Spectral complexity of deep neural networks

TL;DR

This paper introduces a spectral approach to measure the complexity of deep neural networks, revealing how network architecture influences properties like sparsity and disorder, with theoretical insights supported by simulations.

Contribution

It proposes using the angular power spectrum to characterize neural network complexity and classifies networks into different disorder regimes based on spectral analysis.

Findings

Classifies networks as low-disorder, sparse, or high-disorder.

Reveals spectral properties related to activation functions and sparsity.

Validates theoretical predictions with numerical simulations.

Abstract

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 field 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.