On the Use of Unit-Norm Tight Frames to Improve the Average MSE Performance in Compressive Sensing Applications

This letter considers the design of sensing matrices with good expected-case performance for compressive sensing applications. By capitalizing on the mean squared error (MSE) of the oracle estimator, whose performance has been shown to act as a benchmark to the performance of standard sparse recovery algorithms, we demonstrate that a unit-norm tight frame is the closest design-in the Frobenius norm sense-to the solution of a convex relaxation of the optimization problem that relates to the minimization of the MSE of the oracle estimator with respect to the sensing matrix. Simulation results reveal that the MSE performance of a unit-norm tight frame based sensing matrix surpasses that of other standard sensing matrix designs in various scenarios, which include sparse recovery with basis pursuit denoise (BPDN), the Dantzig selector and orthogonal matching pursuit (OMP). This also has important practical implications because a unit-norm tight frame based sensing matrix can be designed very efficiently.