The Restricted Isometry Property for Random Block Diagonal Matrices

In Compressive Sensing, the Restricted Isometry Property (RIP) ensures that robust recovery of sparsevectors is possible from noisy, undersampled measurements via computationally tractable algorithms. Itis by now well-known that Gaussian (or, more generally, sub-Gaussian) random matrices satisfy the RIPunder certain conditions on the number of measurements. Their use can be limited in practice, however,due to storage limitations, computational considerations, or the mismatch of such matrices with certainmeasurement architectures. These issues have recently motivated considerable effort towards studying theRIP for structured random matrices. In this paper, we study the RIP for block diagonal measurementmatrices where each block on the main diagonal is itself a sub-Gaussian random matrix. Our main resultstates that such matrices can indeed satisfy the RIP but that the requisite number of measurements dependson certain properties of the basis in which the signals are sparse. In the best case, these matrices performnearly as well as dense Gaussian random matrices, despite having many fewer nonzero entries. Keywords— Compressive Sensing, Block Diagonal Matrices, Restricted Isometry Property