Polynomial approximation via compressed sensing of high-dimensional functions on lower sets

TL;DR

This paper introduces a novel compressed sensing approach for high-dimensional polynomial approximation of smooth functions, leveraging lower sets and new algorithms to reduce sample complexity and improve recovery guarantees.

Contribution

It presents a weighted $ ext{l}_1$ minimization and iterative hard thresholding method tailored for lower set structures, extending previous work to general orthonormal systems with provable efficiency.

Findings

Reduced sample complexity compared to existing methods

Improved bounds for the restricted isometry property

Numerical results demonstrating computational efficiency

Abstract

This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we present an innovative weighted $\ell_1$ minimization procedure with a precise choice of weights, and a new iterative hard thresholding method, for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices in [5] to the general case of continuous bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted $\ell_1$ minimization strategy.