Eigenvalue distribution of some nonlinear models of random matrices

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Abstract

This paper is concerned with the asymptotic empirical eigenvalue distribution of some non linear random matrix ensemble. More precisely we consider M=1mYY∗M=1mYY∗ with Y=f(WX)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.

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APA

Benigni, L., & Péché, S. (2021). Eigenvalue distribution of some nonlinear models of random matrices. Electronic Journal of Probability, 26. https://doi.org/10.1214/21-EJP699

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