q-Analogue of Hamiltonian Monte Carlo method

TL;DR

This paper introduces a novel $q$-deformed Hamiltonian Monte Carlo ($q$-HMC) method based on $q$-deformed Hamiltonian dynamics, demonstrating improved sampling efficiency in stiff energy landscapes and Bayesian inverse problems.

Contribution

It develops the first computationally feasible $q$-HMC algorithm grounded in $q$-deformed Hamiltonian mechanics, extending traditional HMC to $q$-commutative spaces.

Findings

$q$-HMC satisfies detailed balance.

Demonstrates superior exploration of stiff energy landscapes.

Matches traditional HMC in functional reconstruction tasks.

Abstract

Building upon Lagrangian mechanics on Wess's $q$-commutative spaces, we derive the $q$-deformed Hamiltonian dynamics as formulated by Lavagno et al. (2006). We then develop a computationally tractable scheme and propose a novel Hamiltonian Monte Carlo sampler ($q$-HMC). The proposed $q$-HMC method is shown to satisfy the detailed balance principle. Numerical experiments on distributions with explicit potential functions demonstrate its efficacy, particularly in exploring stiff energy landscapes. This method is also applied to draw samples from the Bayesian posterior distribution of inverse problems. The numerical test for the posterior distribution with stiff potential further shows the advantage of $q$-HMC. And it yields the identical computational implementation process to that of HMC when used to deal with functional reconstruction problems.