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The purpose of the present paper is to study the computation complexity ofdeep ReLU neural networks to approximate functions in H\"older-Nikol'skiispaces of mixed smoothness $H_\infty^\alpha(\mathbb{I}^d)$ on the unit cube$\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\inH_\infty^\alpha(\mathbb{I}^d)$, we explicitly construct nonadaptive andadaptive deep ReLU neural networks having an output that approximates $f$ witha prescribed accuracy $\varepsilon$, and prove dimension-dependent bounds forthe computation complexity of this approximation, characterized by the size andthe depth of this deep ReLU neural network, explicitly in $d$ and$\varepsilon$. Our results show the advantage of the adaptive method ofapproximation by deep ReLU neural networks over nonadaptive one.

Computation complexity of deep ReLU neural networks in high-dimensional approximation