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

We study the Conjugate Kernel associated to a multi-layer linear-width\nfeed-forward neural network with random weights, biases and data. We show that\nthe empirical spectral distribution of the Conjugate Kernel converges to a\ndeterministic limit. More precisely we obtain a deterministic equivalent for\nits Stieltjes transform and its resolvent, with quantitative bounds involving\nboth the dimension and the spectral parameter. The limiting equivalent objects\nare described by iterating free convolution of measures and classical matrix\noperations involving the parameters of the model.\n