COMPUTATION COMPLEXITY OF DEEP RELU NEURAL NETWORKS IN HIGH-DIMENSIONAL APPROXIMATION

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