An Efficient Semismooth Newton Method for Adaptive Sparse Signal Recovery Problems

TL;DR

This paper introduces an adaptive sparse recovery model combining p-_{1-2} regularization with a semismooth Newton method, effectively handling high coherence and noise in compressive sensing.

Contribution

It proposes a novel p-_{1-2} model for sparse recovery and develops a semismooth Newton algorithm with proven fast convergence.

Findings

The model outperforms traditional methods in high coherence scenarios.

The algorithm demonstrates fast local convergence.

Numerical experiments confirm the model's effectiveness.

Abstract

We know that compressive sensing can establish stable sparse recovery results from highly undersampled data under a restricted isometry property condition. In reality, however, numerous problems are coherent, and vast majority conventional methods might work not so well. Recently, it was shown that using the difference between $\ell_1$- and $\ell_2$-norm as a regularization always has superior performance. In this paper, we propose an adaptive $\ell_p$-$\ell_{1-2}$ model where the $\ell_p$-norm with $p\geq 1$ measures the data fidelity and the $\ell_{1-2}$-term measures the sparsity. This proposed model has the ability to deal with different types of noises and extract the sparse property even under high coherent condition. We use a proximal majorization-minimization technique to handle the nonconvex regularization term and then employ a semismooth Newton method to solve the corresponding convex relaxation subproblem. We prove that the sequence generated by the semismooth Newton method admits fast local convergence rate to the subproblem under some technical assumptions. Finally, we do some numerical experiments to demonstrate the superiority of the proposed model and the progressiveness of the proposed algorithm.