Performance Estimation of Compressed Sensing with Oracle Information

This article discusses the performance of the oracle receiver in terms of the nor-malized mean square error in the sparse signal recovery process. The measurement is conducted in completely perturbed scenarios where the system is disturbed simul-taneously by a perturbation matrix exerted on the sensing matrix, a noise vector on the result of measurement and the input noise added directly on the signal. In a bid to achieve concrete results, the entries of the sensing matrix are specified as Bernoulli random variables. The article introduces and proves the lower and upper bounds of the mean square error of the reconstructed signal. Those theoretical bounds hold in high probability for high dimensional signals. Numerical results verified the conclusions, illustrating both the lower and upper bounds as close and meaningful estimations of the mean square error in the Bernoulli case. The result is then compared with previous works in literature on Gaussian sensing matrices. An estimation of the average recovery error is derived as a generalization of the conclusion in [13] for the Gaussian case