A note on the Pennington-Worah distribution

Abstract

This paper is concerned with a new expression of the so-called Pennington-Worah distribution, characterizing the asymptotic empirical eigenvalue distribution of some non linear random matrix ensembles. More precisely consider $M= \frac{1} {m} YY^{*}$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. The function $f$ is applied pointwise and can be seen as an activation function in (random) neural networks. The asymptotic empirical distribution of this ensemble has been computed in [16] and [3]. Here it is related to the Marcenko-Pastur distribution and information plus noise matrices.