Abstract: The Method of Moments is one of the most widely used methods in statistics for parameter estimation, obtained by solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or non-uniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analysis of method of moments are mainly confined to asymptotic normality style of results established under strong assumptions.
In this talk I will present some recent results for estimating Gaussians location mixtures with known or unknown variance. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space before executing the method of moments. Not only does this regularization ensures existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in Wasserstein distance via polynomial interpolation and marjorization, we establish the statistical guarantees and optimality of the proposed procedure. These results can also be viewed as provable algorithms for Generalized Method of Moments which involves non-convex optimization. Extensions to multiple dimensions will be discussed.
This is based on joint work with Pengkun Yang (Illinois).
Bio: Yihong Wu is an assistant professor in the Department of Statistics and Data Science at Yale University. He received his Ph.D. degree from Princeton University in 2011. He was a postdoctoral fellow with the Statistics Department in The Wharton School at the University of Pennsylvania from 2011 to 2012 and an assistant professor in the Department of ECE at the University of Illinois at Urbana-Champaign from 2013 to 2016. His research interests are in the theoretical and algorithmic aspects of high-dimensional statistics and information theory.