# Approximation of Generalized Ridge Functions in High Dimensions

## TL;DR

This paper introduces algorithms for approximating generalized ridge functions, especially linear-sleeve functions, in high-dimensional spaces, with proven error bounds and numerical comparisons.

## Contribution

The paper presents novel algorithms for approximating linear-sleeve functions and extends these methods to general sleeve functions, including error analysis and practical implementation.

## Key findings

- Algorithms effectively approximate linear-sleeve functions.
- Error bounds are established for the proposed algorithms.
- Numerical experiments demonstrate the algorithms' performance.

## Abstract

This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of $\mathbb{R}^N$. 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 $f(x)=g(\text{dist}(x,L)^2)$, when both the linear subspace $L\subset \mathbb{R}^N$ and the function $g\in C^s[0,1]$ 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.