Communications-Inspired Projection Design with Application to Compressive Sensing

We consider the recovery of an underlying signal xᵐ based on projection measurements of the form y=Mx+w, where y^ and w is measurement noise; we are interested in the case m. It is assumed that the signal model p(x) is known and that w(w;0, Σw) for known Σw. The objective is to design a projection matrix M× m to maximize key information-theoretic quantities with operational significance, including the mutual information between the signal and the projections I(x;y) or the Rényi entropy of the projections h_α ( y ) (Shannon entropy is a special case). By capitalizing on explicit characterizations of the gradients of the information measures with respect to the projection matrix, where we also partially extend the well-known results of Palomar and Verdú from the mutual information to the Rényi entropy domain, we reveal the key operations carried out by the optimal projection designs: mode exposure and mode alignment. Experiments are considered for the case of compressive sensing (CS) applied to imagery. In this context, we provide a demonstration of the performance improvement possible through the application of the novel projection designs in relation to conventional ones, as well as justification for a fast online projection design method with which state-of-the-art adaptive CS signal recovery is achieved.