On the benefits of the block-sparsity structure in sparse signal recovery

Abstract

We study the problem of support recovery of block-sparse signals, where nonzero entries occur in clusters, via random noisy measurements. By drawing analogy between the problem of block-sparse signal recovery and the problem of communication over Gaussian multi-input and single-output multiple access channel, we derive the sufficient and necessary condition under which exact support recovery is possible. Based on the results, we show that block-sparse signals can reduce the number of measurements required for exact support recovery, by at least '1/(block size)', compared to conventional or scalar-sparse signals. The minimum gain is guaranteed by increased signal to noise power ratio (SNR) and reduced effective number of entries (i.e., not individual elements but blocks) that are dominant at low SNR and at high SNR, respectively. When the correlation between the elements of each nonzero block is low, a larger gain than '1/(block size)' is expected due to, so called, diversity effect, especially in the moderate and low SNR regime.Index Terms-Support recovery, Block-sparse signals, MISO-MAC channel capacity

Bayesian Compressive Sensing of Sparse Signals with Unknown Clustering Patterns

“…For the MMV case, this means that in addition to the joint sparsity structure, the non-zeros also appear in clusters in each column of in . This feature has been referred to as the clustered structure or block-sparsity pattern in the literature [ 7 , 24 , 25 ]. Applications of clustered sparsity for the SMV cases arise in problems such as gene expression analysis [ 26 ], image reconstruction of hand-written digits [ 27 ], and audio signals using the discrete cosine transform (DCT) basis [ 28 ].…”

“…For example, in magnetoencephalography (MEG), the goal is to investigate the locations where most brain activities are produced. The brain activities exhibit contiguity, meaning that they occur in localized regions [ 25 ]. Therefore, the measured signal at each snapshot can be modeled as a block-sparse SMV problem.…”

AMP-B-SBL: An algorithm for clustered sparse signals using approximate message passing

“…Applications can be found in gene expression analysis [12] and direction of arrival (DOA) [13]. This feature has been referred to as block-sparsity or sparse clustered pattern in the literature [7], [8], [14], [15]. Regarding the SBL algorithms, two main priors have been considered to encourage the sparsity.…”

Distribution agnostic structured sparsity recovery algorithms

“…For example, an ideal sparse channel consisting of a few multi path components could be represented in a block sparse structure [22]. Some other interesting situations where block sparsity arises include gene expression analysis [23], time series data analysis involving lagged variables forming a block, multiple measurement vector (MMV) [24], Peak-to-average-power-ratio (PAPR) reduction in OFDM [25], neural activity [26] and seismic data analysis [27].…”