Neural Network Architecture Beyond Width and Depth

• URL: http://arxiv.org/abs/2205.09459v1
• Date: Thu, 19 May 2022 10:29:11 GMT
• Title: Neural Network Architecture Beyond Width and Depth
• Authors: Zuowei Shen, Haizhao Yang, Shijun Zhang
• Abstract summary: This paper proposes a new neural network architecture by introducing an additional dimension called height beyond width and depth. It is shown that neural networks with three-dimensional architectures are significantly more expressive than the ones with two-dimensional architectures.
• Score: 4.468952886990851
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 hyperparameters 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 hyperparameters), 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 $\le s-1$. 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 $\mathcal{O}(n)$ parameters can approximate Lipschitz continuous functions on $[0,1]^d$ with an error $\mathcal{O}(n^{-(s+1)/d})$, while the optimal approximation error of standard ReLU networks with $\mathcal{O}(n)$ parameters is $\mathcal{O}(n^{-2/d})$. 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, a numerical example is provided to explore the advantages of the super approximation power of ReLU NestNets.

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