# Rapid mixing for Gibbs measures in Riemannian manifolds

arXiv: https://arxiv.org/abs/2606.13453 · HTML: https://mlsanity.com/paper/2606.13453/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 99 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry](https://mlsanity.com/paper/1911.01469/) (2019)
- [From Symmetry to Geometry: Tractable Nonconvex Problems](https://mlsanity.com/paper/2007.06753/) (2020)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Differential Privacy Guarantees for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2201.11980/) (2022)
- [The geometry of the deep linear network](https://mlsanity.com/paper/2411.09004/) (2024)
- [Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization](https://mlsanity.com/paper/2012.15467/) (2020)
- [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)
- [Fast Convergence for Langevin Diffusion with Manifold Structure](https://mlsanity.com/paper/2002.05576/) (2020)
- [On the implicit regularization of Langevin dynamics with projected noise](https://mlsanity.com/paper/2602.12257/) (2026)
- [Online Riemannian Gradient Descent for Quantum State Tomography with Matrix Product Operators](https://mlsanity.com/paper/2605.04533/) (2026)

## Abstract (arXiv)

Langevin dynamics on Riemannian manifolds is analyzed. Conditions ensuring the existence of a suitable logarithmic Sobolev inequality (rapid mixing to the Gibbs measure) are identified. These conditions involve the curvature of the manifold, the inverse temperature, escaping directions from saddle points, and exclude barren plateaus and spurious local minima. We show that when these conditions are met, mixing times polynomial in the dimension of the manifold are achievable. This result is obtained through a relation between Langevin processes in the domain and in the image of a Riemannian submersion. Such a relation can be of independent interest.