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In this paper, we present a probabilistic analysis of iterative node-basedverification-based (NB-VB) recovery algorithms over irregular graphs in thecontext of compressed sensing. Verification-based algorithms are particularlyinteresting due to their low complexity (linear in the signal dimension $n$).The analysis predicts the average fraction of unverified signal elements ateach iteration $\ell$ where the average is taken over the ensembles of inputsignals and sensing matrices. The analysis is asymptotic ($n \rightarrow\infty$) and is similar in nature to the well-known density evolution techniquecommonly used to analyze iterative decoding algorithms. Compared to theexisting technique for the analysis of NB-VB algorithms, which is based onnumerically solving a large system of coupled differential equations, theproposed method is much simpler and more accurate. This allows us to designirregular sensing graphs for such recovery algorithms. The designed irregulargraphs outperform the corresponding regular graphs substantially. For example,for the same recovery complexity per iteration, we design irregular graphs thatcan recover up to about 40% more non-zero signal elements compared to theregular graphs. Simulation results are also provided which demonstrate that theproposed asymptotic analysis matches the performance of recovery algorithms forlarge but finite values of $n$.

Analysis and Design of Irregular Graphs for Node-Based Verification-Based Recovery Algorithms in Compressed Sensing