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This paper is devoted to studying the optimal expressive power of ReLU deepneural networks (DNNs) and its application in approximation via the KolmogorovSuperposition Theorem. We first constructively prove that any continuouspiecewise linear functions on $[0,1]$, comprising $O(N^2L)$ segments, can berepresented by ReLU DNNs with $L$ hidden layers and $N$ neurons per layer.Subsequently, we demonstrate that this construction is optimal regarding theparameter count of the DNNs, achieved through investigating the shatteringcapacity of ReLU DNNs. Moreover, by invoking the Kolmogorov SuperpositionTheorem, we achieve an enhanced approximation rate for ReLU DNNs of arbitrarywidth and depth when dealing with continuous functions in high-dimensionalspaces.

On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theorem