Performance of sparse representation algorithms using randomly generated frames

We consider sparse representations of signals with at most L nonzero coefficients using a frame F of size M in C^N. For any F, we establish a universal numerical lower bound on the average distortion of the representation as a function of the sparsity epsi; = L/N of the representation and redundancy (r - 1) = M/N-1 of F. In low dimensions (e.g., N = 6, 8, 10), this bound is much stronger than the analytical and asymptotic bounds given in another of our papers. In contrast, it is much less straightforward to compute. We then compare the performance of randomly generated frames to this numerical lower bound and to the analytical and asymptotic bounds given in the aforementioned paper. In low dimensions, it is shown that randomly generated frames perform about 2 dB away from the theoretical lower bound, when the optimal sparse representation algorithm is used. In higher dimensions, we evaluate the performance of randomly generated frames using the greedy orthogonal matching pursuit (OMP) algorithm. The results indicate that for small values of ε, OMP performs close to the lower bound and suggest that the loss of the suboptimal search using orthogonal matching pursuit algorithm grows as a function of ε. In all cases, the performance of randomly generated frames hardens about their average as N grows, even when using the OMP algorithm.