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arXiv · 2312.16483

Expressivity and Approximation Properties of Deep Neural Networks with ReLU$^k$ Activation

Juncai He·Tong Mao·Jinchao Xu
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Abstract

In this paper, we investigate the expressivity and approximation properties of deep neural networks employing the ReLU$^k$ activation function for $k \geq 2$. Although deep ReLU networks can approximate polynomials effectively, deep ReLU$^k$ networks have the capability to represent higher-degree polynomials precisely. Our initial contribution is a comprehensive, constructive proof for polynomial representation using deep ReLU$^k$ networks. This allows us to establish an upper bound on both the size and count of network parameters. Consequently, we are able to demonstrate a suboptimal approximation rate for functions from Sobolev spaces as well as for analytic functions. Additionally, through an exploration of the representation power of deep ReLU$^k$ networks for shallow networks, we reveal that deep ReLU$^k$ networks can approximate functions from a range of variation spaces, extending beyond those generated solely by the ReLU$^k$ activation function. This finding demonstrates the adaptability of deep ReLU$^k$ networks in approximating functions within various variation spaces.

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Juncai He, Tong Mao, Jinchao Xu. 2024-01-11. Expressivity and Approximation Properties of Deep Neural Networks with ReLU$^k$ Activation. https://arxiv.org/abs/2312.16483

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