Stereographic Markov Chain Monte Carlo

• URL: http://arxiv.org/abs/2205.12112v2
• Date: Wed, 21 Feb 2024 04:46:20 GMT
• Title: Stereographic Markov Chain Monte Carlo
• Authors: Jun Yang, Krzysztof {\L}atuszy\'nski, Gareth O. Roberts
• Abstract summary: High-dimensional distributions are notoriously difficult for off-the-shelf MCMC samplers. We introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere. In the best scenario, the proposed samplers can enjoy the blessings of dimensionality'' that the convergence is faster in higher dimensions.
• Score: 2.9304381683255945
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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