ℓ1-2 Regularization for Sparse Optimization: Consistency and Global Convergence

The Formula: see text regularization method has a strong sparsity-promoting capability in approaching sparse solutions of linear inverse problems and gains successful applications in various mathematics disciplines and applied science fields. This paper aims to investigate the consistency theory and global convergent algorithms for the Formula: see text regularization problem. In the theoretical aspect, we introduce a notion of restricted eigenvalue condition relative to the Formula: see text penalty and employ it to establish an oracle property and a recovery bound for the global solution of the Formula: see text 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 Formula: see text 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 verify our theoretical convergence results in showing that the 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. Funding: This work was supported by the National Outstanding Youth Foundation of China Grant 12222112. Y. Hu’s work is supported by the National Key Research and Development Program of China Grant 2025YFA1016903, Shenzhen Medical Research Fund Grant B2502001, National Natural Science Foundation of China Grants 12571327 and 12426311, Project of Educational Commission of Guangdong Province Grant 2025ZDZX2059, and Shenzhen Science and Technology Program Grants RCJC20221008092753082 and JCYJ20241202124209011. H. Wang’s work is supported by the National Natural Science Foundation of China Grant 12571326. X. Yang’s work is supported by the Research Grants Council of Hong Kong Grants PolyU 15205223 and NPolyU507/22 and an internal grant Grant 4-ZZPP.