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

Wanling Xie () and Hu Yang ()
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Wanling Xie: Hunan University of Technology and Business
Hu Yang: Chongqing University

AStA Advances in Statistical Analysis, 2023, vol. 107, issue 3, No 5, 469-507

Abstract: Abstract 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 $$\ell _{2,1}+\ell _2$$ ℓ 2 , 1 + ℓ 2 penalty. As a type of square-root-based procedure, one distinct feature is that the estimator is independent of the unknown noise level $$\sigma $$ σ , which is non-trivial to estimate under the high-dimensional setting, especially when $$p\gg n$$ p ≫ 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.

Keywords: Collinearity; Group square-root elastic net; Group sparsity; Noise level; Oracle inequality (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10182-022-00443-x

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