Difference of two norms-regularizations for Q-Lasso. Issue 1 (3rd August 2020) - Moudafi, Abdellatif
Abstract : The focus of this paper is in Q -Lasso introduced in Alghamdi et al. (2013) which extended the Lasso by Tibshirani (1996). The closed convex subset Q belonging in a Euclidean m -space, for m ∈ IN, is the set of errors when linear measurements are taken to recover a signal/image via the Lasso. Based on a recent work by Wang (2013), we are interested in two new penalty methods for Q -Lasso relying on two types of difference of convex functions (DC for short) programming where the DC objective functions are the difference of l 1 and l σ q norms and the difference of l 1 and l r norms with r > 1 . By means of a generalized q -term shrinkage operator upon the special structure of l σ q norm, we design a proximal gradient algorithm for handling the DC l 1 − l σ q model. Then, based on the majorization scheme, we develop a majorized penalty algorithm for the DC l 1 − l r model. The convergence results of our new algorithms are presented as well. We would like to emphasize that extensive simulation results in the case Q = { b } show that these two new algorithms offer improved signal recovery performance and require reduced computational effort relative to state-of-the-art l 1 and l p (p ∈ ( 0, 1 ) ) models, see Wang (2013). We also devise two DC Algorithms on the spirit of a paper where exact DC representation of the cardinality constraint is investigated and which also used the largest- q norm of l σ q and presented numerical results that show the efficiency of our DC Algorithm in comparison with other methods using other penalty terms in the context of quadratic programing, see Jun-ya et al. (2017).