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Variable (feature, gene, model, which we use interchangeably) selections forregression with high-dimensional BIGDATA have found many applications inbioinformatics, computational biology, image processing, and engineering. Oneappealing approach is the L0 regularized regression which penalizes the numberof nonzero features in the model directly. L0 is known as the most essentialsparsity measure and has nice theoretical properties, while the popular L1regularization is only a best convex relaxation of L0. Therefore, it is naturalto expect that L0 regularized regression performs better than LASSO. However,it is well-known that L0 optimization is NP-hard and computationallychallenging. Instead of solving the L0 problems directly, most publications sofar have tried to solve an approximation problem that closely resembles L0regularization.  In this paper, we propose an efficient EM algorithm (L0EM) that directlysolves the L0 optimization problem. $L_0$EM is efficient with high dimensionaldata. It also provides a natural solution to all Lp p in [0,2] problems. Theregularized parameter can be either determined through cross-validation or AICand BIC. Theoretical properties of the L0-regularized estimator are given undermild conditions that permit the number of variables to be much larger than thesample size. We demonstrate our methods through simulation and high-dimensionalgenomic data. The results indicate that L0 has better performance than LASSOand L0 with AIC or BIC has similar performance as computationally intensivecross-validation. The proposed algorithms are efficient in identifying thenon-zero variables with less-bias and selecting biologically important genesand pathways with high dimensional BIGDATA.

Efficient Regularized Regression for Variable Selection with L0 Penalty