Learning Interpolations between Boltzmann Densities

• URL: http://arxiv.org/abs/2301.07388v5
• Date: Tue, 30 May 2023 07:40:03 GMT
• Title: Learning Interpolations between Boltzmann Densities
• Authors: B\'alint M\'at\'e, Fran\c{c}ois Fleuret
• Abstract summary: We introduce a training objective for continuous normalizing flows in the absence of samples. We find a time-dependent vector field $V_t$ that transports samples along the family $p_t$ of densities.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a training objective for continuous normalizing flows that can be used in the absence of samples but in the presence of an energy function. Our method relies on either a prescribed or a learnt interpolation $f_t$ of energy functions between the target energy $f_1$ and the energy function of a generalized Gaussian $f_0(x) = ||x/\sigma||_p^p$. The interpolation of energy functions induces an interpolation of Boltzmann densities $p_t \propto e^{-f_t}$ and we aim to find a time-dependent vector field $V_t$ that transports samples along the family $p_t$ of densities. The condition of transporting samples along the family $p_t$ is equivalent to satisfying the continuity equation with $V_t$ and $p_t = Z_t^{-1}e^{-f_t}$. Consequently, we optimize $V_t$ and $f_t$ to satisfy this partial differential equation. We experimentally compare the proposed training objective to the reverse KL-divergence on Gaussian mixtures and on the Boltzmann density of a quantum mechanical particle in a double-well potential.

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