Minimum cross-entropy distributions on Wasserstein balls and their applications

TL;DR

This paper characterizes and efficiently computes the minimum relative-entropy distribution within Wasserstein balls around a discrete measure, with applications in probabilistic modeling.

Contribution

It introduces a novel characterization and algorithm for minimum cross-entropy distributions constrained by Wasserstein distances.

Findings

Provides an explicit characterization of the minimum cross-entropy distribution

Develops an efficient algorithm for computation

Demonstrates applications in probabilistic modeling and inference

Abstract

Given a prior probability density $p$ on a compact set $K$ we characterize the probability distribution $q_{\delta}^*$ on $K$ contained in a Wasserstein ball $B_{\delta}(\mu)$ centered in a given discrete measure $\mu$ for which the relative-entropy $H(q,p)$ achieves its minimum. This characterization gives us an algorithm for computing such distributions efficiently