Gaussian Mixture Identifiability from degree 6 Moments

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.