关于近似消息传递的一个简明说明
A Concise Tutorial on Approximate Message Passing
摘要：
标准线性回归的高维信号恢复在通信、压缩传感、图像处理等许多工程领域都是一个重要的挑战。Donoho \textit{et al}提出的近似消息传递（AMP）算法是解决这类问题的一种有效算法，该算法在独立同分布（IID）亚高斯随机矩阵区域内具有贝叶斯最优性能。AMP的一个显著特点是AMP的动力学行为可以完全由一个标量方程来预测，称为站演变（SE）。虽然AMP在IID亚高斯随机矩阵中是最优的，但当测量矩阵超过IID亚高斯时，AMP可能无法收敛。为了扩展随机测量矩阵的区域，提出了一种与期望传播（EP）、期望一致（EC）和向量AMP（VAMP）算法相同的EP相关算法正交AMP（OAMP）。本文旨在对这些算法进行综述。我们从最坏的情况开始，即，最小绝对收缩和选择算子（LASSO）推理问题，并给出了基于消息传递的AMP的详细推导过程。此外，在贝叶斯最优设置中，我们给出了贝叶斯最优AMP，它与LASSO的AMP略有不同。此外，我们回顾了一些AMP相关的算法：OAMP，VAMP和内存AMP（MAMP），它可以应用于更一般的随机矩阵。
英文摘要：
High-dimensional signal recovery of standard linear regression is a key challenge in many engineering fields, such as, communications, compressed sensing, and image processing. The approximate message passing (AMP) algorithm proposed by Donoho \textit{et al} is a computational efficient method to such problems, which can attain Bayes-optimal performance in independent identical distributed (IID) sub-Gaussian random matrices region. A significant feature of AMP is that the dynamical behavior of AMP can be fully predicted by a scalar equation termed station evolution (SE). Although AMP is optimal in IID sub-Gaussian random matrices, AMP may fail to converge when measurement matrix is beyond IID sub-Gaussian. To extend the region of random measurement matrix, an expectation propagation (EP)-related algorithm orthogonal AMP (OAMP) was proposed, which shares the same algorithm with EP, expectation consistent (EC), and vector AMP (VAMP). This paper aims at giving a review for those algorithms. We begin with the worst case, i.e., least absolute shrinkage and selection operator (LASSO) inference problem, and then give the detailed derivation of AMP derived from message passing. Also, in the Bayes-optimal setting, we give the Bayes-optimal AMP which has a slight difference from AMP for LASSO. In addition, we review some AMP-related algorithms: OAMP, VAMP, and Memory AMP (MAMP), which can be applied to more general random matrices.
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