Deformations of Boltzmann Distributions

TL;DR

This paper introduces a method to transform samples from one Boltzmann distribution to another using a derived equation relating the transformation to the distribution's log-likelihoods, demonstrated on lattice field theory.

Contribution

It derives a new equation linking transformations and unnormalized log-likelihoods for Boltzmann distributions, enabling improved sampling techniques.

Findings

Transformation improves sampling efficiency for $p_\tau$

Method outperforms traditional approaches in lattice field theory

Extends the action $S_0$ to a family $S_t$ for better sampling

Abstract

Consider a one-parameter family of Boltzmann distributions $p_t(x) = \tfrac{1}{Z_t}e^{-S_t(x)}$. This work studies the problem of sampling from $p_{t_0}$ by first sampling from $p_{t_1}$ and then applying a transformation $\Psi_{t_1}^{t_0}$ so that the transformed samples follow $p_{t_0}$. We derive an equation relating $\Psi$ and the corresponding family of unnormalized log-likelihoods $S_t$. The utility of this idea is demonstrated on the $\phi^4$ lattice field theory by extending its defining action $S_0$ to a family of actions $S_t$ and finding a $\tau$ such that normalizing flows perform better at learning the Boltzmann distribution $p_\tau$ than at learning $p_0$.