Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry

We deal with the task of sampling from an unnormalized Boltzmann density $\rho_D$by learning a Boltzmann curve given by energies $f_t$ starting in a simple density $\rho_Z$.First, we examine conditions under which Fisher-Rao flows are absolutely continuous in the Wasserstein geometry.Second, we address specific interpolations $f_t$ and the learning of the related density/velocity pairs $(\rho_t,v_t)$.It was numerically observed that the linear interpolation, which requires only a parametrization of the velocity field $v_t$,suffers from a "teleportation-of-mass" issue.Using tools from the Wasserstein geometry,we give an analytical example,where we can precisely measure the explosion of the velocity field.Inspired by Máté and Fleuret, who parametrize both $f_t$ and $v_t$, we propose aninterpolation which parametrizes only $f_t$ and fixes an appropriate $v_t$. This corresponds tothe Wasserstein gradient flow of the Kullback-Leibler divergence related to Langevin dynamics. We demonstrate by numerical examples that our model provides a well-behaved flow field which successfully solves the above sampling task.