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.

中文翻译：

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具有欠定正则化矩阵的广义 LASSO

本文研究了广义 LASSO 和基本 LASSO 公式之间的内在联系。前者是后者的扩展版本，通过向系数引入正则化矩阵。我们表明，当正则化矩阵在满秩条件下为偶数或欠定时，广义 LASSO 可以通过拉格朗日框架转换为 LASSO 形式。此外，我们表明，LASSO 的一些已发表结果可以扩展到广义 LASSO，并且 LASSO 的一些变体，例如稳健 LASSO，可以重写为广义 LASSO 形式，从而可以转化为基本 LASSO。基于这种联系，许多关于 LASSO 的现有结果，例如高效 LASSO 求解器，可以用于广义 LASSO。