On the Global-Local Dichotomy in Sparsity Modeling
This addresses the global-local gap in sparsity modeling for inverse problems in imaging, offering a theoretical and computational improvement, though it appears incremental as it builds on existing sparse modeling frameworks.
The paper tackles the suboptimality of traditional sparse modeling for large data like images by proposing a global model constructed from local sparsity assumptions, showing it leads to a constrained underdetermined linear system solvable by methods like ADMM, with numerical evidence supporting the theory.
The traditional sparse modeling approach, when applied to inverse problems with large data such as images, essentially assumes a sparse model for small overlapping data patches. While producing state-of-the-art results, this methodology is suboptimal, as it does not attempt to model the entire global signal in any meaningful way - a nontrivial task by itself. In this paper we propose a way to bridge this theoretical gap by constructing a global model from the bottom up. Given local sparsity assumptions in a dictionary, we show that the global signal representation must satisfy a constrained underdetermined system of linear equations, which can be solved efficiently by modern optimization methods such as Alternating Direction Method of Multipliers (ADMM). We investigate conditions for unique and stable recovery, and provide numerical evidence corroborating the theory.