Density evolution analysis of node-based verification-based algorithms in compressed sensing

Abstract

Abstract-In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressive sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension n). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration in the asymptotic regime where n → ∞. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of non-trivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of . Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of n. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is much simpler and more accurate.

Compressed Sensing using sparse binary measurements: A rateless coding perspective

“…For simplicity and convenience of analysis, our work in this paper is based on the noiseless assumption. This is consistent with the nature of recovery process since LM1/LM2 methods cannot cope with noisy coefficients [4] [5]. Noisy extensions have been considered elsewhere [3] [14], however, we believe that there is a room for improvement in their simplicity and efficiency.…”

“…However, CSBP is rather complex both from the analysis and implementation perspective, since the messages exchanged across factor graph represent continuous probability distributions. As lowcomplexity versions of CSBP amenable to rigorous analysis, VB decoding algorithms (such as LM1 and LM2) have been investigated in noiseless CS scenario in [4] [5]. Using the CS setup analogue to LDPC codes, in these works, codingtheoretic tools such as the density evolution (DE) and stopping set analysis are applied to assess the CS recovery performance.…”

Iterative Reweighted <formula formulatype="inline"><tex Notation="TeX">$\ell_{2}/\ell_{1}$</tex> </formula> Recovery Algorithms for Compressed Sensing of Block Sparse Signals

“…Particularly, when the 2-norm of nonzero blocks decays fast enough if sorted in descending order, δΓ << δ. This would imply that the upper bound in (42) can be smaller than (6). Thus, IR (2) -2/ 1 may outperform the 2/ 1 minimization.…”

“…In the same work, conditions for the recovery of the signal based on the notions of mutual and cumulative subspace coherence were also derived. 2 A popular category of recovery algorithms for compressed sensing are those based on message-passing algorithms, see, e.g., [31][32][33][34][35][36][37][38][39][40][41][42]. In particular, the approximate message-passing (AMP) algorithm of [36] has attracted much attention due to its remarkable performance/complexity trade-off [43].…”

Ultra Low-Complexity Detection of Spectrum Holes in Compressed Wideband Spectrum Sensing

“…In the context of block sparse signals, it is demonstrated in [13] that AMP with James-Steins shrinkage estimator (AMP-JS) can outperform the existing block sparse recovery algorithms. Another subcategory of iterative message-passing recovery algorithms are verification-based ones [14]- [16], which are based on sparse sensing matrices and are thus of lower complexity compared to the message-passing algorithms on dense matrices (graphs). These algorithms however are very sensitive to noise.…”