Speaker: Xiuyuan Cheng, Princeton University

Title: The Limiting Spectrum of Random Inner-product Kernel Matrices

Time: Thursday, 5/17, 10am

Location: 北京国际数学研究中心 全斋全29教室 上午10点

Abstract: We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1, ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a real-valued function. The eigenvalue distribution of these random kernel matrices is studied at the "large p, large n" regime. It is shown that, when p and n go to infinity, p/n = gamma which is a constant, and f is properly scaled so that Var(f(X_i^T X_j)) is O(p^{-1}), the spectral density converges weakly to a limiting density on R. The limiting density is dictated by a cubic equation involving its Stieltjes transform. While for smooth kernel functions the limiting spectral density has been previously shown to be the Marcenko-Pastur distribution, our analysis is applicable to non-smooth kernel functions, resulting in a new family of limiting densities.

Bio: Xiuyuan Cheng received B.S. in 2009 from the School of athematical Sciences, Peking University, and is currently Ph.D. candidate at the Program of Applied and Mathematical Sciences (PACM), Princeton University.