Past Event: Oden Institute Seminar

Tensor moments of gaussian mixture models: theory and applications

João M. Pereira, Postdoctoral fellow, Oden Institute, UT Austin

Abstract

Gaussian mixture models (GMM) are fundamental tools in statistical and data sciences. We study the moments of multivariate Gaussians and GMMs. The dth moment of an n-dimensional random variable is a symmetric d-way tensor of size nd, so working with moments is assumed to be prohibitively expensive for d > 2 and larger values of n. In this work, we develop theory and numerical methods for implicit computations with moment tensors of GMMs, reducing the storage costs to O(n2) for general covariance matrices and O(n) for diagonal ones. We derive concise analytic expressions for the moments in terms of symmetrized tensor products, relying on the correspondence between symmetric tensors and homogeneous polynomials. The primary application of this theory is estimating GMM parameters from a set of observations, which can be formulated as a moment-matching optimization problem. If there is a known and common covariance matrix, then it is possible to debias the data observations, in which case and the problem of estimating the unknown means reduces to symmetric tensor decomposition. Numerical results illustrate the numerical efficiency of these approaches. This is joint work with Joe Kileel and Tamara G Kolda.

Biography

João is a postdoc in the Oden Institute at UT Austin, working with Joe Kileel and Rachel Ward. Previously, he was a postdoc at Duke University, working with Vahid Tarokh, and obtained is Ph.D. degree in Applied Mathematics at Princeton University, advised by Amit Singer and Emmanuel Abbe. He is broadly interested in tensor decompositions, information theory and applied mathematics.