Sparse Representations Are Most Likely to Be the Sparsest Possible

Given a signal and a full-rank matrix with, we define the signal's overcomplete representations as all satisfying. Among all the possible solutions, we have special interest in the sparsest one—the one minimizing. Previous work has established that a representation is unique if it is sparse enough, requiring. The measure stands for the minimal number of columns from that are linearly dependent. This bound is tight—examples can be constructed to show that with or more nonzero entries, uniqueness is violated. In this paper we study the behavior of overcomplete representations beyond the above bound. While tight from a worst-case standpoint, a probabilistic point of view leads to uniqueness of representations satisfying. Furthermore, we show that even beyond this point, uniqueness can still be claimed with high confidence. This new result is important for the study of the average performance of pursuit algorithms—when trying to show an equivalence between the pursuit result and the ideal solution, one must also guarantee that the ideal result is indeed the sparsest.