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In high-dimensional data analysis, bi-level sparsity is often assumed whencovariates function group-wisely and sparsity can appear either at the grouplevel or within certain groups. In such cases, an ideal model should be able toencourage the bi-level variable selection consistently. Bi-level variableselection has become even more challenging when data have heavy-taileddistribution or outliers exist in random errors and covariates. In this paper,we study a framework of high-dimensional M-estimation for bi-level variableselection. This framework encourages bi-level sparsity through acomputationally efficient two-stage procedure. In theory, we provide sufficientconditions under which our two-stage penalized M-estimator possessessimultaneous local estimation consistency and the bi-level variable selectionconsistency if certain nonconvex penalty functions are used at the group level.Both our simulation studies and real data analysis demonstrate satisfactoryfinite sample performance of the proposed estimators under different irregularsettings.

A High-dimensional M-estimator Framework for Bi-level Variable Selection