Eigenvalue distribution of nonlinear models of random matrices

04/05/2019

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by   Lucas Benigni, et al.

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This paper is concerned with the asymptotic empirical eigenvalue distribution of a non linear random matrix ensemble. More precisely we consider M= 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. 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 previous result 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 applications.