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 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 $\Omega(n^4)$. The constant hidden in the 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.