angular Step-size Decay
Jun 22, 2026Florian Hübler, Kai Lion, Antonio Orvieto +1Learning Rate Scheduling Riemannian Optimization
Department of Computer Science · ETH Zurich, Switzerland · ELLIS Institute Tübingen, MPI-IS +1
LoRA-Muon: Spectral Steepest Descent on the Low-Rank Manifold
Jun 11, 2026Franz Louis Cesista, Katherine Crowson, Cédric Simal +1Deep Learning Optimization Riemannian Optimization
Ateneo de Manila University · EleutherAI · NaXys +1
A Riemannian Approach to Low-Rank Optimal Transport
Jun 10, 2026Pratik Jawanpuria, Bamdev MishraRiemannian Optimization Second-Order Optimization
Centre for Machine Intelligence and Data Science, IIT Bombay · Microsoft India
LieIPM: Lie Group Interior Point Method for Direct Trajectory Optimization of Rigid Bodies
Jun 9, 2026Sangli Teng, Ruiqi Zhang, Tzu-Yuan Lin +5Trajectory Optimization Riemannian Optimization
University of California, Berkeley · MIT · Ohio University +1
Generalized-CVO: Fast and Correspondence-Free Local Point Cloud Registration with Second Order Riemannian Optimization
Jun 8, 2026Ray Zhang, Marcus Greiff, Thomas Lew +1Riemannian Optimization LiDAR Odometry
Toyota Research Institute, 4440 El Camino Real, Los Altos, California, USA
Riemannian Gradient Descent for Low-Rank Architectures
Jun 1, 2026Nicholas KnightRiemannian Optimization Low-Rank Approximation
NVIDIA
Riemannian Optimization for Hadamard Products of Low-Rank Matrices
May 31, 2026Pratik Jawanpuria, Ankish Chandresh, Bamdev MishraRiemannian Optimization Low-Rank Approximation
Centre for Machine Intelligence and Data Science, Indian Institute of Technology Bombay, India · Microsoft India
Riemannian Stochastic Optimization for Sufficient Dimension Reduction
May 29, 2026Thibault Pautrel, François PortierStochastic Optimization Riemannian Optimization
Laboratoire des Signaux et Systèmes (L2S), CentraleSupélec, Université Paris-Saclay, Gif-sur-Yvette, France · ENSAI, CREST, Bruz, France
Information-Geometric Optimization on Spheres
May 27, 2026Vladimir Ja\' cimovićInformation Geometry Kuramoto Model
Faculty of Natural Sciences and Mathematics University of Montenegro Cetinjski put bb., 81000 Podgorica Montenegro
Optimization over the intersection of manifolds
May 21, 2026Yan Yang, Bin Gao, Ya-xiang YuanRiemannian Optimization
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China · State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China
CMC-Opt: Constraint Manifold with Corners for Inequality-Constrained Optimization
May 20, 2026Yetong Zhang, Frank DellaertKinodynamic Motion Planning Riemannian Optimization
College of Computing Georgia Institute of Technology Atlanta, USA
Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization
May 18, 2026Chuansen Peng, Xiaojing ShenRiemannian Optimization Dynamic Graph Learning
School of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China
Distributed Pose Graph Optimization via Continuous Riemannian Dynamics
May 11, 2026Jaeho Shin, Maani Ghaffari, Yulun TianRiemannian Optimization Multi-Robot Systems
University of Michigan Ann Arbor, MI 48109, USA
Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds
May 10, 2026Yibang Li, Bihari Lal Pandey, Ravi Sah +4Stiefel Manifold Optimization Riemannian Optimization
Nanyang Technological University, Singapore. · Indian Institute of Technology Bombay, India. · University of Sydney, Australia. +1
Demystifying Manifold Constraints in LLM Pre-training
May 6, 2026Kang An, Jiaxiang Li, Donald Goldfarb +1Language Model Pretraining Riemannian Optimization
Rice University · Independent Researcher · Columbia University
Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations
May 4, 2026Benyamin GhojoghRiemannian Optimization Riemannian Geometry
Waterloo, Ontario, Canada
Model Compression with Exact Budget Constraints via Riemannian Manifolds
May 1, 2026Michael Helcig, Dan AlistarhModel Compression Riemannian Optimization
ETH Zürich · IST Austria
Safety-Critical Contextual Control via Online Riemannian Optimization with World Models
Apr 21, 2026Tongxin LiRiemannian Optimization Safety-Critical Control
Member, IEEE
Natural gradient descent with momentum
Apr 16, 2026Anthony Nouy, Agustín SomacalDeep Learning Optimization Riemannian Optimization
Centrale Nantes, Nantes Universit´e, Laboratoire de Math´ematiques Jean Leray UMR CNRS 6629
Natural Riemannian gradient for learning functional tensor networks
Apr 10, 2026Nikolas Klug, Michael Ulbrich, André Uschmajew +1Tensor Networks Riemannian Optimization
Institute of Mathematics, University of Augsburg, 86159 Augsburg, Germany · Department of Mathematics, Technical University of Munich, 85748 Garching b. München, Germany · Institute of Mathematics & Centre for Advanced Analytics and Predictive Sciences, University of Augsburg, 86159 Augsburg, Germany

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RODR: Riemannian Orthogonally Decoupled Regularization for Disentangled Manifold Representation

Jiayu Zhu, Wenlai Zhao

Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, China

Riemannian Optimization Point Cloud Denoising Point Cloud Learning Geometric Representation Learning Manifold Learning

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Point cloud denoising is essentially a geometric recovery task that aims to reconstruct the intrinsic structure of a smooth 2D Riemannian manifold embedded in R^3 from noisy, discrete ambient-space samples. Despite the remarkable progress of modern manifold-aware encoders and generative transport models in geometric representation learning, a fundamental objective-geometry mismatch remains underexplored. Theoretically, we identified that this mismatched coupling leads to geometric gradient interference, where conflicting optimization objectives result in structural degradation and point clustering. We introduce Riemannian Orthogonally Decoupled Regularization (RODR) to reformulate the optimization trajectory by disentangling the normal (fitting) and tangential (distribution) components. Guided by a vector-attention and entropy-aware adaptive strategy, RODR effectively preserves high-fidelity geometric details while maintaining sampling uniformity. Experiments demonstrate that RODR reaches performance comparable to state-of-the-art baselines and suggests improved distribution regularity and reduced local aggregation effectively. Our work establishes a generic and interpretable framework for disentangled geometric optimization in point cloud processing.