This paper is devoted to studying the optimal expressive power of ReLU deep neural networks (DNNs) and its application in approximation via the Kolmogorov Superposition Theorem. We first constructively prove that any continuous piecewise linear functions on $[0,1]$, comprising $O(N^2L)$ segments, can be represented by ReLU DNNs with $L$ hidden layers and $N$ neurons per layer. Subsequently, we demonstrate that this construction is optimal regarding the parameter count of the DNNs, achieved through investigating the shattering capacity of ReLU DNNs. Moreover, by invoking the Kolmogorov Superposition Theorem, we achieve an enhanced approximation rate for ReLU DNNs of arbitrary width and depth when dealing with continuous functions in high-dimensional spaces.

翻译：本文致力于研究ReLU深度神经网络（DNNs）的最优表达能力及其在基于Kolmogorov叠加定理逼近中的应用。我们首先构造性地证明，$[0,1]$上由$O(N^2L)$段组成的任意连续分段线性函数，可由每层具有$L$个隐藏层和$N$个神经元的ReLU DNNs表示。随后，通过探究ReLU DNNs的分片容量，我们证明该构造在DNN参数数量方面是最优的。此外，借助Kolmogorov叠加定理，我们实现了处理高维空间中连续函数时任意宽度和深度ReLU DNNs的增强逼近率。