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We study the Conjugate Kernel associated to a multi-layer linear-widthfeed-forward neural network with random weights, biases and data. We show thatthe empirical spectral distribution of the Conjugate Kernel converges to adeterministic limit. More precisely we obtain a deterministic equivalent forits Stieltjes transform and its resolvent, with quantitative bounds involvingboth the dimension and the spectral parameter. The limiting equivalent objectsare described by iterating free convolution of measures and classical matrixoperations involving the parameters of the model.

Deterministic equivalent of the Conjugate Kernel matrix associated to Artificial Neural Networks