A comparison of typical ℓp minimization algorithms

Recently, compressed sensing has been widely applied to various areas such as signal processing, machine learning, and pattern recognition. To find the sparse representation of a vector w.r.t. a dictionary, an l1 minimization problem, which is convex, is usually solved in order to overcome the computational difficulty. However, to guarantee that the l1 minimizer is close to the sparsest solution, such as those with the lp ð0opo1Þ penalties require much weaker incoherence conditions and smaller signal to noise ratio to guarantee a successful recovery. Hence the lp ð0opo1Þ regularization serves as a better alternative to the popular l1 one. In this paper, we review some typical algorithms, Iteratively Reweighted l1 minimization (IRL1), Iteratively Reweighted Least Squares (IRLS) (and its general form General Iteratively Reweighted Least Squares (GIRLS)), and Iteratively Thresholding Method (ITM), for lp minimization and do comprehensive comparison among them, in which IRLS is identified as having the best performance and being the fastest as well. & 2013 Elsevier B.V. All rights reserved.