Analysis of generalized ridge functions in high dimensions

Abstract

The approximation of functions in many variables suffers from the so-called “curse of dimensionality”. Namely, functions on RN with smoothness of order s can be recovered at most with an accuracy of n−s/N applying n-dimensional spaces for linear or nonlinear approximation. However, there is a common belief that functions arising as solutions of real world problems have more structure than usual TV-variate functions. This has led to the introduction of different models for those functions. One of the most popular models is that of so-called ridge functions, which are of the form R N ⊇ Ω  x ↦ f(x) = g(Ax) (1) where A e Rm, N is a matrix and m is considerably smaller than N. The approximation of such functions was for example studied in [1], [2], [3], and [4]. However, by considering functions of the form (1), we assume that real world problems can be described by functions that are constant along certain linear subspaces. Such assumption is quite restrictive and we, therefore, want to study a more generalized form of ridge functions, namely functions which are constant along certain submanifolds of R. Hence, we introduce the notion of generalized ridge functions, which are defined to be functions of the form R N  x ↦ f(x) = g(dist(x, M)), (2) where M is a d-dimensional, smooth submanifold of RN and g e Cs(R). Note that if M is an (N−1)-dimensional, affine subspace of RN and we consider the signed distance in equation (2), we indeed have the case of a usual ridge function. We will analyze how the methods to approximate usual ridge functions apply to generalized ridge functions and investigate new algorithms for their approximation.