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Estimation of a sparse group of sparse vectors

We consider a problem of estimating a sparse group of sparse normal mean vectors. The proposed approach is based on penalized likelihood estimation with complexity penalties on the number of nonzero mean vectors and the numbers of their "significant" components, and can be performed by a computationally fast algorithm. The resulting estimators are developed within Bayesian framework and can be viewed as MAP estimators. We establish their adaptive minimaxity over a wide range of sparse and dense settings. The presented short simulation study demonstrates the efficiency of the proposed approach that successfully competes with the recently developed sparse group lasso estimator.