The shape of a Gaussian mixture is characterized by the probability density of the distance between two samples

Abstract

Let $\bf{x}$ be a random variable with density $ρ(x)$ taking values in ${\mathbb R}^d$. We are interested in finding a representation for the shape of $ρ(x)$, i.e. for the orbit $\{ ρ(g\cdot x) | g\in E(d) \}$ of $ρ$ under the Euclidean group. Let $x_1$ and $x_2$ be two random samples picked, independently, following $ρ(x)$, and let $Δ$ be the squared Euclidean distance between $x_1$ and $x_2$. We show, if $ρ(x)$ is a mixture of Gaussians whose covariance matrix is the identity, and if the means of the Gaussians are in generic position, then the density $ρ(x)$ is reconstructible, up to a rigid motion in $E(d)$, from the density of $\bfΔ$. In other words, any two such Gaussian mixtures $ρ(x)$ and $\barρ (x)$ with the same distribution of distances are guaranteed to be related by a rigid motion $g\in E(d)$ as $ρ(x)=\barρ (g\cdot x)$. We also show that a similar result holds when the distance is defined by a symmetric bilinear form.