Rapid mixing for Gibbs measures in Riemannian manifolds
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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.
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References · 10
- 2019Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetrycited by 34
- 2020From Symmetry to Geometry: Tractable Nonconvex Problemscited by 29
- 2022Convergence of the Riemannian Langevin Algorithmcited by 25
- 2022Differential Privacy Guarantees for Stochastic Gradient Langevin Dynamicscited by 19
- 2024The geometry of the deep linear networkcited by 11
- 2020Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initializationcited by 9
- 2022Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspectivecited by 4
- 2020Fast Convergence for Langevin Diffusion with Manifold Structurecited by 2
- 2026On the implicit regularization of Langevin dynamics with projected noisecited by 1
- 2026Online Riemannian Gradient Descent for Quantum State Tomography with Matrix Product Operatorscited by 1
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