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

• URL: http://arxiv.org/abs/2410.03282v1
• Date: Fri, 4 Oct 2024 09:54:11 GMT
• Title: Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry
• Authors: Jannis Chemseddine, Christian Wald, Richard Duong, Gabriele Steidl,
• Abstract summary: We deal with the task of sampling from an unnormalized Boltzmann density $rho_D$ by learning a Boltzmann curve given by $f_t$. Inspired by M'at'e and Fleuret, we propose an which parametrizes only $f_t$ and fixes an appropriate $v_t$. This corresponds to the Wasserstein flow of the Kullback-Leibler divergence related to Langevin dynamics.
• Score: 1.609940380983903
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 M\'at\'e 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.

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