title: Generalized LASSO with under-determined regularization matrices.
abstract: This paper studies the intrinsic connection between a generalized LASSO and a basic LASSO formulation. The former is the extended version of the latter by introducing a regularization matrix to the coefficients. We show that when the regularization matrix is even- or under-determined with full rank conditions, the generalized LASSO can be transformed into the LASSO formviathe Lagrangian framework. In addition, we show that some published results of LASSO can be extended to the generalized LASSO, and some variants of LASSO,e.g., robust LASSO, can be rewritten into the generalized LASSO form and hence can be transformed into basic LASSO. Based on this connection, many existing results concerning LASSO,e.g., efficient LASSO solvers, can be used for generalized LASSO.
authors.affiliation: Key Laboratory of Biomedical Information Engineering of Ministry of Education and Department of Biomedical Engineering, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, China.
authors.forename: Junbo
authors.initials: J
authors.lastname: Duan
authors.affiliation: Université de Lorraine and the CNRS at the CRAN (Centre de Recherche en Automatique de Nancy), UMR 7039, Campus Sciences, BP 70239, 54506 Vandoeuvre-lés-Nancy Cedex, France.
authors.forename: Charles
authors.initials: C
authors.lastname: Soussen
authors.affiliation: Université de Lorraine and the CNRS at the CRAN (Centre de Recherche en Automatique de Nancy), UMR 7039, Campus Sciences, BP 70239, 54506 Vandoeuvre-lés-Nancy Cedex, France.
authors.forename: David
authors.initials: D
authors.lastname: Brie
authors.affiliation: L'UNAM Université, Ecole Centrale de Nantes and the CNRS at the Institut de Recherche en Communications et Cybernétique de Nantes (IRCCyN), UMR 6597, 1 rue de la Noë, BP 92101, F-44321 Nantes Cedex 3, France.
authors.forename: Jérôme
authors.initials: J
authors.lastname: Idier
authors.affiliation: Key Laboratory of Biomedical Information Engineering of Ministry of Education and Department of Biomedical Engineering, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, China.
authors.forename: Mingxi
authors.initials: M
authors.lastname: Wan
authors.affiliation: Department of Biomedical Engineering, and the Department of Biostatistics and Bioinformatics, Lindy Boggs Center, Tulane University, New Orleans, LA 70118, USA.
authors.forename: Yu-Ping
authors.initials: YP
authors.lastname: Wang
keywords: LASSO; deconvolution; diagonally dominant; generalized LASSO; robust LASSO; solution path; total variation