Properties and Iterative Methods for the -Lasso

We introduce the -lasso which generalizes the well-known lasso of Tibshirani (1996) with a closed convex subset of a Euclidean m-space for some integer . This set 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 -lasso depend on a tuning parameter . In this paper, we obtain basic properties of the solutions as a function of . Because of ill posedness, we also apply regularization to the -lasso. In addition, we discuss iterative methods for solving the -lasso which include the proximal-gradient algorithm and the projection-gradient algorithm.

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