The Structured Smooth Adjustment for Square-root Regularization: Theory, algorithm and applications

摘要

In this paper, a novel method called Structured Smooth Adjustment for Square-root Regularization (SSASR) is proposed to simultaneously select grouped variables and encourage piecewise smoothness within each group. This approach is based on square-root regularization with a joint norm regularizer that, like the group lasso, shrinks a group of coefficients to identically zero and, additionally, involves an additional IGTV regularizer to enforce certain structural constraints – instead of pure sparsity – on the coefficients. We show the SSASR estimator can achieve optimal estimation and prediction, which is adaptive to the unknown noise level, under some mild conditions on the design matrix. To implement, an efficient algorithm termed Scaled Dual Forward–backward Splitting is proposed with proved convergence. Furthermore, we carry out an experimental evaluation on both synthetic data and real data obtained from glioblastoma multiforme samples and gray images.