Thresholding-based Iterative Selection Procedures for Model Selection and Shrinkage

TL;DR

This paper introduces Thresholding-based Iterative Selection Procedures (TISP) for model selection and shrinkage, providing a unified, efficient approach that extends orthogonal design results to general matrices, with a novel Hybrid-TISP method balancing sparsity and shrinkage.

Contribution

It develops a new thresholding-based framework for nonconvex penalized regression, including a Hybrid-TISP method that combines hard and ridge thresholding for improved performance.

Findings

Hybrid-TISP outperforms existing methods in test error.

Theoretical convergence and properties are established nonasymptotically.

The approach simplifies computation and analysis for nonorthogonal designs.

Abstract

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 $l_1$-constraint (or the soft-thresholding) 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 nonconvex 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 $l_0$-penalty and the $l_2$-penalty, and adaptively achieves the right balance between shrinkage and selection in statistical modeling. In practice, Hybrid-TISP shows superior performance in test-error and is parsimonious.