Deformations of Boltzmann Distributions

Abstract

Consider a one-parameter family of Boltzmann distributions $p_t(x)=\frac{1}{Z_t}{e}^{-S_t(x)}$. 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 ${\varphi }^4$ 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 $p_0$.

Comments: Machine Learning for the Physical Sciences Workshop at NeurIPS '22