Efficient Sampling on Riemannian Manifolds via Langevin MCMC
This provides a theoretical foundation for sampling in non-Euclidean spaces, which is incremental as it extends known Euclidean results to manifolds.
The paper tackles the problem of efficiently sampling from Gibbs distributions on Riemannian manifolds using Langevin MCMC, proving that the algorithm achieves an ε-Wasserstein distance to the target distribution in Õ(ε⁻²) steps, matching Euclidean complexity.
We study the task of efficiently sampling from a Gibbs distribution $d π^* = e^{-h} d {vol}_g$ over a Riemannian manifold $M$ via (geometric) Langevin MCMC; this algorithm involves computing exponential maps in random Gaussian directions and is efficiently implementable in practice. The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming $\nabla h$ is Lipschitz and $M$ has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence. Combined with a contraction guarantee for the geometric Langevin Diffusion under Kendall-Cranston coupling, we prove that the Langevin MCMC iterates lie within $ε$-Wasserstein distance of $π^*$ after $\tilde{O}(ε^{-2})$ steps, which matches the iteration complexity for Euclidean Langevin MCMC. Our results apply in general settings where $h$ can be nonconvex and $M$ can have negative Ricci curvature. Under additional assumptions that the Riemannian curvature tensor has bounded derivatives, and that $π^*$ satisfies a $CD(\cdot,\infty)$ condition, we analyze the stochastic gradient version of Langevin MCMC, and bound its iteration complexity by $\tilde{O}(ε^{-2})$ as well.