Thresholding-based Iterative Selection Procedures for Model Selection and Shrinkage

This paper discusses a class of thresholding-based iterative selection procedures (TISP) for model selection and shrinkage. People have long before noticed the weakness of the convex l1-constraint (or the softthresholding) in wavelets and have designed many different forms of nonconvex penalties to increase model sparsity and accuracy. But for a nonorthogonal regression matrix, there is great difficulty in both investigating the performance in theory and solving the problem in computation. TISP provides a simple and efficient way to tackle this so that we successfully borrow the rich results in the orthogonal design to solve the (nonconvexly) penalized regression for a general design matrix. Our starting point is, however, thresholding rules rather than penalty functions. Indeed, there is a universal connection between them. But a drawback of the latter is its non-unique form, and our approach greatly facilitates the computation and the analysis. In fact, we are able to build the convergence theorem and explore theoretical properties of the selection and estimation via TISP nonasymptotically. More importantly, a novel Hybrid-TISP is proposed based on hard-thresholding and ridge-thresholding. It provides a fusion between the l0-penalty and the l2-penalty, and adaptively achieves the right balance between shrinkage and selection in statistical modeling. In practice, HybridTISP shows superior performance in test-error and is parsimonious. AMS 2000 subject classifications: Primary 62J07, 62J05.