Interpolation via weighted $l_1$ minimization

TL;DR

This paper introduces a weighted $l_1$ minimization method that simultaneously promotes sparsity and smoothness in function interpolation, extending compressive sensing concepts to structured sparse expansions.

Contribution

It proposes specific weight choices in $l_1$ minimization to improve function reconstruction, integrating smoothness and sparsity priors, and extends theoretical properties to weighted sparse expansions.

Findings

Achieves interpolation rates for functions with weighted $l_p$ coefficients.

Extends restricted isometry and null space properties to weighted sparse settings.

Applicable to spherical harmonic and polynomial interpolation in multiple dimensions.

Abstract

Functions of interest are often smooth and sparse in some sense, and both priors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but cannot incorporate nonlinear sparsity structure. At the same time, nonlinear methods such as $l_1$ minimization can reconstruct sparse functions from very few samples, but do not necessarily encourage smoothness. Here we show that weighted $l_1$ minimization effectively merges the two approaches, promoting both sparsity and smoothness in reconstruction. More precisely, we provide specific choices of weights in the $l_1$ objective to achieve rates for functions with coefficient sequences in weighted $l_p$ spaces, $p<=1$. We consider the implications of these results for spherical harmonic and polynomial interpolation, in the univariate and multivariate setting. Along the way, we extend concepts from compressive sensing such as the restricted isometry property and null space property to accommodate weighted sparse expansions; these developments should be of independent interest in the study of structured sparse approximations and continuous-time compressive sensing problems.