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

Abstract

A multivariate ridge function is a function of the form f(x)=g(aTx), where g is univariate and a∈Rd. 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∞-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.