Natural Thresholding Algorithms for Signal Recovery with Sparsity

TL;DR

This paper introduces natural thresholding algorithms for sparse signal recovery, offering a computationally efficient alternative to optimal $k$-thresholding with proven performance guarantees and empirical robustness.

Contribution

The paper develops the natural thresholding family via first-order approximation, reducing computational cost while maintaining recovery guarantees and robustness.

Findings

NT algorithms are computationally faster than OT-based algorithms.

NT algorithms perform comparably to mainstream sparse recovery methods.

Theoretical guarantees are established under restricted isometry property.

Abstract

The algorithms based on the technique of optimal $k$-thresholding (OT) were recently proposed for signal recovery, and they are very different from the traditional family of hard thresholding methods. However, the computational cost for OT-based algorithms remains high at the current stage of their development. This stimulates the development of the so-called natural thresholding (NT) algorithm and its variants in this paper. The family of NT algorithms is developed through the first-order approximation of the so-called regularized optimal $k$-thresholding model, and thus the computational cost for this family of algorithms is significantly lower than that of the OT-based algorithms. The guaranteed performance of NT-type algorithms for signal recovery from noisy measurements is shown under the restricted isometry property and concavity of the objective function of regularized optimal $k$-thresholding model. Empirical results indicate that the NT-type algorithms are robust and very comparable to several mainstream algorithms for sparse signal recovery.