Approximation of Smoothness Classes by Deep Rectifier Networks

Abstract

We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B_q^{\alpha }(L^p)$ in arbitrary dimension $d$, on general domains. We show that \alert{deep rectifier} networks with a fixed activation function attain optimal or near to optimal approximation rates for functions in the Besov space $B_{\tau }^{\alpha }(L^{\tau })$ on the critical embedding line $1/\tau =\alpha /d+1/p$ for \emph{arbitrary} smoothness order $\alpha >0$. Using interpolation theory, this implies that the entire range of smoothness classes at or above the critical line is (near to) optimally approximated by deep ReLU/RePU networks.

To appear in SIAM Journal on Numerical Analysis

References in corpus (2)

• Nonlinear Approximation and (Deep) ReLU Networks
• Expressivity of Deep Neural Networks

Cited by in corpus (2)

• Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part I
• Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part II