A RANDOM MATRIX APPROACH TO NEURAL NETWORKS

Abstract : This article studies the Gram random matrix model G = 1 T Σ T Σ, Σ = σ(W X), classically found in the analysis of random feature maps and random neural networks, where X = [x1,. .. , xT ] ∈ 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(W X) 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 + γIT) −1 , for γ > 0, has a similar behavior as that met in sample covariance matrix models, involving notably the moment Φ = T n E[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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Cosme Louart, Zhenyu Liao, Romain Couillet. A RANDOM MATRIX APPROACH TO NEURAL NETWORKS. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2018, 28 (2), pp.1190-1248. ⟨hal-01957656⟩

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