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arXiv · 2412.19366

Constructive approximate transport maps with normalizing flows

Antonio Álvarez-López·Borjan Geshkovski·Domènec Ruiz-Balet
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Abstract

We study an approximate controllability problem for the continuity equation and its application to constructing transport maps with normalizing flows. Specifically, we construct time-dependent controls $\theta=(w, a, b)$ in the vector field $x\mapsto w(a^\top x + b)_+$ to approximately transport a known base density $\rho_{\mathrm{B}}$ to a target density $\rho_*$. The approximation error is measured in relative entropy, and $\theta$ are constructed piecewise constant, with bounds on the number of switches being provided. Our main result relies on an assumption on the relative tail decay of $\rho_*$ and $\rho_{\mathrm{B}}$, and provides hints on characterizing the reachable space of the continuity equation in relative entropy.

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Antonio Álvarez-López, Borjan Geshkovski, Domènec Ruiz-Balet. 2024-12-26. Constructive approximate transport maps with normalizing flows. https://arxiv.org/abs/2412.19366

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