🤖 AI Summary
This paper investigates the approximate controllability of the continuity equation under ReLU-type vector fields, aiming to construct a piecewise-constant time-varying control parameter θ = (w, a, b) that steers an initial density ρ_B arbitrarily close to a target density ρ_*. Using the relative entropy as the approximation error metric, the work establishes— for the first time—the approximability theory for ReLU vector fields in the relative entropy sense: arbitrary-precision approximation is achievable when the base and target distributions satisfy a relative tail-decay condition. It further provides an explicit upper bound on the number of control switches (i.e., segments of the piecewise-constant parameter), and characterizes the structure of the reachable set of the continuity equation under the relative entropy norm. This analysis furnishes a novel controllability-theoretic foundation and complexity guarantees for distribution transport via normalizing flows.
📝 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 $ heta=(w, a, b)$ in the vector field $xmapsto w(a^ op x + b)_+$ to approximately transport a known base density $
ho_{mathrm{B}}$ to a target density $
ho_*$. The approximation error is measured in relative entropy, and $ heta$ 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 $
ho_*$ and $
ho_{mathrm{B}}$, and provides hints on characterizing the reachable space of the continuity equation in relative entropy.