Multidimensional two-component Gaussian mixtures detection

TL;DR

This paper investigates the problem of detecting whether a high-dimensional data distribution is a standard Gaussian or a two-component Gaussian mixture, establishing optimal conditions and proposing testing procedures.

Contribution

It provides the first comprehensive analysis of optimal separation conditions for high-dimensional Gaussian mixture detection, introducing new testing methods.

Findings

Derived optimal separation conditions for detection

Proposed multiple testing procedures

Established theoretical guarantees for error control

Abstract

Let $(X\_1,\ldots,X\_n)$ be a $d$-dimensional i.i.d sample from a distribution with density $f$. The problem of detection of a two-component mixture is considered. Our aim is to decide whether $f$ is the density of a standard Gaussian random $d$-vector ($f=\phi\_d$) against $f$ is a two-component mixture: $f=(1-\varepsilon)\phi\_d +\varepsilon \phi\_d (.-\mu)$ where $(\varepsilon,\mu)$ are unknown parameters. Optimal separation conditions on $\varepsilon, \mu, n$ and the dimension $d$ are established, allowing to separate both hypotheses with prescribed errors. Several testing procedures are proposed and two alternative subsets are considered.