Apply tangent-bundle Riemannian metrics to construct and analyze kinetic Langevin on general manifolds

Develop a methodology to apply Riemannian metrics on the tangent bundle of a Riemannian manifold (for example, the Sasaki metric) to construct kinetic Langevin dynamics and kinetic Langevin Monte Carlo (KLMC) algorithms on general manifolds, and rigorously establish convergence guarantees for the resulting kinetic Langevin SDEs and their MCMC discretizations (e.g., under f-divergences or Wasserstein distances).

Background

The paper explains that convergence analyses based on f-divergences require gradients on the tangent bundle, which in turn need a Riemannian metric defined there. While standard constructions such as the Sasaki metric exist for tangent bundles, the authors note that incorporating these metrics to build kinetic Langevin dynamics and analyze their convergence on general manifolds remains unresolved.

This issue is distinct from the Lie group setting treated in the paper, where left-trivialization enables comparisons of velocities across tangent spaces. For general manifolds, path-dependent parallel transport complicates defining uniform comparisons of velocities, thereby challenging both algorithm design and analysis.