Deformations of Boltzmann Distributions

Consider a one-parameter family of Boltzmann distributions pt(x) = tfrac1Zte-St(x). This work studies the problem of sampling from pt0 by first sampling from pt1 and then applying a transformation Psit1t0 so that the transformed samples follow pt0. We derive an equation relating Psi and the corresponding family of unnormalized log-likelihoods St. The utility of this idea is demonstrated on the φ4 lattice field theory by extending its defining action S0 to a family of actions St and finding a τ such that normalizing flows perform better at learning the Boltzmann distribution pτ than at learning p0.