A multivariate ridge function is a function of the form @math , where @math is univariate and @math . We show that the recovery of an unknown ridge function defined on the hypercube @math with Lipschitz-regular profile @math suffers from the curse of dimensionality when the recovery error is measured in the @math -norm, even if we allow randomized algorithms. If a limited number of components of @math is substantially larger than the others, then the curse of dimensionality is not present and the problem is weakly tractable provided the profile @math is sufficiently regular.
An alternative to the component-wise positivity of the ridge vector that also guarantees polynomial tractability is to assume that for some given @math . This assumption works both for ridge functions defined on the hypercube and for ridge functions defined on the Euclidean ball @math whereas does not lead to a polynomially tractable problem for ridge functions defined on the hypercube, see @cite_25 . Assumption has been studied in @cite_22 @cite_1 @cite_25 , where @cite_1 studies also the effect of noisy measurements. @cite_25 , it has been shown that recovery of ridge functions defined on the Euclidean ball in general suffers from the curse of dimensionality. This finding is based on novel two-sided estimates that reduce the decay behavior of the worst-case recovery error to the decay behavior of entropy numbers of @math -balls. whereas @math implies weak tractability for sufficiently large @math .
@cite_22 @cite_1 @cite_25
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Let us assume that f is a continuous function defined on the unit ball of ℝ d , 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(x i ), 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 a sampling choice of the points xi drawn at random for each function approximation; algorithms (Algorithm 1 and Algorithm 2) for computing the approximating function, whose complexity is at most polynomial in the dimension d and in the number m of points. Due to the arbitrariness of A, the sampling points will be chosen according to suitable random distributions, and our results hold with overwhelming probability. Our approach uses tools taken from the compressed sensing framework, recent Chernoff bounds for sums of positive semidefinite matrices, and classical stability bounds for invariant subspaces of singular value decompositions. We present effective algorithms for uniform approximation of multivariate functions satisfying some prescribed inner structure. We extend, in several directions, the analysis of recovery of ridge functions f ( x ) = g ( a , x ) as performed earlier by one of the authors and his coauthors. We consider ridge functions defined on the unit cube - 1 , 1 ] d as well as recovery of ridge functions defined on the unit ball from noisy measurements. We conclude with the study of functions of the type f ( x ) = g ( ? a - x ? l 2 d 2 ) . We study the properties of ridge functions (f(x)=g(a 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 ( >0 ) (including infinite smoothness) and the ridge direction (a ) has (p )-norm ( a _p 1 ). First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in (L_ ). 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 ( ) and (p ). Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Woźniakowski.