Session II.4 - Foundations of Data Science and Machine Learning
Poster
Identifiability of Gaussian Mixtures from their sixth moments
Alexander Taveira Blomenhofer
CWI, Amsterdam, The Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.
I give a partial answer to an identifiability question from algebraic statistics: When do the truncated moments of a Gaussian mixture distribution uniquely determine the parameters (means and covariance matrices)? We will see that generically, a mixture of at most $m=\theta(n^4)$ Gaussian distributions in n variables is uniquely determined by its moments of degree 6. The proof relies on recent advances in both the theory of secant varieties and on Fröberg's conjecture, which jointly allow to reduce identifiability to a simple combinatorial problem. I show that the resulting Gaussian Moment variety of degree 6 is identifiable up to rank $m=\theta(n^4)$, which asymptotically matches the bound from counting parameters, with the constant hidden in the O-notation being optimal.