Gaussian Mixture Separation and Denoising on Parameterized Varieties

Abstract

This dissertation examines identifiability questions for mixtures of Gaussians: When are the mean vectors and covariance matrices of some Gaussian mixture uniquely determined by the mixture moments of a certain, fixed degree? Assuming generality of the parameters (i.e. the mean vectors and covariance matrices), the problem admits an efficient treatment by the theory of secant varieties, and the answer is determined by some combinatorial constrains between the rank (i.e. the number of Gaussians), the degree of the moments and the number of variables. For mixtures of centered Gaussians, it is shown that identifiability holds true in a range of ranks which is asymptotically optimal in the number of variables. This solves the problem for all degrees and "most" ranks. Nontrivial identifiability results can be obtained from degree 6 onwards. For mixtures of arbitrary Gaussians, it is shown with a similar argument that "most" secants of the degree-6 Gaussian moment variety are identifiable. The dissertation then derives an algorithm to compute low-rank powers-of-forms decompositions, a topic which is closely related to mixtures of centered Gaussians. This algorithm manages to recover the addends of a power sum decomposition, using semidefinite programming and conditions on the Gram spectrahedron of the second-order power sum.

The fourth chapter deals with semi-local projection towards parameterized varieties and is motivated by estimation problems in statistics. It is shown that a variant of Lasserre's hierarchy for these problems has finite convergence with an explicit degree bound. The last chapter shows a finite-convergence type result for a hierarchy of relaxations to detect spaces of singular matrices.