Gaussian Mixture Identifiability from degree 6 Moments

TL;DR

This paper proves that Gaussian mixture parameters can be uniquely identified from sixth-order moments in high-dimensional spaces, establishing optimal bounds and exploring the minimal degree needed for identifiability.

Contribution

It provides the first comprehensive resolution of identifiability from degree-6 moments for high-dimensional Gaussian mixtures, with optimal bounds and insights into lower degrees.

Findings

Parameters of Gaussian mixtures are identifiable from degree-6 moments in high dimensions.

Degree-4 moments are insufficient for identifiability in nontrivial cases.

Numerical experiments suggest degree-5 moments may be minimal for identifiability.

Abstract

We resolve most cases of identifiability from sixth-order moments for Gaussian mixtures on spaces of large dimensions. Our results imply that the parameters of a generic mixture of $ m\leq\mathcal{O}(n^4) $ Gaussians on $ \mathbb R^n $ can be uniquely recovered from the mixture moments of degree 6. The constant hidden in the $ \mathcal{O} $-notation is optimal and equals the one in the upper bound from counting parameters. We give an argument that degree-4 moments never suffice in any nontrivial case, and we conduct some numerical experiments indicating that degree 5 is minimal for identifiability.