# Moment Identifiability of Homoscedastic Gaussian Mixtures

## TL;DR

This paper investigates the identifiability of homoscedastic Gaussian mixture models using moment varieties, providing classification results and explicit parameter recovery methods for certain cases.

## Contribution

It introduces a geometric approach to Gaussian mixture identifiability via secant varieties and classifies cases with defective moment varieties for moments up to order three.

## Key findings

- Identifiability determined for mixtures with fewer components than the space dimension.
- Closed-form parameter recovery for two-component one-dimensional mixtures using moments up to order four.
- Connection between rank estimation and secant varieties of rational normal curves.

## Abstract

We consider the problem of identifying a mixture of Gaussian distributions with same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander-Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two component setting we provide a closed form solution for parameter recovery based on moments up to order four, while in the one dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves.