Abstract-Group-based sparsity models are instrumental in linear and non-linear regression problems. The main premise of these models is the recovery of "interpretable" signals through the identification of their constituent groups, which can also provably translate in substantial savings in the number of measurements for linear models in compressive sensing. In this paper, we establish a combinatorial framework for group-model selection problems and highlight the underlying tractability issues. In particular, we show that the group-model selection problem is equivalent to the well-known NP-hard weighted maximum coverage problem. Leveraging a graph-based understanding of group models, we describe group structures that enable correct model selection in polynomial time via dynamic programming. Furthermore, we show that popular group structures can be explained by linear inequalities involving totally unimodular matrices, which afford other polynomial time algorithms based on relaxations. We also present a generalization of the group model that allows for within group sparsity, which can be used to model hierarchical sparsity. Finally, we study the Pareto frontier between approximation error and sparsity budget of group-sparse approximations for two tractable models, among which the tree sparsity model, and illustrate selection and computation tradeoffs between our framework and the existing convex relaxations.

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Abstract-Group-based sparsity models are proven instrumental in linear regression problems for recovering signals from much fewer measurements than standard compressive sensing. A promise of these models is to lead to "interpretable" signals for which we identify its constituent groups, however we show that, in general, claims of correctly identifying the groups with convex relaxations would lead to polynomial time solution algorithms for an NP-hard problem. Instead, leveraging a graph-based understanding of group models, we describe group structures which enable correct model identification in polynomial time via dynamic programming. We also show that group structures that lead to totally unimodular constraints have tractable relaxations. Finally, we highlight the non-convexity of the Pareto frontier of group-sparse approximations and what it means for tractability.

Abstract-Group-based sparsity models are proven instrumental in linear regression problems for recovering signals from much fewer measurements than standard compressive sensing. A promise of these models is to lead to "interpretable" signals for which we identify its constituent groups, however we show that, in general, claims of correctly identifying the groups with convex relaxations would lead to polynomial time solution algorithms for an NP-hard problem. Instead, leveraging a graph-based understanding of group models, we describe group structures which enable correct model identification in polynomial time via dynamic programming. We also show that group structures that lead to totally unimodular constraints have tractable relaxations. Finally, we highlight the non-convexity of the Pareto frontier of group-sparse approximations and what it means for tractability.