The restricted isometry property for random block diagonal matrices

Abstract

In Compressive Sensing, the Restricted Isometry Property (RIP) ensures that robust recovery of sparse vectors is possible from noisy, undersampled measurements via computationally tractable algorithms. It is by now well-known that Gaussian (or, more generally, sub-Gaussian) random matrices satisfy the RIP under 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 certain measurement architectures. These issues have recently motivated considerable effort towards studying the RIP for structured random matrices. In this paper, we study the RIP for block diagonal measurement matrices where each block on the main diagonal is itself a sub-Gaussian random matrix. Our main result states that such matrices can indeed satisfy the RIP but that the requisite number of measurements depends on certain properties of the basis in which the signals are sparse. In the best case, these matrices perform nearly as well as dense Gaussian random matrices, despite having many fewer nonzero entries.

Signal Recovery and System Calibration from Multiple Compressive Poisson Measurements

“…More importantly, as we elaborate on later, such a Rademacher configuration facilitates an easy construction of a nonnegative sensing matrix necessary for Poisson sensing, by simply adding a DC offset. The Gaussian configuration proposed in [8,21] cannot be easily adapted to guarantee such a nonnegativity constraint. Nevertheless, in addition to their value for Poisson sensing, our concentration-of-measure results also shed light on CS for multiple linear measurements; in the next section, we present some applications of our concentration-of-measure results in that case.…”

“…The block-diagonal measurement matrix poses a more challenging (and practical) problem. Hence, the proof techniques from [22,24] cannot be applied to the multiple-measurements case, and the nonnegativity constraint on the sensing matrix also invalidates adaptation of the results in [8,21].…”

“…Compressive sensing with multiple measurements has only been addressed in [8,21], and there for the Gaussian measurement model. The case of a Poisson single-measurement model (without perturbation on the sensing matrix) has been considered in [22,24].…”

“…CS with multiple measurements has only been addressed in [8,21], and there for the Gaussian measurement model. Furthermore, randomly constituted sensing matrices employed in the aforementioned work violate the nonnegativity constraint of the Poisson measurement model.…”

Hierarchical CoSaMP for compressively sampled sparse signals with nested structure

“…Moreover, in general, the acquisition phase itself could in principle be designed so as to reduce the approximation error on the inner nested subsets, by organizing the sensing matrix so as to relate subsets of the measurements to subsets of the signal sparsity domain, much in the same way as block diagonal matrices [17,18]; further analysis is needed to investigate on the RIP of a CS matrix inducing a nested structure on the acquired CS measurements.…”