A unified algorithm for mixed $$l_{2,p}$$ l 2 , p -minimizations and its application in feature selection

Recently, matrix norm $$l_{2,1}$$ l 2 , 1 has been widely applied to feature selection in many areas such as computer vision, pattern recognition, biological study and etc. As an extension of $$l_1$$ l 1 norm, $$l_{2,1}$$ l 2 , 1 matrix norm is often used to find jointly sparse solution. Actually, computational studies have showed that the solution of $$l_p$$ l p -minimization ( $$0>p>1$$ 0 > p > 1 ) is sparser than that of $$l_1$$ l 1 -minimization. The generalized $$l_{2,p}$$ l 2 , p -minimization ( $$p\in (0,1]$$ p ∈ ( 0 , 1 ] ) is naturally expected to have better sparsity than $$l_{2,1}$$ l 2 , 1 -minimization. This paper presents a type of models based on $$l_{2,p}\ (p\in (0, 1])$$ l 2 , p ( p ∈ ( 0 , 1 ] ) matrix norm which is non-convex and non-Lipschitz continuous optimization problem when $$p$$ p is fractional ( $$0>p>1$$ 0 > p > 1 ). For all $$p$$ p in $$(0, 1]$$ ( 0 , 1 ] , a unified algorithm is proposed to solve the $$l_{2,p}$$ l 2 , p -minimization and the convergence is also uniformly demonstrated. In the practical implementation of algorithm, a gradient projection technique is utilized to reduce the computational cost. Typically different $$l_{2,p}\ (p\in (0,1])$$ l 2 , p ( p ∈ ( 0 , 1 ] ) are applied to select features in computational biology. Copyright Springer Science+Business Media New York 2014