Group-based sparsity models are proven instrumental in linear regression
problems for recovering signals from much fewer measurements than standard
compressive sensing. The main promise of these models is the recovery of
"interpretable" signals through the identification of their constituent groups.
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 (WMC). Leveraging a graph-based
understanding of group models, we describe group structures which enable
correct model selection in polynomial time via dynamic programming.
Furthermore, group structures that lead to totally unimodular constraints have
tractable discrete as well as convex 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 of group-sparse approximations for two tractable models, among which
the tree sparsity model, and illustrate selection and computation trade-offs
between our framework and the existing convex relaxations.

本研究通过图形模型的理解和动态规划技术，基于组稀疏模型的组结构，解决了组模型选择问题，提出了一种修正的组模型，并研究了两个可操作的模型的群稀疏逼近的 Pareto 前沿以及选择和计算的权衡。