Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure

Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page