Spectral Complexity of Deep Neural Networks

It is well known that randomly initialized, push-forward, fully connected neural networks weakly converge to isotropic Gaussian processes in the limit where the width of all layers goes to infinity. In this paper, we propose to use the angular power spectrum of the limiting fields to characterize the complexity of the network architecture. In particular, we define sequences of random variables associated with the angular power spectrum and provide a full characterization of the network complexity in terms of the asymptotic distribution of these sequences as the depth diverges. On this basis, we classify neural networks as low-disorder, sparse, or high-disorder; we show how this classification highlights a number of distinct features for standard activation functions and, in particular, sparsity properties of ReLU networks. Our theoretical results are also validated by numerical simulations.