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

Abstract

Let $\mathbf{x}$ 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 $\mathbf{\Delta }$. 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.

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