ReLU神经网络可精准逼近带未知间断面的分段常数函数。
ReLU neural network approximation to piecewise constant functions
- 三层数神经网络即可实现高精度逼近，误差有理论保证。
- 当间断面为凸集时，可给出权重和偏置的精确解析表达式。
- 适合研究神经网络逼近理论或需要高精度分段函数建模的场景。
本文研究了在 $\bR^d$ 有界区域内，具有未知间断界面 $Γ$ 的分段常数函数的 ReLU 神经网络逼近性质。假设间断界面 $Γ$ 可被一组连接的超平面以精度 $\varepsilon > 0$ 近似，则证明了三层 ReLU 网络足以准确逼近任意此类函数，并给出了误差上界。此外，若间断界面为凸集，可提供具有精确权重与偏置的 ReLU 网络逼近的解析公式。
原文摘要 · Abstract (English)
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 $Γ$ 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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