# Escaping from saddle points on Riemannian manifolds

arXiv: https://arxiv.org/abs/1906.07355 · HTML: https://mlsanity.com/paper/1906.07355/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2019.

## Built on by (most cited first, 30 of 33)

- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold](https://mlsanity.com/paper/2209.09211/) (2022)
- [Efficiently Escaping Saddle Points in Bilevel Optimization](https://mlsanity.com/paper/2202.03684/) (2022)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Riemannian stochastic optimization methods avoid strict saddle points](https://mlsanity.com/paper/2311.02374/) (2023)
- [Escaping strict saddle points of the Moreau envelope in nonsmooth optimization](https://mlsanity.com/paper/2106.09815/) (2021)
- [Towards a Theoretical Foundation of Policy Optimization for Learning Control Policies](https://mlsanity.com/paper/2210.04810/) (2022)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)
- [Efficient Sampling on Riemannian Manifolds via Langevin MCMC](https://mlsanity.com/paper/2402.10357/) (2024)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Riemannian Optimization on Relaxed Indicator Matrix Manifold](https://mlsanity.com/paper/2503.20505/) (2025)
- [No-go Theorem for Acceleration in the Hyperbolic Plane](https://mlsanity.com/paper/2101.05657/) (2021)
- [Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms](https://mlsanity.com/paper/2102.02837/) (2021)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective](https://mlsanity.com/paper/2209.15130/) (2022)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded Geometric Penalties](https://mlsanity.com/paper/2305.16186/) (2023)
- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)
- [On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization](https://mlsanity.com/paper/2110.12121/) (2021)
- [Understanding Riemannian Acceleration via a Proximal Extragradient Framework](https://mlsanity.com/paper/2111.02763/) (2021)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Riemannian stochastic approximation algorithms](https://mlsanity.com/paper/2206.06795/) (2022)
- [Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation](https://mlsanity.com/paper/2306.16617/) (2023)
- [Stochastic Modified Flows for Riemannian Stochastic Gradient Descent](https://mlsanity.com/paper/2402.03467/) (2024)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Escaping Saddle Points with Compressed SGD](https://mlsanity.com/paper/2105.10090/) (2021)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [Avoiding strict saddle points of nonconvex regularized problems](https://mlsanity.com/paper/2401.09274/) (2024)
- [Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions](https://mlsanity.com/paper/2607.20316/) (2026)
- [Asymptotic Escape of Spurious Critical Points on the Low-rank Matrix Manifold](https://mlsanity.com/paper/2107.09207/) (2021)

## 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.