A Measurement Rate-MSE Tradeoff for Compressive Sensing Through Partial Support Recovery

We consider the problem of estimating sparse vectors from noisy linear measurements in the high dimensionality regime. For a fixed number k of nonzero entries, we study the fundamental relationship between two relevant quantities: the measurement rate, which characterizes the asymptotic behavior of the dimensions of the measurement matrix in terms of the ratio m/log n (with m being the number of measurements and n the dimension of the sparse vector), and the estimation mean square error. First, we use an information-theoretic approach to derive sufficient conditions on the measurement rate to reliably recover a part of the support set that represents a certain fraction of the total vector power. Second, we characterize the mean square error of an estimator that uses partial support set information. Using these two parts, we derive a tradeoff between the measurement rate and the mean-square error. This tradeoff is achievable using a two-step approach: first support set recovery, and then estimation of the active components. Finally, for both deterministic and random vectors, we perform a numerical evaluation to verify the advantages of the methods based on partial support set recovery.