Sparse group selection is the process of selecting a small part of nonoverlapping groups to achieve the good interpretability and prediction on the response, and it has recently seen increasing applications in machine learning, image processing and bio-medical fields. However, developing robust and efficient algorithms for group selection remains a challenging research topic due to the computational complexity and potential outliers in high-dimensional settings. Motivated by the superior performance of rank-based methodology, we design a fast and efficient algorithm based on the \(\ell _{2,0}\) penalty to achieve the goal of robust group selection for a given size of active groups s. This new algorithm can iteratively detect the active groups and exclude the irrelevant ones. When s is not less than \(s^*\) (the true size of active groups), we theoretically prove that the proposed algorithm covers the true subset of active groups with high probability and the estimation error of the solution sequence generated by our algorithm decays to the optimal error bound in a few iterations. Moreover, coupled with the group Bayesian information criterion, an adaptive algorithm is further introduced to determine the optimal s. Theoretically, without any prior knowledge of \(s^*\) , the proposed adaptive algorithm is able to exactly identify the true subset of active groups with probability approaching to one. Finally, extensive simulation examples show that our method outperforms existing competitors, resulting in significant improvements in terms of efficiency and accuracy of group selection and parametric estimation. The Bardet-Biedl syndrome gene expression data set is also analyzed to illustrate the application of our proposed method.
