$\ell_{1\text{-}2}$ Regularization for Sparse Optimization: Consistency and Global Convergence

TL;DR

This paper studies the theoretical properties and develops algorithms for $\, ext{ extlbrackdbl}1 ext{-}2 ext{ extrbrack}$ regularization, demonstrating its effectiveness in promoting sparsity and recovering true sparse solutions in inverse problems.

Contribution

It introduces a restricted eigenvalue condition for $\, ext{ extlbrackdbl}1 ext{-}2 ext{ extrbrack}$ regularization, and proposes convergent algorithms with theoretical guarantees for sparse recovery.

Findings

Algorithms effectively recover true sparse solutions.

Proposed methods outperform existing algorithms in sparsity recovery.

Theoretical guarantees hold under restricted isometry property.

Abstract

The $\ell_{1\text{-}2}$ regularization method has a strong sparsity promoting capability in approaching sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. This paper aims to investigate the consistency theory and global convergent algorithms for the $\ell_{1\text{-}2}$ regularization problem. In the theoretical aspect, we introduce a notion of restricted eigenvalue condition relative to the $\ell_{1\text{-}2}$ penalty, and employ it to establish an oracle property and a recovery bound for the global solution of the $\ell_{1\text{-}2}$ regularization problem. In the algorithmic aspect, we propose two types of iterative thresholding algorithms with the truncation technique and the continuation technique, respectively, to solve the $\ell_{1\text{-}2}$ regularization problem. Moreover, under the assumption of the well-known restricted isometry property, we establish the convergence of the proposed algorithms to the ground true sparse solution within a tolerance relevant to the noise level and the recovery bound. Preliminary numerical results show that our proposed algorithms can approach the ground true sparse solution and significantly enhance the sparsity recovery capability, compared with the popular sparse optimization algorithms in the literature.