Algebraic Identifiability of Gaussian Mixtures

TL;DR

This paper investigates the algebraic identifiability of univariate Gaussian mixtures, proving that their moment varieties have expected dimension, but showing higher-dimensional cases are defective, with implications for Gaussian mixture models.

Contribution

It establishes the expected dimension of moment varieties for univariate Gaussian mixtures and demonstrates defectiveness in higher dimensions, extending algebraic geometry results to Gaussian models.

Findings

Moment varieties of univariate Gaussian mixtures have expected dimension.

Higher-dimensional Gaussian mixtures are defective in their moments.

Extension of Alexander-Hirschowitz Theorem to Gaussian setting suggested.

Abstract

We prove that all moment varieties of univariate Gaussian mixtures have the expected dimension. Our approach rests on intersection theory and Terracini's classification of defective surfaces. The analogous identifiability result is shown to be false for mixtures of Gaussians in dimension three and higher. Their moments up to third order define projective varieties that are defective. Our geometric study suggests an extension of the Alexander-Hirschowitz Theorem for Veronese varieties to the Gaussian setting.