The Restricted Isometry Property for block diagonal matrices

In compressive sensing (CS), the Restricted Isometry Property (RIP) is a powerful condition on measurement operators which ensures robust recovery of sparse vectors is possible from noisy, undersampled measurements via computationally tractable algorithms. Early papers in CS showed that Gaussian random matrices satisfy the RIP with high probability, but such matrices are usually undesirable in practical applications due to storage limitations, computational considerations, or the mismatch of such matrices with certain measurement architectures. To alleviate some or all of these difficulties, recent research efforts have focused on structured random matrices. In this paper, we study block diagonal measurement matrices where each block on the main diagonal is itself a Gaussian random matrix. The main result of this paper shows that such matrices can indeed satisfy the RIP but that the requisite number of measurements depends on the coherence of the basis in which the signals are sparse. In the best case—for signals that are sparse in the frequency domain—these matrices perform nearly as well as dense Gaussian random matrices despite having many fewer nonzero entries.