# Fast Markov Chain Monte Carlo Algorithms via Lie Groups

## TL;DR

This paper introduces a new MCMC algorithm derived from Lie group principles, demonstrating faster convergence on complex models like spin glasses compared to existing methods.

## Contribution

It presents a novel MCMC algorithm based on Lie group theory, expanding the toolkit for efficient sampling in complex probability spaces.

## Key findings

- The new algorithm converges faster than traditional MCMC methods on spin glass models.
- Explicit numerical examples illustrate the effectiveness of the proposed algorithms.
- Derivation of multiple MCMC variants from Lie group considerations.

## Abstract

From basic considerations of the Lie group that preserves a target probability measure, we derive the Barker, Metropolis, and ensemble Markov chain Monte Carlo (MCMC) algorithms, as well as variants of waste-recycling Metropolis-Hastings and an altogether new MCMC algorithm. We illustrate these constructions with explicit numerical computations, and we empirically demonstrate on a spin glass that the new algorithm converges more quickly than its siblings.