Continuous transformations of probability measures and their transport representations

TL;DR

This paper investigates when and how a transformation of probability measures can be represented as a push-forward via a map, focusing on regularity conditions and the impact of Lipschitz continuity in Wasserstein space.

Contribution

It characterizes the existence and regularity of transport representations of measure transformations, especially under Lipschitz conditions, with illustrative examples.

Findings

Transport representations exist under certain conditions.

Lipschitz continuity ensures continuous transport maps.

Examples demonstrate the sharpness of assumptions.

Abstract

Given a function $F$ transforming a probability measure $\mu$ into another one $F(\mu)$, we study the existence and regularity of a transport representation of it. That is, we ask whether we can represent the image $F(\mu)$ of the input probability measure $\mu$ as the push-forward of $\mu$ by a map $f(\cdot,\mu)$ which may depend on $\mu$; and furthermore, how regular $f$ can be chosen depending on $F$. Even if $F$ is continuous and a transport representative exists, it cannot necessarily be chosen in a continuous way; however, if $F$ is Lipschitz continuous with respect to the Wasserstein distance, then $f$ can be chosen continuous. We provide several examples to illustrate the sharpness of our assumptions. This question is motivated by approximation results for transformations of probability distributions with transformers.