On some aspects of approximation of ridge functions

TL;DR

This paper develops algorithms for the uniform approximation of multivariate ridge functions, extending previous analyses to new domains and noisy settings, and exploring functions based on Euclidean norms.

Contribution

It introduces effective algorithms for approximating ridge functions on various domains, including noisy measurements and functions involving Euclidean norms.

Findings

Algorithms successfully approximate ridge functions on the unit cube and ball.

Extensions to noisy data improve robustness of recovery.

Analysis broadens understanding of ridge function approximation in high dimensions.

Abstract

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 $f(x)=g(\langle a,x\rangle)$ 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 $f(x)=g(\|a-x\|_{l_2^d}^2)$.