A diffusion approach to Stein's method on Riemannian manifolds

Abstract

We detail an approach to develop Stein's method for bounding integral metrics on probability measures defined on a Riemannian manifold $\mathbf{M}$. Our approach exploits the relationship between the generator of a diffusion on $\mathbf{M}$ with target invariant measure and its characterising Stein operator. We consider a pair of such diffusions with different starting points, and through analysis of the distance process between the pair, derive Stein factors, which bound the solution to the Stein equation and its derivatives. The Stein factors contain curvature-dependent terms and reduce to those currently available for ${\mathbb{R}}^m$, and moreover imply that the bounds for ${\mathbb{R}}^m$ remain valid when $\mathbf{M}$ is a flat manifold

References in corpus (6)

• Formulae for the derivatives of heat semigroups
• A short survey of Stein's method
• The reproducing Stein kernel approach for post-hoc corrected sampling
• Multivariate stable approximation in Wasserstein distance by Stein's method
• Approximation of Riemannian measures by Stein's method
• A Stein Goodness-of-fit Test for Directional Distributions

Cited by in corpus (5)

• Optimal quantisation of probability measures using maximum mean discrepancy
• A Unifying and Canonical Description of Measure-Preserving Diffusions
• Constructing exchangeable pairs by diffusion on manifolds and its application
• Interpretable Stein Goodness-of-fit Tests on Riemannian Manifolds
• Stein's Method for Probability Distributions on ${\mathbb{S}}^1$