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 ${\ell }_{2,1}$ 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.

在本文中，提出了一种新的方法，称为平方根正则化的结构平滑调整（SSASR），可以同时选择分组变量并鼓励每个组内的分段平滑。此方法基于联合的平方根正则化${\ell }_{2，\mathrm{1个}}$像套索一样，将一组系数缩小为相同零的范数正则化器，此外，还涉及一个额外的IGTV正则化器，以对系数实施某些结构性约束，而不是纯粹的稀疏性。我们表明，在设计矩阵的某些温和条件下，SSASR估计器可以实现最佳估计和预测，从而可以适应未知噪声水平。为了实现该算法，提出了一种有效的算法，该算法具有可证明的收敛性，该算法称为可缩放双前向后拆分。此外，我们对合成数据和从胶质母细胞瘤样样和灰度图像获得的真实数据进行了实验评估。