Approximation of Generalized Ridge Functions in High Dimensions

2017/01/24 by Keiper, Sandra
#FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1701.07018

Abstract

This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of ℝ^N. We introduce the notion of linear-sleeve functions, whose function values only depend on the distance to some unknown linear subspace L. We propose two effective algorithms to approximate linear-sleeve functions f(x)=g(dist(x,L)²), when both the linear subspace L⊂ ℝ^N and the function g∈ C^s[0,1] are unknown. We will prove error bounds for both algorithms and provide an extensive numerical comparison of both. We further propose an approach of how to apply these algorithms to capture general sleeve functions, which are constant along some lower dimensional submanifolds.

Related

• Approximation of Curve-based Sleeve Functions in High Dimensions
• On some aspects of approximation of ridge functions
• Notes on ridge functions and neural networks
• Entropy and sampling numbers of classes of ridge functions
• On the approximation by weighted ridge functions