Stereographic Markov Chain Monte Carlo

TL;DR

This paper introduces a novel class of MCMC algorithms that map high-dimensional distributions onto a sphere, improving mixing and convergence properties, especially for heavy-tailed distributions, and demonstrating faster convergence in higher dimensions.

Contribution

The paper develops sphere-mapped MCMC algorithms, including Metropolis and Bouncy Particle variants, that are uniformly ergodic for diverse distributions and exhibit rapid high-dimensional convergence.

Findings

Algorithms are uniformly ergodic for a broad class of distributions.

Empirical results show rapid convergence in high dimensions.

Proposed methods outperform traditional MCMC in challenging scenarios.

Abstract

High-dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves results in empirically observed ``stickiness'' and poor theoretical mixing properties -- lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the convergence is faster in higher dimensions.