Bayes-Optimal Convolutional AMP
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This paper proposes Bayes-optimal convolutional approximate message-passing (CAMP) for signal recovery in compressed sensing. CAMP uses the same low-complexity matched filter (MF) for interference suppression as approximate message-passing (AMP). To improve the convergence property of AMP for ill-conditioned sensing matrices, the so-called Onsager correction term in AMP is replaced by a convolution of all preceding messages. The tap coefficients in the convolution are determined so as to realize asymptotic Gaussianity of estimation errors via state evolution (SE) under the assumption of orthogonally invariant sensing matrices. An SE equation is derived to optimize the sequence of thresholding functions in CAMP. The optimized CAMP is proved to be Bayes-optimal for all orthogonally invariant sensing matrices if the SE equation has a unique fixed-point. This implies that CAMP can cover the same class of sensing matrices as high-complexity orthogonal/vector AMP that requires the linear minimum mean-square error (LMMSE) filter instead of the MF.
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