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This paper discusses a class of thresholding-based iterative selectionprocedures (TISP) for model selection and shrinkage. People have long beforenoticed the weakness of the convex $l_1$-constraint (or the soft-thresholding)in wavelets and have designed many different forms of nonconvex penalties toincrease model sparsity and accuracy. But for a nonorthogonal regressionmatrix, there is great difficulty in both investigating the performance intheory and solving the problem in computation. TISP provides a simple andefficient way to tackle this so that we successfully borrow the rich results inthe orthogonal design to solve the nonconvex penalized regression for a generaldesign matrix. Our starting point is, however, thresholding rules rather thanpenalty functions. Indeed, there is a universal connection between them. But adrawback of the latter is its non-unique form, and our approach greatlyfacilitates the computation and the analysis. In fact, we are able to build theconvergence theorem and explore theoretical properties of the selection andestimation via TISP nonasymptotically. More importantly, a novel Hybrid-TISP isproposed based on hard-thresholding and ridge-thresholding. It provides afusion between the $l_0$-penalty and the $l_2$-penalty, and adaptively achievesthe right balance between shrinkage and selection in statistical modeling. Inpractice, Hybrid-TISP shows superior performance in test-error and isparsimonious.

Thresholding-based Iterative Selection Procedures for Model Selection and Shrinkage