On the Rate-Distortion Performance of Compressed Sensing

Encouraging recent results in compressed sensing or compressive sampling suggest that a set of inner products with random measurement vectors forms a good representation of a source vector that is known to be sparse in some fixed basis. With quantization of these inner products, the encoding can be considered universal for sparse signals with known sparsity level. We analyze the operational rate-distortion performance of such source coding both with genie-aided knowledge of the sparsity pattern and maximum likelihood estimation of the sparsity pattern. We show that random measurements induce an additive logarithmic rate penalty, i.e., at high rates the performance with rate R + O(log R) and random measurements is equal to the performance with rate R and deterministic measurements matched to the source.