# An accelerated first-order method for non-convex optimization on manifolds

arXiv: https://arxiv.org/abs/2008.02252 · HTML: https://mlsanity.com/paper/2008.02252/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2020.

## Built on by (most cited first, 15 of 15)

- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Optimization on manifolds: A symplectic approach](https://mlsanity.com/paper/2107.11231/) (2021)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2502.01512/) (2025)
- [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)
- [Understanding Riemannian Acceleration via a Proximal Extragradient Framework](https://mlsanity.com/paper/2111.02763/) (2021)
- [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)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Quantitative Convergences of Lie Group Momentum Optimizers](https://mlsanity.com/paper/2405.20390/) (2024)
- [Differentially Private Geodesic Regression](https://mlsanity.com/paper/2504.11304/) (2025)
- [Stable Coresets via Posterior Sampling: Aligning Induced and Full Loss Landscapes](https://mlsanity.com/paper/2511.17399/) (2025)

## Abstract (arXiv)

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.