Group sparse recovery via group square-root elastic net and the iterative multivariate thresholding-based algorithm

In this work, we propose a novel group selection method called Group Square-Root Elastic Net. It is based on square-root regularization with a group elastic net penalty, i.e., a \(\ell _{2,1}+\ell _2\) penalty. As a type of square-root-based procedure, one distinct feature is that the estimator is independent of the unknown noise level \(\sigma \), which is non-trivial to estimate under the high-dimensional setting, especially when \(p\gg n\). In many applications, the estimator is expected to be sparse, not in an irregular way, but rather in a structured manner. It makes the proposed method very attractive to tackle both high-dimensionality and structured sparsity. We study the correct subset recovery under a Group Elastic Net Irrepresentable Condition. Both the slow rate bounds and fast rate bounds are established, the latter under the Restricted Eigenvalue assumption and Gaussian noise assumption. To implement, a fast algorithm based on the scaled multivariate thresholding-based iterative selection idea is introduced with proved convergence. A comparative study examines the superiority of our approach against alternatives.