Properties and Iterative Methods for the Q -Lasso

Abstract

We introduce the $Q$-lasso which generalizes the well-known lasso of Tibshirani (1996) with $Q$ a closed convex subset of a Euclidean m-space for some integer $m\ge 1$. This set $Q$ can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the lasso. Solutions of the $Q$-lasso depend on a tuning parameter $\gamma$. In this paper, we obtain basic properties of the solutions as a function of $\gamma$. Because of ill posedness, we also apply $l_1\text{-}{l}_2$ regularization to the $Q$-lasso. In addition, we discuss iterative methods for solving the $Q$-lasso which include the proximal-gradient algorithm and the projection-gradient algorithm.