# Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature

arXiv: https://arxiv.org/abs/1609.02901 · HTML: https://mlsanity.com/paper/1609.02901/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2016.

## Built on by (most cited first, 7 of 7)

- [An Introduction to Hamiltonian Monte Carlo Method for Sampling](https://mlsanity.com/paper/2108.12107/) (2021)
- [Stereographic Markov Chain Monte Carlo](https://mlsanity.com/paper/2205.12112/) (2022)
- [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)
- [Sampling and estimation on manifolds using the Langevin diffusion](https://mlsanity.com/paper/2312.14882/) (2023)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions](https://mlsanity.com/paper/2607.20316/) (2026)

## Abstract (arXiv)

We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold M, which we call the textitgeodesic walk. We prove that the mixing time of this walk on any manifold with positive sectional curvature Cx(u,v) bounded both above and below by 0 < mathfrakm2 ≤ Cx(u,v) ≤ mathfrakM2 < ∞ is O^*left(fracmathfrakM2mathfrakm2right). In particular, this bound on the mixing time does not depend explicitly on the dimension of the manifold. In the special case that M is the boundary of a convex body, we give an explicit and computationally tractable algorithm for approximating the exact geodesic walk. As a consequence, we obtain an algorithm for sampling uniformly from the surface of a convex body that has running time bounded solely in terms of the curvature of the body.