This talk discusses how to recover parameters in diagonal Gaussian mixture models using tensor decompositions. High-order moments of the Gaussian mixture model are estimated from samples. They form incomplete symmetric tensors generated by hidden parameters in the model. We propose to use generating polynomials to compute incomplete symmetric tensor approximations. The obtained decomposition is utilized to recover parameters in the model. We prove that our recovered parameters are accurate when the estimated moments are accurate. Using high-order moments enables our algorithm to learn Gaussian mixtures with more components. For a given model dimension and order, we provide an upper bound of the number of components in the Gaussian mixture model that our algorithm can compute.

I will present computational methods to estimate certain statistical models from data samples. The models are conditionally-independent mixture models, and cover a wide range of applications. The optimization methods are based on tensor decomposition and completion, but crucially they avoid the explicit formation of tensors which enables scalability to high dimensions. Time permitting, I’ll also discuss the algebraic varieties underlying the problem. This is based on joint works with Yifan Zhang, with Joao Pereira and Tammy Kolda, and with Yulia Alexandr and Bernd Sturmfels.

We describe computationally tractable relaxations based on spectral methods for polynomial optimization. Concretely, our relaxations yield bounds based on minimum eigenvalues of matrices that are obtained as linear functions of the problem data.
Spectral methods are computationally much less expensive than semidefinite programs, which makes them an attractive candidate for obtaining bounds on large-scale polynomial optimization problems. We identify the algebraic structure underlying a polynomial optimization problem that makes it amenable to spectral methods, and we demonstrate the efficiency and applicability of our framework with numerical examples.