mlsanityData to 7 Oct 2026

A Brief Introduction to Manifold Optimization

arXiv 1906.05450 ↗First posted 13 Jun 2019math.OCcited by 35

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Abstract (arXiv)

Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry and etc. One of the main challenges usually is the non-convexity of the manifold constraints. By utilizing the geometry of manifold, a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold. From this perspective, intrinsic structures, optimality conditions and numerical algorithms for manifold optimization are investigated. Some recent progress on the theoretical results of manifold optimization are also presented.

Key facts. Cited by 35 papers in the map (data through 7 Oct 2026). First posted on arXiv 13 Jun 2019.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 7; 2022: 10; 2023: 7; 2024: 1; 2025: 3; 2026: 7.

35 papers.

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  1. 2022Tutorial on amortized optimizationcited by 83
  2. 2022Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifoldcited by 50
  3. 2022Bayesian Learning for Neural Networks: an algorithmic surveycited by 22
  4. 2023A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifoldcited by 21
  5. 2021Fast and Efficient MMD-based Fair PCA via Optimization over Stiefel Manifoldcited by 18
  6. 2023SDDM: Score-Decomposed Diffusion Models on Manifolds for Unpaired Image-to-Image Translationcited by 13
  7. 2026Mano: Restriking Manifold Optimization for LLM Trainingcited by 12
  8. 2021Dynamical Wasserstein Barycenters for Time-series Modelingcited by 10
  9. 2022Momentum Stiefel Optimizer, with Applications to Suitably-Orthogonal Attention, and Optimal Transportcited by 10
  10. 2022First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spacescited by 10
  11. 2025Landing with the Score: Riemannian Optimization through Denoisingcited by 7
  12. 2021Simulation-based Bayesian inference for multi-fingered robotic graspingcited by 5
23 more
  1. 2022Riemannian Natural Gradient Methodscited by 5
  2. 2024Nonconvex Federated Learning on Compact Smooth Submanifolds With Heterogeneous Datacited by 5
  3. 2025Riemannian Optimization on Relaxed Indicator Matrix Manifoldcited by 5
  4. 2022Optimization on Manifolds via Graph Gaussian Processescited by 4
  5. 2021Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimizationcited by 3
  6. 2021On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimizationcited by 3
  7. 2022A Variance-Reduced Stochastic Gradient Tracking Algorithm for Decentralized Optimization with Orthogonality Constraintscited by 2
  8. 2023A Block Coordinate Descent Method for Nonsmooth Composite Optimization under Orthogonality Constraintscited by 2
  9. 2025RL-DAUNCE: Reinforcement Learning-Driven Data Assimilation with Uncertainty-Aware Constrained Ensemblescited by 2
  10. 2021Tensor completion using geodesics on Segre manifoldscited by 1
  11. 2021Vector Transport Free Riemannian LBFGS for Optimization on Symmetric Positive Definite Matrix Manifoldscited by 1
  12. 2022Riemannian Stochastic Gradient Method for Nested Composition Optimizationcited by 1
  13. 2023Decentralized Riemannian natural gradient methods with Kronecker-product approximationscited by 1
  14. 2023Decentralized Weakly Convex Optimization Over the Stiefel Manifoldcited by 1
  15. 2026GeoNorm: Unify Pre-Norm and Post-Norm with Geodesic Optimizationcited by 1
  16. 2026Geodesic Gradient Descent: A Generic and Learning-rate-free Optimizer on Objective Function-induced Manifoldscited by 1
  17. 2022Changes from Classical Statistics to Modern Statistics and Data Sciencecited by 0
  18. 2023Tame Riemannian Stochastic Approximationcited by 0
  19. 2023Deep Metric Tensor Regularized Policy Gradientcited by 0
  20. 2026A second-order method landing on the Stiefel manifold via Newtonunicodex2013Schulz iterationcited by 0
  21. 2026Fitting Unknown Number of Hyperplanes with Manifold Optimizationcited by 0
  22. 2026Retraction-Free Optimization over the Stiefel Manifold for the LoRA Fine-Tuningcited by 0
  23. 2026Incremental Aggregation on the Grassmannian for Asynchronous Eigenspace Computationcited by 0