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A greedy forward-backward algorithm for atomic norm constrained minimization

In many applications in signal and image processing, communications, and system identification, one aims to recover a signal that has a simple representation in a given basis or frame. Key devices for obtaining such representations are objects called atoms, and functions called atomic norms. These concepts unify the idea of simple representations across several known applications, and motivate extensions to new problem classes of interest. In important special cases, fast and efficient algorithms are available to solve the reconstruction problems, but an approach that works well for the general atomic-norm paradigm has not been forthcoming to date. In this paper, we combine a greedy selection scheme with a backward step that sparsifies the basis by removing less significant elements that were included at earlier iterations. We show that the overall scheme achieves the same convergence rate as the forward greedy scheme alone, provided that backward steps are taken only when they do not degrade the solution quality too badly. Finally, we validate our method by describing applications to three problems of interest.