Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry
Keywords: Sampling, Boltzmann densities, Fisher-Rao Curves, Wasserstein Gradient Flows, Diffusion, Interpolations
Abstract: 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.
Supplementary Material: zip
Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
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Submission Number: 11046
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