no code implementations • 19 Jan 2023 • Lukas Gonon, Robin Graeber, Arnulf Jentzen
In particular, it is a key contribution of this work to reveal that for all $a, b\in\mathbb{R}$ with $b-a\geq 7$ we have that the functions $[a, b]^d\ni x=(x_1,\dots, x_d)\mapsto\prod_{i=1}^d x_i\in\mathbb{R}$ for $d\in\mathbb{N}$ as well as the functions $[a, b]^d\ni x =(x_1,\dots, x_d)\mapsto\sin(\prod_{i=1}^d x_i) \in \mathbb{R} $ for $ d \in \mathbb{N} $ can neither be approximated without the curse of dimensionality by means of shallow ANNs nor insufficiently deep ANNs with ReLU activation but can be approximated without the curse of dimensionality by sufficiently deep ANNs with ReLU activation.
