Approximation of Curve-based Sleeve Functions in High Dimensions

TL;DR

This paper introduces a novel method for approximating high-dimensional sleeve functions based on curves, involving profile recovery and curve representation, with theoretical guarantees under certain conditions.

Contribution

It presents the first approach to approximate curve-based sleeve functions in high dimensions, including a two-step method and analysis of well-separation conditions.

Findings

Method successfully recovers sleeve functions with guaranteed quality.

The approach is effective under well-separation conditions.

Theoretical analysis supports the method's convergence and success.

Abstract

Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function - the profile - is recovered, and second the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximate the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.