The benefits of acting locally: Reconstruction algorithms for sparse in levels signals with stable and robust recovery guarantees

TL;DR

This paper develops new stable and robust recovery guarantees for iterative algorithms tailored to sparse in levels signals, enhancing computational efficiency and theoretical understanding in compressive imaging and related fields.

Contribution

It introduces the first stable, robust guarantees for iterative hard thresholding and CoSaMP algorithms in the sparse in levels setting, extending existing theory.

Findings

New theoretical recovery guarantees for iterative algorithms.

Numerical validation of an extended orthogonal matching pursuit.

Comparison showing improved guarantees over weighted  minimization.

Abstract

The sparsity in levels model recently inspired a new generation of effective acquisition and reconstruction modalities for compressive imaging. Moreover, it naturally arises in various areas of signal processing such as parallel acquisition, radar, and the sparse corruptions problem. Reconstruction strategies for sparse in levels signals usually rely on a suitable convex optimization program. Notably, although iterative and greedy algorithms can outperform convex optimization in terms of computational efficiency and have been studied extensively in the case of standard sparsity, little is known about their generalizations to the sparse in levels setting. In this paper, we bridge this gap by showing new stable and robust uniform recovery guarantees for sparse in level variants of the iterative hard thresholding and the CoSaMP algorithms. Our theoretical analysis generalizes recovery guarantees currently available in the case of standard sparsity and favorably compare to sparse in levels guarantees for weighted $\ell^1$ minimization. In addition, we also propose and numerically test an extension of the orthogonal matching pursuit algorithm for sparse in levels signals.