{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/minimization-of-transformed-l-1-penalty","title":"Minimization of Transformed $L_1$ Penalty: Theory, Difference of Convex Function Algorithm, and Robust Application in Compressed Sensing","arxiv_id":"1411.5735","date":"2014-11-21","proceeding":null,"authors":["Shuai Zhang","Jack Xin"],"abstract":"We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed $l_1$ (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates $l_0$ and $l_1$ norms through a nonnegative parameter $a \\in (0,+\\infty)$, similar to $l_p$ with $p \\in (0,1]$, and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem and discuss the exact recovery of $l_0$ norm minimal solution based on the null space property (NSP). We then prove the stable recovery of $l_0$ norm minimal solution if the sensing matrix $A$ satisfies a restricted isometry property (RIP). Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The inner loop concerns an $l_1$ minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value $a=1$, and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with $l_1$ minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix $A$. DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.","url_abs":"http://arxiv.org/abs/1411.5735v4","url_pdf":"http://arxiv.org/pdf/1411.5735v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"minimization-of-transformed-l-1-penalty","repo_url":"https://github.com/zsivine/DCATL1","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}