Efficient Bi-level Variable Selection and Application to Estimation of Multiple Covariance Matrices

Abstract

Variable selection plays an important role in analyzing high dimensional data. When the data possesses certain group structures in which individual variables are also meaningful scientifically, we are naturally interested in selecting important groups as well as important variables. We introduce a new regularization by combining the $\ell _{p,0}$-norm and $\ell _0$-norm for bi-level variable selection. Using an appropriate DC (Difference of Convex functions) approximation, the resulting problem can be solved by DC Algorithm. As an application, we implement the proposed algorithm for estimating multiple covariance matrices sharing some common structures such as the locations or weights of non-zero elements. The experimental results on both simulated and real datasets demonstrate the efficiency of our algorithm.