Let us assume that f is a continuous function defined on the unit ball of Rd , of the form f(x) = g(Ax), where A is a k×d matrix and g is a function of k variables for k ≪ d. We are given a budget m ∈ N of possible point evaluations f(xi), i = 1, . . . , m, of f, which we are allowed to query in order to construct a uniform approximating function. Under certain smoothness and variation assumptions on the function g, and an arbitrary choice of the matrix A, we present in this paper