Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry
<|reference_start|>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 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.<|reference_end|>
@article{chemseddine2024neural,
    title={Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the
  Wasserstein Geometry},
    author={Jannis Chemseddine, Christian Wald, Richard Duong, Gabriele Steidl},
    journal={arXiv preprint arXiv:2410.03282},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.03282},
    primaryClass={cs.LG math.AP math.PR}
}