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A Random Matrix Approach to Neural Networks
Cosme Louart, Zhenyu Liao, Romain Couillet
This article studies the Gram random matrix model G=1/TΣᵀΣ, Σ=σ(WX), classically found in the analysis of random feature maps and random neural networks, where X=[x₁,…,x_T]∈ℝ^(p×T) is a (data) matrix of bounded norm, W∈ℝ^(n×p) is a matrix of independent zero-mean unit variance entries, and σ:ℝ→ℝ is a Lipschitz continuous (activation) function --- σ(WX) being understood entry-wise. By means of a key concentration of measure lemma arising from non-asymptotic random matrix arguments, we prove that, as n,p,T grow large at the same rate, the resolvent Q=(G+γI_T)⁻¹, for γ>0, has a similar behavior as that met in sample covariance matrix models, involving notably the moment Φ=T/n𝔼[G], which provides in passing a deterministic equivalent for the empirical spectral measure of G. Application-wise, this result enables the estimation of the asymptotic performance of single-layer random neural networks. This in turn provides practical insights into the underlying mechanisms into play in random neural networks, entailing several unexpected consequences, as well as a fast practical means to tune the network hyperparameters.
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