Recovery Bounds for Cardinality Regularized Optimization Problem

Mixed sparse optimization has been extensively applied in the modeling of many important problems in various disciplines, in which the sparse structure appears as the inter-group and intra-group manners simultaneously. In this paper, we consider the ₀ ℓ 0 regularization problem for mixed sparse optimization and investigate its consistency theory. In particular, we first introduce the notions of sparse eigenvalue conditions, one of the weakest regularity conditions in the literature, and discuss their relations with the uniquely solvable property and restricted eigenvalue conditions. Then we establish the oracle property without any regularity condition and provide a recovery bound for the global solutions of the ₀ ℓ 0 regularized mixed sparse optimization problem under the assumption of sparse eigenvalue condition. Moreover, by virtue of the notion of epi-convergence, an asymptotic analysis is provided to advance the understanding of the convergence of the 2,p ℓ 2 , p regularization to the 2,0 ℓ 2 , 0 regularization as p→ 0₊ p → 0 + , in terms of regularity condition, global solution set and recovery bound, respectively.