Limiting Spectrum of Random Kernel Matrices

Tuesday, September 20, 2016 - 1:25pm - 2:25pm
Lind 305
Xiuyuan Cheng (Yale University)
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 kernel random matrices is studied in the large p, large n regime. It is shown that with suitable normalization the spectral density converges weakly, and we identify the limit. Our analysis applies as long as the rescaled kernel function is generic, and particularly, this includes non-smooth functions, e.g. Heaviside step function. The limiting densities interpolate between the Marcenko-Pastur density and the semi-circle density.

Xiuyuan Cheng is currently a Gibbs Assistant Professor at the Program of Applied Mathematics of Yale University. Before joining Yale, she was a Postdoctoral Researcher in Département d'Informatique of École normale supérieure, France, from 2013 to 2015. She received her Ph.D. from the Program of Applied and Computational Mathematics at Princeton University in 2013, advised jointly by Prof. Amit Singer and Prof. Weinan E. Her research focuses on high dimensional data analysis and mathematical theories of machine learning.