Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published December 2014 | Published + Submitted
Journal Article Open

Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions

Abstract

We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis–Hastings algorithm that adds an accept–reject mechanism to a Markov chain proposal in order to make the chain reversible with respect to the target measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein–Uhlenbeck proposal (pCN) for which the reference measure is invariant. Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by exhibiting a dimension-independent Wasserstein spectral gap for pCN algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an L^2-spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM algorithm applied to the reference measures degenerates as the dimension tends to infinity.

Additional Information

© Institute of Mathematical Statistics, 2014. Received December 2011; revised February 2013. [MR is] supported by EPSRC, the Royal Society, and the Leverhulme Trust. [AMS is] Supported by EPSRC and ERC. [SJV is] Supported by ERC.

Attached Files

Published - stuart112.pdf

Submitted - 1112.1392.pdf

Files

1112.1392.pdf
Files (782.2 kB)
Name Size Download all
md5:712117ed3ada2249dab89da1eee272ad
398.9 kB Preview Download
md5:b7f35505bcdbfaa4d7062618a4aea334
383.3 kB Preview Download

Additional details

Created:
August 20, 2023
Modified:
March 5, 2024