Computation complexity of deep relu neural networks in high-dimensional approximation - Dũng, Dinh; Nguyen, Van Kien; Thao, Mai Xuan (Journal of Computer Science and Cybernetics, 2021)
The purpose of the present paper is to study the computation complexity of deep ReLU neural networks to approximate functions in H\"older-Nikol'skii spaces of mixed smoothness $H_∞^α(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in H_∞^α(\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive deep ReLU neural networks having an output that approximates $f$ with a prescribed accuracy $ε$, and prove dimension-dependent bounds for the computation complexity of this approximation, characterized by the size and the depth of this deep ReLU neural network, explicitly in $d$ and $ε$. Our results show the advantage of the adaptive method of approximation by deep ReLU neural networks over nonadaptive one.