Oracle performance estimation of Bernoulli-distributed sparse vectors

Compressed Sensing (CS) is now a well-established research area and a plethora of applications has emerged in the last decade. In this context, assuming N available noisy measurements, lower bounds on the Bayesian Mean Square Error (BMSE) for the estimated entries of a sparse amplitude vector are derived in the proposed work for (i) a Gaussian overcomplete measurement matrix and (ii) for a random support, assuming that each entry is modeled as the product of a continuous random variable and a Bernoulli random variable indicating that the current entry is non-zero with probability P. A closed-form expression of the Expected CRB (ECRB) is proposed. In the second part, the BMSE of the Linear Minimum MSE (LMMSE) estimator is derived and it is proved that the LMMSE estimator tends to be statistically efficient in asymptotic conditions, i.e., if product (1- P)^2 SNR is maximized. This means that in the context of the Gaussian CS problem, the LMMSE estimator gathers together optimality in the low noise variance regime and a simple derivation (as opposed to the derivation of the MMSE estimator). This result is original because the LMMSE estimator is generally sub-optimal for CS when the measurement matrix is a single realizationof a given random process.