Learning multivariate functions with low-dimensional structures using polynomial bases

TL;DR

This paper introduces a polynomial-based method for approximating high-dimensional functions with low-dimensional structures, utilizing ANOVA decomposition to reconstruct functions from scattered data and analyze variable interactions.

Contribution

It presents a novel approach combining polynomial bases and ANOVA decomposition for efficient high-dimensional function approximation with low superposition dimension.

Findings

Successful reconstruction of functions from scattered data

Enhanced understanding of variable interactions in high dimensions

Applicable to functions with low superposition dimension

Abstract

In this paper we propose a method for the approximation of high-dimensional functions over finite intervals with respect to complete orthonormal systems of polynomials. An important tool for this is the multivariate classical analysis of variance (ANOVA) decomposition. For functions with a low-dimensional structure, i.e., a low superposition dimension, we are able to achieve a reconstruction from scattered data and simultaneously understand relationships between different variables.