The Restricted Isometry Property for block diagonal matrices

Abstract

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.

CCS: Energy-efficient data collection in clustered wireless sensor networks utilizing block-wise compressive sensing

“…Fortunately the matrix satisfies the restricted isometry property (RIP) to be able to work with CS. The RIP of BDMs has been studied in [38], [39] and it has been shown that BDMs can satisfy RIP and therefore can be used as efficient measurement matrices. The required number of the measurements though 2 A Rademacher random variable takes a value of +1 or -1 with equal probability.…”

“…According to (4) [38], we can see that for canonical and wavelet bases, the number of required measurements is a linear function of N c . Based on this, we can state the following lemma to find the optimal number of clusters N * c for minimizing the power consumption.…”

“…According to [38], [40], the number of measurements required for a BDM, consisting of N c blocks with Gaussian entries, to satisfy RIP with high probability is given as [38] …”

Spatial-temporal Correlation-Based Low-Latency Compressed Sensing in WSNs

“…Yap et al [10] have shown that block-diagonal random measurement matrices can perform as good as dense random measurement matrix in CS signal acquisition and recovery. So we also use block-diagonal measurement matrix.…”

Partitioned compressive sensing with neighbor-weighted decoding

“…In the compressive sensing field, partitioned encoding has been studied for block encoding of natural images [19], with basis-specific enhancements used to improve reconstruction quality. Authors have also recently proven sufficiency of blockdiagonal matrices for signal recovery [20], and extended these results to analyze signals heterogeneous across partitions [21].…”