$l_{2,p}$ Matrix Norm and Its Application in Feature Selection
It addresses feature selection in computational biology, offering an incremental improvement over existing methods.
This paper tackles the problem of feature selection by introducing a new $l_{2,p}$ matrix norm as an extension of $l_{2,1}$-norm to nonconvex cases, with experimental results showing that some choices of $0<p<1$ improve sparsity patterns compared to $p=1$.
Recently, $l_{2,1}$ matrix norm has been widely applied to many areas such as computer vision, pattern recognition, biological study and etc. As an extension of $l_1$ vector norm, the mixed $l_{2,1}$ matrix norm is often used to find jointly sparse solutions. Moreover, an efficient iterative algorithm has been designed to solve $l_{2,1}$-norm involved minimizations. Actually, computational studies have showed that $l_p$-regularization ($0<p<1$) is sparser than $l_1$-regularization, but the extension to matrix norm has been seldom considered. This paper presents a definition of mixed $l_{2,p}$ $(p\in (0, 1])$ matrix pseudo norm which is thought as both generalizations of $l_p$ vector norm to matrix and $l_{2,1}$-norm to nonconvex cases $(0<p<1)$. Fortunately, an efficient unified algorithm is proposed to solve the induced $l_{2,p}$-norm $(p\in (0, 1])$ optimization problems. The convergence can also be uniformly demonstrated for all $p\in (0, 1]$. Typical $p\in (0,1]$ 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 $p=1$.