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~Θ(n^4) Gaussians on ℝ^n 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.

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Optimization for and with Machine Learning
Networks and Optimization

Taveira Blomenhofer, F. A. (2023). Gaussian mixture identifiability from degree 6 moments.