Deformation Theory of Boltzmann Distributions

Consider a one-parameter family of Boltzmann distributions $p_t(x) = \tfrac{1}{Z_t}e^{-S_t(x)}$. We study the problem of sampling from $p_{t_0}$ by first sampling from $p_{t_1}$ and then applying a transformation $\Psi_{t_1}^{t_0}$ to the samples so that to they follow $p_{t_0}$. We derive an equation relating $\Psi$ and the corresponding family of unnormalized log-likelihoods $S_t$. We demonstrate the utility of this idea 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 $p_0$.