Constructive approximate transport maps with normalizing flows

• URL: http://arxiv.org/abs/2412.19366v3
• Date: Sat, 16 Aug 2025 14:50:38 GMT
• Title: Constructive approximate transport maps with normalizing flows
• Authors: Antonio Álvarez-López, Borjan Geshkovski, Domènec Ruiz-Balet,
• Abstract summary: We study an approximate controllability problem for the continuity equation and its application to constructing transport maps with normalizing flows.<n>We construct time-dependent controls $theta=(w, a, b)$ in the vector field $xmapsto w(atop x + b)_+$ to approximately transport a known base density.<n>Our main result relies on an assumption on the relative tail decay of $rho_*$ and $rho_mathrmB$, and provides hints on characterizing the reachable space of the continuity equation in relative entropy.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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