Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912058343161856 |
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| author | Chemseddine, Jannis Wald, Christian Duong, Richard Steidl, Gabriele |
| author_facet | Chemseddine, Jannis Wald, Christian Duong, Richard Steidl, Gabriele |
| contents | We deal with the task of sampling from an unnormalized Boltzmann density $ρ_D$ by learning a Boltzmann curve given by energies $f_t$ starting in a simple density $ρ_Z$. 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 $(ρ_t,v_t)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03282 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry Chemseddine, Jannis Wald, Christian Duong, Richard Steidl, Gabriele Machine Learning Analysis of PDEs Probability We deal with the task of sampling from an unnormalized Boltzmann density $ρ_D$ by learning a Boltzmann curve given by energies $f_t$ starting in a simple density $ρ_Z$. 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 $(ρ_t,v_t)$. 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. |
| title | Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry |
| topic | Machine Learning Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2410.03282 |