Efficient Regularized Regression with L₀ Penalty for Variable Selection and Network Construction

Variable selections for regression with high-dimensional big data have found many applications in bioinformatics and computational biology. One appealing approach is the L₀ regularized regression which penalizes the number of nonzero features in the model directly. However, it is well known that L₀ optimization is NP-hard and computationally challenging. In this paper, we propose efficient EM (L₀EM) and dual L₀EM (DL₀EM) algorithms that directly approximate the L₀ optimization problem. While L₀EM is efficient with large sample size, DL₀EM is efficient with high-dimensional (n ≪ m) data. They also provide a natural solution to all L   p ∈ [0,2] problems, including lasso with p = 1 and elastic net with p ∈ [1,2]. The regularized parameter λ can be determined through cross validation or AIC and BIC. We demonstrate our methods through simulation and high-dimensional genomic data. The results indicate that L₀ has better performance than lasso, SCAD, and MC+, and L₀ with AIC or BIC has similar performance as computationally intensive cross validation. The proposed algorithms are efficient in identifying the nonzero variables with less bias and constructing biologically important networks with high-dimensional big data.