mlsanityData to 7 Oct 2026

Escaping from saddle points on Riemannian manifolds

arXiv 1906.07355 ↗First posted 18 Jun 2019math.OC cs.LG stat.MLcited by 33

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Abstract (arXiv)

We consider minimizing a nonconvex, smooth function f on a Riemannian manifold M. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as 1/ε2 on the accuracy ε, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension d, hence is almost dimension-free. The rate also has a polynomial dependence on the parameters describing the curvature of the manifold and the smoothness of the function. While the unconstrained problem (Euclidean setting) is well-studied, our result is the first to prove such a rate for nonconvex, manifold-constrained problems.

Key facts. Cited by 33 papers in the map (data through 7 Oct 2026). First posted on arXiv 18 Jun 2019.

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Citing papers per year: 2021: 8; 2022: 10; 2023: 6; 2024: 4; 2025: 4; 2026: 1.

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  1. 2022Riemannian Score-Based Generative Modellingcited by 216
  2. 2022Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifoldcited by 50
  3. 2022Efficiently Escaping Saddle Points in Bilevel Optimizationcited by 19
  4. 2022First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spacescited by 10
  5. 2023Riemannian stochastic optimization methods avoid strict saddle pointscited by 8
  6. 2021Escaping strict saddle points of the Moreau envelope in nonsmooth optimizationcited by 7
  7. 2022Towards a Theoretical Foundation of Policy Optimization for Learning Control Policiescited by 6
  8. 2023Minimizing Dynamic Regret on Geodesic Metric Spacescited by 6
  9. 2024Efficient Sampling on Riemannian Manifolds via Langevin MCMCcited by 6
  10. 2022Riemannian accelerated gradient methods via extrapolationcited by 5
  11. 2025Riemannian Optimization on Relaxed Indicator Matrix Manifoldcited by 5
  12. 2021No-go Theorem for Acceleration in the Hyperbolic Planecited by 4
21 more
  1. 2021Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithmscited by 4
  2. 2022Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penaltiescited by 4
  3. 2023Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded Geometric Penaltiescited by 4
  4. 2025Decentralized Online Riemannian Optimization Beyond Hadamard Manifoldscited 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. 2021Understanding Riemannian Acceleration via a Proximal Extragradient Frameworkcited by 3
  8. 2022Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspectivecited by 3
  9. 2024A Framework for Bilevel Optimization on Riemannian Manifoldscited by 3
  10. 2022Riemannian stochastic approximation algorithmscited by 2
  11. 2023Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computationcited by 2
  12. 2024Stochastic Modified Flows for Riemannian Stochastic Gradient Descentcited by 2
  13. 2021Escaping Saddle Points with Compressed SGDcited by 1
  14. 2023Extragradient Type Methods for Riemannian Variational Inequality Problemscited by 1
  15. 2024Avoiding strict saddle points of nonconvex regularized problemscited by 1
  16. 2025Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifoldscited by 1
  17. 2026Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functionscited by 1
  18. 2021Asymptotic Escape of Spurious Critical Points on the Low-rank Matrix Manifoldcited by 0
  19. 2022Accelerated Multiplicative Weights Update Avoids Saddle Points almost alwayscited by 0
  20. 2023Faster Riemannian Newton-type Optimization by Subsampling and Cubic Regularizationcited by 0
  21. 2025Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Managementcited by 0