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≤O(n4) Gaussians on Rn can be uniquely recovered from the mixture moments of degree 6.
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.Comment: 22 page