We investigate the computation of a sparse solution to an underdetermined system of linear equations using the Huber loss function as a proxy for the 1-norm and a quadratic error term à la Lasso. The approach is termed “penalized Huber loss”. The results of the paper allow to calculate a sparse solution using a simple extrapolation formula under a sign constancy condition that can be removed if one works with extreme points. Conditions leading to sign constancy, as well as necessary and sufficient conditions for computation of a sparse solution by penalized Huber loss, and ties among different solutions are presented.

中文翻译：

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欠定Huber损失的线性方程组的稀疏解

我们研究使用Huber损失函数作为1-范数和二次误差项àla Lasso的代理，对欠定线性方程组的稀疏解的计算。该方法称为“惩罚性胡贝尔损失”。本文的结果允许在符号恒定性条件下使用简单的外推公式计算稀疏解，如果使用极点，则可以将其删除。给出了导致符号恒定的条件，以及通过惩罚性的Huber损失计算稀疏解的必要和充分条件，以及不同解之间的联系。