Beyond sparsity: Recovering structured representations by ${\ell}^1$ minimization and greedy algorithms

Finding a sparse approximation of a signal from an arbitrary dictionary is a very useful tool to solve many problems in signal processing. Several algorithms, such as Basis Pursuit (BP) and Matching Pursuits (MP, also known as greedy algorithms), have been introduced to compute sparse approximations of signals, but such algorithms a priori only provide sub-optimal solutions. In general, it is difficult to estimate how close a computed solution is from the optimal one. In a series of recent results, several authors have shown that both BP and MP can successfully recover a sparse representation of a signal provided that it is sparse enough, that is to say if its support (which indicates where are located the nonzero coefficients) is of sufficiently small size. In this paper we define identifiable structures that support signals that can be recovered exactly by minimization (Basis Pursuit) and greedy algorithms. In other words, if the support of a representation belongs to an identifiable structure, then the representation will be recovered by BP and MP. In addition, we obtain that if the output of an arbitrary decomposition algorithm is supported on an identifiable structure, then one can be sure that the representation is optimal within the class of signals supported by the structure. As an application of the theoretical results, we give a detailed study of a family of multichannel dictionaries with a special structure (corresponding to the representation problem ) often used in, e.g., under-determined source sepa-ration problems or in multichannel signal processing. An identifiable structure for such dictionaries is defined using a generalization of Tropp’s Babel function which combines the coherence of the mixing matrix with that of the time-domain dictionary , and we obtain explicit structure conditions which ensure that both minimization and a multi-channel variant of Matching Pursuit can recover structured multichannel representations. The multichannel Matching Pursuit algorithm is described in detail and we conclude with a discussion of some implications of our results in terms of blind source separation based on sparse decompositions. Communicated by Yuesheng Xu