Statistics Talks: Hong Chang Ji, David Renfrew

#1: Hong Chang Ji, Van Vleck Visiting Assistant Professor, Department of Mathematics, University of Wisconsin-Madison (website:  https://sites.google.com/view/hongchangji)

 

Title: Spectral estimators in general linear models   (Hong Chang Ji)

Abstract: General linear model concerns the statistical problem of estimating a vector x from the vector of measurements y=Ax+e, where A is a given design matrix whose rows correspond to individual measurements and represents errors in measurements. Popular iterative algorithms used in this context, e.g. message passing, require a "warm start", meaning they must be initialized better than a random guess. In practice, it is often the case that a spectral estimator, i.e. a principal component of certain matrix built from Y, serves as such an initialization. In this talk, we discuss the theoretical aspect of the spectral estimator and present a theorem on its performance guarantee. The theorem gives a threshold for the sample complexity, that is, how many measurements are needed for a warm start to exist.

 

#2: David Renfrew, Associate Professor, Department of Mathematics and Statistics, SUNY Bunghampton (website: https://people.math.binghamton.edu/renfrew/web.html) 

 

Title: Universality for roots of derivatives of entire functions  (David Renfrew)

Abstract: We show for a large class of entire functions, f, that after proper rescaling, on compact sets, the derivatives of converge to cosine, in particular their roots become evenly spaced. This proves a conjecture of Farmer and Rhoades [Trans. Amer. Math. Soc., 357(9):3789–3811, 2005] and Farmer [Adv. Math., 411:Paper No. 108781, 14, 2022] for our class of entire functions. A main ingredient of our proof is to show that high derivatives of high degree polynomials behave like Hermite polynomials, which we prove using the techniques from the newly developed field of finite free probability. This is joint work with Andrew Campbell and Sean O'Rourke.