# The recovery of ridge functions on the hypercube suffers from the curse of dimensionality

## TL;DR

This paper demonstrates that recovering ridge functions on the hypercube in the uniform norm is generally hindered by the curse of dimensionality, but becomes feasible when the function's structure is sufficiently sparse and regular.

## Contribution

It establishes the curse of dimensionality for ridge function recovery in high dimensions and identifies conditions under which the problem becomes weakly tractable.

## Key findings

- Recovery suffers from curse of dimensionality in general.
- Sparsity in the ridge vector alleviates the curse.
- Regularity of the profile function enables weak tractability.

## Abstract

A multivariate ridge function is a function of the form $f(x) = g(a^{\scriptscriptstyle T} x)$, where $g$ is univariate and $a \in \mathbb{R}^d$. We show that the recovery of an unknown ridge function defined on the hypercube $[-1,1]^d$ with Lipschitz-regular profile $g$ suffers from the curse of dimensionality when the recovery error is measured in the $L_\infty$-norm, even if we allow randomized algorithms. If a limited number of components of $a$ is substantially larger than the others, then the curse of dimensionality is not present and the problem is weakly tractable provided the profile $g$ is sufficiently regular.