On the design of linear projections for compressive sensing with side information

In this paper, we study the problem of projection kernel design for the reconstruction of high-dimensional signals from low-dimensional measurements in the presence of side information, assuming that the signal of interest and the side information signal are described by a joint Gaussian mixture model (GMM). In particular, we consider the case where the projection kernel for the signal of interest is random, whereas the projection kernel associated to the side information is designed. We then derive sufficient conditions on the number of measurements needed to guarantee that the minimum mean-squared error (MMSE) tends to zero in the low-noise regime. Our results demonstrate that the use of a designed kernel to capture side information can lead to substantial gains in relation to a random one, in terms of the number of linear projections required for reliable reconstruction.