Rapid mixing for Gibbs measures in Riemannian manifolds
Provides theoretical guarantees for sampling on manifolds, relevant for Bayesian inference and machine learning on constrained or curved spaces.
The paper identifies conditions for rapid mixing of Langevin dynamics to Gibbs measures on Riemannian manifolds, achieving polynomial mixing times in dimension when curvature, temperature, and saddle point escape conditions are met.
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.