Compressed sensing approaches for polynomial approximation of high-dimensional functions

TL;DR

This paper surveys recent advances in using sparse recovery and weighted  minimization to efficiently approximate high-dimensional functions with polynomial expansions, reducing the curse of dimensionality.

Contribution

It highlights the structured sparsity in polynomial expansions and demonstrates how weighted  minimization mitigates the curse of dimensionality in high-dimensional approximation.

Findings

Sample complexity depends logarithmically on dimension d

Structured sparsity can be exploited for efficient approximation

Weighted  minimization improves high-dimensional polynomial approximation

Abstract

In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the computation of polynomial approximations of high-dimensional functions on $d$-dimensional hypercubes. We show that smooth, multivariate functions possess expansions in orthogonal polynomial bases that are not only approximately sparse, but possess a particular type of structured sparsity defined by so-called lower sets. This structure can be exploited via the use of weighted $\ell^1$ minimization techniques, and, as we demonstrate, doing so leads to sample complexity estimates that are at most logarithmically dependent on the dimension $d$. Hence the curse of dimensionality - the bane of high-dimensional approximation - is mitigated to a significant extent. We also discuss several practical issues, including unknown noise (due to truncation or numerical error), and highlight a number of open problems and challenges.