贝叶斯最优卷积AMP
Bayes-Optimal Convolutional AMP
摘要：
本文提出了贝叶斯最优卷积近似消息传递（CAMP）的压缩感知信号恢复。CAMP使用与近似消息传递（AMP）相同的低复杂度匹配滤波器（MF）进行干扰抑制。为了改善AMP的病态传感矩阵的收敛特性，AMP中所谓的Onsager校正项被替换为所有先前消息的卷积。在感知矩阵正交不变的假设下，通过状态演化（SE）确定卷积中的抽头系数，以实现估计误差的渐近高斯性。推导了一个用于优化CAMP算法中去噪器序列的SE方程。如果SE方程收敛于一个不动点且该不动点是唯一的，则证明了优化后的CAMP对所有正交不变的感知矩阵都是贝叶斯最优的。对于具有低到中等条件数的感测矩阵，CAMP可以实现与需要线性最小均方误差（LMMSE）滤波器而不是MF的高复杂度正交/矢量AMP相同的性能。
英文摘要：
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 denoisers in CAMP. The optimized CAMP is proved to be Bayes-optimal for all orthogonally invariant sensing matrices if the SE equation converges to a fixed-point and if the fixed-point is unique. For sensing matrices with low-to-moderate condition numbers, CAMP can achieve the same performance as high-complexity orthogonal/vector AMP that requires the linear minimum mean-square error (LMMSE) filter instead of the MF.
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