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≥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 γ. In this paper, we obtain basic properties of the solutions as a function of γ. Because of ill posedness, we also apply l1-l2 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.