Quasi-Equivalence of Width and Depth of Neural Networks

Abstract: While classic studies proved that wide networks allow universal approximation, recent research and successes of deep learning demonstrate the power of deep networks. Based on a symmetric consideration, we investigate if the design of artificial neural networks should have a directional preference, and what the mechanism of interaction is between the width and depth of a network. Inspired by the De Morgan law, we address this fundamental question by establishing a quasi-equivalence between the width and depth of ReLU networks in two aspects. First, we formulate two transforms for mapping an arbitrary ReLU network to a wide network and a deep network respectively for either regression or classification so that the essentially same capability of the original network can be implemented. Then, we replace the mainstream artificial neuron type with a quadratic counterpart, and utilize the factorization and continued fraction representations of the same polynomial function to construct a wide network and a deep network, respectively. Based on our findings, a deep network has a wide equivalent, and vice versa, subject to an arbitrarily small error.