A projection matrix design method for MSE deduction in adaptive compressive sensing

In adaptive compressive sensing, measurement matrix is adaptively designed to recover the sparse signal. After the support of the sparse signal is recovered, the projection measurement is designed to reduce the mean squared error of the estimated sparse signal. This paper proposes a projection matrix design method using the support set of the sparse signal estimated via previous measurements. The projection matrix design problem is formulated as an optimization problem, which minimizes the mean squared error of the estimated sparse signal with the energy of the projection matrix being constrained. A water filling like algorithm is proposed to solve the optimization problem. It is proved that the Water Filling algorithm gives the globally optimal solution once the sufficient condition is satisfied. A method is given to check whether the result given by the Water Filling algorithm is globally optimal or not. Numerical experiments demonstrate the effectiveness of the designed algorithm.