Subspace Thresholding Pursuit: A Reconstruction Algorithm for Compressed Sensing

TL;DR

This paper introduces Subspace Thresholding Pursuit (STP), an iterative greedy algorithm for sparse signal reconstruction in compressed sensing, demonstrating theoretical guarantees and superior empirical performance over existing methods.

Contribution

The paper presents STP, a novel algorithm combining subspace pursuit and iterative hard thresholding, with theoretical guarantees and improved empirical results for various signal types.

Findings

STP has theoretical guarantees comparable to $\, ext{l}_1$ minimization.

STP outperforms state-of-the-art algorithms for Gaussian signals.

STP improves empirical performance for large undersampling ratios.

Abstract

We propose a new iterative greedy algorithm for reconstructions of sparse signals with or without noisy perturbations in compressed sensing. The proposed algorithm, called \emph{subspace thresholding pursuit} (STP) in this paper, is a simple combination of subspace pursuit and iterative hard thresholding. Firstly, STP has the theoretical guarantee comparable to that of $\ell_1$ minimization in terms of restricted isometry property. Secondly, with a tuned parameter, on the one hand, when reconstructing Gaussian signals, it can outperform other state-of-the-art reconstruction algorithms greatly; on the other hand, when reconstructing constant amplitude signals with random signs, it can outperform other state-of-the-art iterative greedy algorithms and even outperform $\ell_1$ minimization if the undersampling ratio is not very large. In addition, we propose a simple but effective method to improve the empirical performance further if the undersampling ratio is large. Finally, it is showed that other iterative greedy algorithms can improve their empirical performance by borrowing the idea of STP.