Open Access
2021 Eigenvalue distribution of some nonlinear models of random matrices
Lucas Benigni, Sandrine Péché
Author Affiliations +
Electron. J. Probab. 26: 1-37 (2021). DOI: 10.1214/21-EJP699


This paper is concerned with the asymptotic empirical eigenvalue distribution of some non linear random matrix ensemble. More precisely we consider M=1mYY 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. We compute the asymptotic empirical distribution of this ensemble in the case where W and X have sub-Gaussian tails and f is real analytic. This extends a result of [32] where the case of Gaussian matrices W and X is considered. We also investigate the same questions in the multi-layer case, regarding neural network and machine learning applications.

Funding Statement

Research was accomplished while Sandrine Péché was supported by the Institut Universitaire de France.


The authors would like to thank D. Schröder and Z. Fan for pointing out errors in a previous version of the article as well as anonymous referees for helpful suggestions on how to improve the present paper.


Download Citation

Lucas Benigni. Sandrine Péché. "Eigenvalue distribution of some nonlinear models of random matrices." Electron. J. Probab. 26 1 - 37, 2021.


Received: 13 April 2021; Accepted: 3 September 2021; Published: 2021
First available in Project Euclid: 3 December 2021

Digital Object Identifier: 10.1214/21-EJP699

Primary: 15B52 , 62M45

Keywords: machine learning , neural networks , random matrices

Vol.26 • 2021
Back to Top