A diffusion approach to Stein's method on Riemannian manifolds

Le, Huiling; Lewis, Alexander; Bharath, Karthik; Fallaize, Christopher

Abstract

We detail an approach to developing Stein’s method for bounding integral metrics on probability measures defined on a Riemannian manifold M. Our approach exploits the relationship between the generator of a diffusion on M having a 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 Rm, and moreover imply that the bounds for Rm remain valid when M is a flat manifold.

Citation

Le, H., Lewis, A., Bharath, K., & Fallaize, C. (2024). A diffusion approach to Stein's method on Riemannian manifolds. Bernoulli, 30(2), 1079-1104. https://doi.org/10.3150/23-bej1625

• Journal Article Type: Article
• Acceptance Date: Apr 21, 2023
• Publication Date: May 1, 2024
• Deposit Date: May 10, 2023
• Publicly Available Date: May 1, 2024
• Journal: Bernoulli
• Print ISSN: 1350-7265
• Publisher: Bernoulli Society for Mathematical Statistics and Probability
• Peer Reviewed: Peer Reviewed
• Volume: 30
• Issue: 2
• Pages: 1079-1104
• DOI: https://doi.org/10.3150/23-bej1625
• Keywords: Coupling; integral metrics; Stein equation; stochastic flow; Wasserstein distance
• Public URL: https://nottingham-repository.worktribe.com/output/20560259
• Publisher URL: https://projecteuclid.org/journals/bernoulli/volume-30/issue-2/A-diffusion-approach-to-Steins-method-on-Riemannian-manifolds/10.3150/23-BEJ1625.full