Departmental Colloquium - Dr. João M. Pereira

Method of Moments: From Sample Complexity to Efficient Implicit Computations

Abstract:

The focus of this talk is the multivariate method of moments for parameter estimation. First from a theoretical standpoint, we show that in problems where the noise is high, the sample complexity, that is, the number of observations necessary to estimate parameters, is dictated by the moments of the distribution. This follows from a Taylor expansion of the KL divergence that holds in the low signal-to-noise ratio regime. Second from a computational standpoint, we develop a method of moments to estimate the parameters of Gaussian Mixture Models (GMMs) implicitly, that is, without explicitly forming the moments. This addresses the curse of dimensionality: while the number of entries of higher-order moments of multivariate random variables scale exponentially with the order of the moments, our implicit approach has computational and storage costs similar to those of expectation- maximization (EM), and opens the door to the competitiveness between the two methods.

Bio:

João recently joined the University of Georgia as an Assistant Professor in the Department of Mathematics. He is an applied mathematician studying the Mathematics of Data Science and Machine Learning. He obtained his Ph.D. in Applied Mathematics from Princeton University, advised by Emmanuel Abbe and Amit Singer. Afterwards, he worked as a postdoc with Vahid Tarokh at Duke University, and with Joe Kileel and Rachel Ward at University of Texas at Austin. Before joining UGA, he was an Assistant Professor at Instituto de Matemática Pura e Aplicada (IMPA), in Rio de Janeiro, Brazil.