A diffusion approach to Stein's method on Riemannian manifolds
This work provides a theoretical extension of Stein's method to Riemannian manifolds, which is incremental as it builds on known techniques for Euclidean spaces.
The authors tackled the problem of bounding integral metrics for probability measures on Riemannian manifolds by extending Stein's method using diffusion processes, deriving curvature-dependent Stein factors that generalize existing results for Euclidean spaces to flat manifolds.
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