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The OSCAR (octagonal selection and clustering algorithm for regression)regularizer consists of a L_1 norm plus a pair-wise L_inf norm (responsible forits grouping behavior) and was proposed to encourage group sparsity inscenarios where the groups are a priori unknown. The OSCAR regularizer has anon-trivial proximity operator, which limits its applicability. We reformulatethis regularizer as a weighted sorted L_1 norm, and propose its groupingproximity operator (GPO) and approximate proximity operator (APO), thus makingstate-of-the-art proximal splitting algorithms (PSAs) available to solveinverse problems with OSCAR regularization. The GPO is in fact the APO followedby additional grouping and averaging operations, which are costly in time andstorage, explaining the reason why algorithms with APO are much faster thanthat with GPO. The convergences of PSAs with GPO are guaranteed since GPO is anexact proximity operator. Although convergence of PSAs with APO is may not beguaranteed, we have experimentally found that APO behaves similarly to GPO whenthe regularization parameter of the pair-wise L_inf norm is set to anappropriately small value. Experiments on recovery of group-sparse signals(with unknown groups) show that PSAs with APO are very fast and accurate.

Solving OSCAR regularization problems by proximal splitting algorithms