Generalized LASSO with under-determined regularization matrices

• A theoretical study is conducted on the generalized LASSO.
• A condition is derived to guarantee the equivalence between the generalized LASSO and LASSO.
• An upper bound is given on the complexity of the LARS algorithm.
• A closed-form solution of the generalized LASSO is derived for several cases.

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 form via the 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.