Random Matrix Theory and its applications to machine learning, high dimensional statistics and finance

Abstract

Random Matrix Theory (RMT) is the study of the spectra of large random matrices. It has been pioneered by Eugene Wigner, motivated by nuclear physics and more precisely the spectroscopy of heavy atoms. The goal of this talk is to present various modern applications. 1) First I will present a previous work on the use of RMT in machine learning. The idea is to model the Jacobian of a neural network at initialization as the product of large random matrices. The spectrum naturally allows us to infer stability, which is strongly related to performance. And the operation encoding the spectrum of the Jacobian is the so-called free multiplicative convolution. 2) Second I will present an ongoing work in high dimensional statistics. Consider the basic operation of estimating the spectrum of large covariance matrices. This estimation has an inherent "large dimensional bias", when one observes a multivariate sample whose size is comparable to the dimension. Solving this issue amounts to understanding free multiplicative deconvolution. And our work shows how to obtain a computable and statistically consistent estimation. 3) Finally, I will sketch an application to Markowitz's portfolio theory, showing the relevance of free deconvolution when the number of periods and the number of assets are comparable. This will be the topic of future works. We will focus on point 2). Items 1) and 3) will be discussed in a more panoramic way. In any case, we thus hope to illustrate Wigner's famous maxim on "the unreasonable effectiveness of mathematics in natural sciences".