Gaussian mixtures closest to a given measure via optimal transport

TL;DR

This paper characterizes Gaussian mixtures closest to a given measure using optimal transport, providing a numerical scheme to approximate the minimal Wasserstein distance and detect if the measure is a Gaussian mixture within a parameter set.

Contribution

It introduces a novel optimal transport framework for Gaussian mixture approximation and develops a mesh-free semidefinite relaxation scheme for practical computation.

Findings

The scheme can approximate the Wasserstein distance arbitrarily closely.

It can detect when the measure is a Gaussian mixture within the parameter set.

It allows extraction of mixture components when the measure is a finite atomic Gaussian mixture.

Abstract

Given a determinate (multivariate) probability measure $\mu$, we characterize Gaussian mixtures $\nu\_\phi$ which minimize the Wasserstein distance $W\_2(\mu,\nu\_\phi)$ to $\mu$ when the mixing probability measure $\phi$ on the parameters $(m,\Sigma)$ of the Gaussians is supported on a compact set $S$.(i) We first show that such mixtures are optimal solutions of a particular optimal transport (OT) problem where the marginal $\nu\_{\phi}$ of the OT problem is also unknown via the mixing measure variable $\phi$. Next (ii) by using a well-known specific property of Gaussian measures, this optimal transport is then viewed as a Generalized Moment Problem (GMP) and if the set $S$ of mixture parameters $(m,\Sigma)$ is a basic compact semi-algebraic set, we provide a "mesh-free" numerical scheme to approximate as closely as desired the optimal distance by solving a hierarchy of semidefinite relaxations of increasing size. In particular, we neither assume that the mixing measure is finitely supported nor that the variance is the same for all components. If the original measure $\mu$ is not a Gaussian mixture with parameters $(m,\Sigma)\in S$, then a strictly positive distance is detected at a finite step of the hierarchy. If the original measure $\mu$ is a Gaussian mixture with parameters $(m,\Sigma)\in S$, then all semidefinite relaxations of the hierarchy have same zero optimal value. Moreover if the mixing measure is atomic with finite support, its components can sometimes be extracted from an optimal solution at some semidefinite relaxation of the hierarchy when Curto & Fialkow's flatness condition holds for some moment matrix.