Approximation of Nonlinear Functionals Using Deep ReLU Networks

• URL: http://arxiv.org/abs/2304.04443v1
• Date: Mon, 10 Apr 2023 08:10:11 GMT
• Title: Approximation of Nonlinear Functionals Using Deep ReLU Networks
• Authors: Linhao Song, Jun Fan, Di-Rong Chen and Ding-Xuan Zhou
• Abstract summary: We investigate the approximation power of functional deep neural networks associated with the rectified linear unit (ReLU) activation function. In addition, we establish rates of approximation of the proposed functional deep ReLU networks under mild regularity conditions.
• Score: 7.876115370275732
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on $L^p([-1, 1]^s)$ for integers $s\ge1$ and $1\le p<\infty$. However, their theoretical properties are largely unknown beyond universality of approximation or the existing analysis does not apply to the rectified linear unit (ReLU) activation function. To fill in this void, we investigate here the approximation power of functional deep neural networks associated with the ReLU activation function by constructing a continuous piecewise linear interpolation under a simple triangulation. In addition, we establish rates of approximation of the proposed functional deep ReLU networks under mild regularity conditions. Finally, our study may also shed some light on the understanding of functional data learning algorithms.

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