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