This online event will take place via Zoom. Please contact the APAM Department for the Zoom link.

Speaker: Jose Israel Rodriguez, University of Wisconsin, Madison

Title: Estimating Gaussian mixtures using sparse polynomial moment systems 

Abstract: A fundamental problem in statistics is to estimate the parameters of a density from samples. There are several approaches that one can take to solve this problem, but we will focus on the method of moments. Here, one takes sample moments as estimates for the moments of the unknown distribution. It turns out, the moments of the distribution can often be expressed as polynomials in the density’s parameters. Therefore, if we are given sample moments we can recover parameter estimates by solving a nonlinear system of equations. In fact, we are solving a system of polynomial equations and are able to leverage the tools of Algebraic Statistics and Applied Algebraic Geometry.In this talk we consider a prominent class of models: mixtures of Gaussian distributions. Using sample moment data we recover the mean and variance of each component the mixture by solving a system of equations. The first part focuses on bounding the degree of these polynomial systems. Pearson studied the 2-mixture case by solving a degree nine polynomial. We study the k-mixture case, which gives polynomial systems with much larger degree. Our techniques involve polyhedral methods for solving sparse polynomial systems. A nice consequence of our proofs is that they yield a continuation method for finding the estimate to recover the parameters. The second part of the talk does density estimation for mixtures of high dimensional Gaussians. 

Biography: Jose Israel Rodriguez is an Assistant Professor of Mathematics at the University of Wisconsin --- Madison. Jose received his Ph.D in mathematics from the University of California, Berkeley and bachelors degree from the University of Texas. Prior to his time at Madison, he had postdoc positions at the University of Chicago, University of Notre Dame, and the Simons Institute for the Theory of Computing. His research interests include applied algebraic geometry and algebraic statistics. He is an active member of the SIAM's Algebraic Geometry Activity Group and co-organizes the SIAM SAGA (Seminar in Applied Geometry and Algebra).