ReLU neural network approximation to piecewise constant functions

• URL: http://arxiv.org/abs/2410.16506v1
• Date: Mon, 21 Oct 2024 20:58:34 GMT
• Title: ReLU neural network approximation to piecewise constant functions
• Authors: Zhiqiang Cai, Junpyo Choi, Min Liu,
• Abstract summary: We show that a three-layer ReLU NN is sufficient to accurately approximate any piecewise constant function. If the discontinuity interface is convex, an analytical formula of the ReLU NN approximation with exact weights and biases is provided.
• Score: 3.5928501649873326
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• Abstract: This paper studies the approximation property of ReLU neural networks (NNs) to piecewise constant functions with unknown interfaces in bounded regions in $\mathbb{R}^d$. Under the assumption that the discontinuity interface $\Gamma$ may be approximated by a connected series of hyperplanes with a prescribed accuracy $\varepsilon >0$, we show that a three-layer ReLU NN is sufficient to accurately approximate any piecewise constant function and establish its error bound. Moreover, if the discontinuity interface is convex, an analytical formula of the ReLU NN approximation with exact weights and biases is provided.

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