Greedy-Like Algorithms for the Cosparse Analysis Model

TL;DR

This paper introduces new greedy-like algorithms for the cosparse analysis model, providing theoretical guarantees and demonstrating their effectiveness through empirical evaluation.

Contribution

It proposes a new family of pursuit algorithms for the cosparse analysis model, with performance guarantees based on a restricted isometry property.

Findings

Algorithms perform well empirically with plain thresholding projection.

Theoretical guarantees are established assuming a near optimal projection scheme.

The methods extend greedy pursuit techniques to the cosparse analysis framework.

Abstract

The cosparse analysis model has been introduced recently as an interesting alternative to the standard sparse synthesis approach. A prominent question brought up by this new construction is the analysis pursuit problem -- the need to find a signal belonging to this model, given a set of corrupted measurements of it. Several pursuit methods have already been proposed based on $\ell_1$ relaxation and a greedy approach. In this work we pursue this question further, and propose a new family of pursuit algorithms for the cosparse analysis model, mimicking the greedy-like methods -- compressive sampling matching pursuit (CoSaMP), subspace pursuit (SP), iterative hard thresholding (IHT) and hard thresholding pursuit (HTP). Assuming the availability of a near optimal projection scheme that finds the nearest cosparse subspace to any vector, we provide performance guarantees for these algorithms. Our theoretical study relies on a restricted isometry property adapted to the context of the cosparse analysis model. We explore empirically the performance of these algorithms by adopting a plain thresholding projection, demonstrating their good performance.