Neural Network Architecture Beyond Width and Depth

This paper proposes a new neural network architecture by introducing an additional dimension called height beyond width and depth. Neural network architectures with height, width, and depth as hyper-parameters are called three-dimensional architectures. It is shown that neural networks with three-dimensional architectures are significantly more expressive than the ones with two-dimensional architectures (those with only width and depth as hyper-parameters), e.g., standard fully connected networks. The new network architecture is constructed recursively via a nested structure, and hence we call a network with the new architecture nested network (NestNet). A NestNet of height s is built with each hidden neuron activated by a NestNet of height ≤ s-1. When s=1, a NestNet degenerates to a standard network with a two-dimensional architecture. It is proved by construction that height-s ReLU NestNets with O(n) parameters can approximate 1-Lipschitz continuous functions on [0,1]d with an error O(n-(s+1)/d), while the optimal approximation error of standard ReLU networks with O(n) parameters is O(n-2/d). Furthermore, such a result is extended to generic continuous functions on [0,1]d with the approximation error characterized by the modulus of continuity. Finally, we use numerical experimentation to show the advantages of the super-approximation power of ReLU NestNets.