Constructive approximate transport maps with normalizing flows

TL;DR

This paper develops a method to construct approximate transport maps using normalizing flows by controlling the continuity equation, with error bounds in relative entropy and conditions on tail decay.

Contribution

It introduces a piecewise constant control approach for transport maps with bounds on switches, advancing the understanding of approximate controllability in the continuity equation.

Findings

Provides bounds on the number of control switches.

Establishes conditions on tail decay for approximate transport.

Offers insights into the reachable space in relative entropy.

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.