Entropy and Sampling Numbers of Classes of Ridge Functions

Abstract

We study the properties of ridge functions $f(x)=g(a\cdot x)$ in high dimensions $d$ from the viewpoint of approximation theory. The function classes considered consist of ridge functions such that the profile $g$ is a member of a univariate Lipschitz class with smoothness $\alpha >0$ (including infinite smoothness) and the ridge direction $a$ has $p$-norm $\Vert a\Vert _p\le 1$. First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in $L_{\infty }$. We show that they are essentially as compact as the class of univariate Lipschitz functions. Second, we examine sampling numbers and consider two extreme cases. In the case $p=2$, sampling ridge functions on the Euclidean unit ball suffers from the curse of dimensionality. Moreover, it is as difficult as sampling general multivariate Lipschitz functions, which is in sharp contrast to the result on entropy numbers. When we additionally assume that all feasible profiles have a first derivative uniformly bounded away from zero at the origin, the complexity of sampling ridge functions reduces drastically to the complexity of sampling univariate Lipschitz functions. In between, the sampling problem’s degree of difficulty varies, depending on the values of $\alpha$ and $p$. Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Woźniakowski.