Convergence of Preconditioned Hamiltonian Monte Carlo on Hilbert Spaces
This work provides theoretical convergence guarantees for pHMC in infinite-dimensional spaces, which is important for researchers and practitioners using MCMC methods in complex, high-dimensional settings.
This paper investigates the preconditioned Hamiltonian Monte Carlo (pHMC) algorithm in an infinite-dimensional Hilbert space setting. The authors prove convergence bounds for adjusted pHMC in the 1-Wasserstein distance, assuming a condition similar to strong log-concavity of the target measure.
In this article, we consider the preconditioned Hamiltonian Monte Carlo (pHMC) algorithm defined directly on an infinite-dimensional Hilbert space. In this context, and under a condition reminiscent of strong log-concavity of the target measure, we prove convergence bounds for adjusted pHMC in the standard 1-Wasserstein distance. The arguments rely on a synchronous coupling of two copies of pHMC, which is controlled by adapting elements from arXiv:1805.00452.