Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry
Jannis Chemseddine Christian Wald Richard Duong Gabriele Steidl
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 an
interpolation which parametrizes only $f_t$ and fixes an appropriate $v_t$. 
This corresponds to
the 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.
Sampling, Boltzmann densities, Fisher-Rao Curves, Wasserstein Gradient Flows, Diffusion, Interpolations