Mixing-time guarantees for ManifoldMALA

Establish non-asymptotic mixing-time guarantees for the Metropolis-adjusted Weighted Langevin Algorithm (ManifoldMALA) in both unconstrained and constrained sampling settings, quantifying the dependence on dimension and problem parameters, and under appropriate regularity assumptions on the target potential and the Riemannian metric.

Background

The paper analyzes MAPLA and compares its mixing-time scalings with other constrained samplers. While DikinWalk has proven bounds, the authors note a gap for ManifoldMALA: there are no established non-asymptotic mixing-time guarantees in either unconstrained or constrained regimes. Given ManifoldMALA’s widespread use and its reliance on the divergence correction term (∇·M{-1}), providing rigorous mixing-time results would fill an important gap in the literature.

Establishing these guarantees would enable principled comparisons between MAPLA, ManifoldMALA, DikinWalk, RHMC, and other samplers that leverage geometric information, and clarify the role of the divergence correction term in high-dimensional performance.

References

We begin by remarking that \textsf{ManifoldMALA} has no known mixing time guarantees for either the unconstrained or the constrained sampling problem.

— High-accuracy sampling from constrained spaces with the Metropolis-adjusted Preconditioned Langevin Algorithm  (2412.18701 - Srinivasan et al., 2024) in Section 1 (Introduction), paragraph following Table 1 (mixing-time-scalings)

Hence, most literature discussing sampling on manifolds do not consider the high-dimensional limit or optimise for well-behaved dimensional scaling, and it is fair to say that dimensionally independent sampling on arbitrary manifolds is an unsolved problem.

— Optimal scaling and curvature correction for constrained MCMC on spheres  (2610.17946 - Shkeir et al., 13 Oct 2026) in Introduction, Section 1