  This paper proposes a low-computational Bayesian algorithm for noisy sparse
recovery (NSR), called BHT-BP. In this framework, we consider an LDPC-like
measurement matrices which has a tree-structured property, and additive white
Gaussian noise. BHT-BP has a joint detection-and-estimation structure
consisting of a sparse support detector and a nonzero estimator. The support
detector is designed under the criterion of the minimum detection error
probability using a nonparametric belief propagation (nBP) and composite binary
hypothesis tests. The nonzeros are estimated in the sense of linear MMSE, where
the support detection result is utilized. BHT-BP has its strength in noise
robust support detection, effectively removing quantization errors caused by
the uniform sampling-based nBP. Therefore, in the NSR problems, BHT-BP has
advantages over CS-BP which is an existing nBP algorithm, being comparable to
other recent CS solvers, in several aspects. In addition, we examine impact of
the minimum nonzero value of sparse signals via BHT-BP, on the basis of the
results of the recent literature. Our empirical result shows that variation of
x_min is reflected to recovery performance in the form of SNR shift.
