$l_{2,p}$ Matrix Norm and Its Application in Feature Selection

Recently, l2,1 matrix norm has been widely applied to many areas such as computer vision, pattern recognition, biological study and etc. As an extension of l1 vector norm, the mixed l2,1 matrix norm is often used to find jointly sparse solutions. Moreover, an efficient iterative algorithm has been designed to solve l2,1-norm involved minimizations. Actually, computational studies have showed that lp-regularization (0 < p < 1) is sparser than l1-regularization, but the extension to matrix norm has been seldom considered. This paper presents a definition of mixed l2,p (p ∈ (0, 1]) matrix pseudo norm which is thought as both generalizations of lp vector norm to matrix and l2,1-norm to nonconvex cases (0 < p < 1). Fortunately, an efficient unified algorithm is proposed to solve the induced l2,pnorm (p ∈ (0, 1]) optimization problems. The convergence can also be uniformly demonstrated for all p ∈ (0, 1]. Typical p ∈ (0, 1] are applied to select features in computational biology and the experimental results show that some choices of 0 < p < 1 do improve the sparse pattern of using p = 1.