When over-complete dictionary is used, the search for the sparsest of all representations amounts to the need to solve a highly complicated combinatorial problem, with complexity growing exponenentially with the number of atoms in the dictionary. The Basis Pursuit algorithm (BP) (by Chen, Donoho, and Saundrers, 1995) was shown to be a highly effective tool for approximating the solution for the above problem.
In this talk we describe a fascinating analysis that explains the surprising empirical success of the PB. We start by presenting Donoho and Huo’s results, and then show how these results could be further improved. We first discuss the case of pair of ortho-bases and show tighter bounds on the required sparsity of the signal representation that guarantees BP-optimality. We then turn to extend previous results for arbitrary dictionaries, showing that all previous work falls as a special case of the new theory. Finally, we show how the obtained analysis results can be used to solve the problem of separating a volume of voxels into a sparse set of points, lines, and planes.