Abstract

We resolve most cases of identifiability from sixth-order moments for Gaussian mixtures on spaces of large dimensions. Our results imply that for a mixture of $m$ Gaussians on an $n$-dimensional space, the means and covariances can be uniquely recovered from the mixture moments of degree 6, as long as $m$ is bounded by some function in $\mathrm{\Omega }(n^4)$. 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.

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