Approximate message passing (AMP) is a scalable, iterative approach to signal recovery. For structured random measurement ensembles, including independent and identically distributed (i.i.d.) Gaussian and rotationally-invariant matrices, the performance of AMP can be characterized by a scalar recursion called state evolution (SE). The pseudo-Lipschitz (polynomial) smoothness is conventionally assumed. In this work, we extend the SE for AMP to a new class of measurement matrices with independent (not necessarily identically distributed) entries. We also extend it to a general class of functions, called controlled functions which are not constrained by the polynomial smoothness; unlike the pseudo-Lipschitz function that has polynomial smoothness, the controlled function grows exponentially. The lack of structure in the assumed measurement ensembles is addressed by leveraging Lindeberg-Feller. The lack of smoothness of the assumed controlled function is addressed by a proposed conditioning technique leveraging the empirical statistics of the AMP instances. The resultants grant the use of the SE to a broader class of measurement ensembles and a new class of functions.

翻译：近似消息传递（AMP）是一种可扩展的迭代式信号恢复方法。对于结构化随机测量集合（包括独立同分布高斯矩阵和旋转不变矩阵），AMP的性能可通过称为状态演化（SE）的标量递归过程来刻画。传统研究中通常假设伪Lipschitz（多项式）光滑性。本文首先将AMP的状态演化方法拓展至一类新的测量矩阵（元素独立但不一定同分布），同时将其推广至更一般的函数类——控制函数，该函数不受多项式光滑性约束，与具有多项式光滑性的伪Lipschitz函数不同，控制函数呈指数增长。针对假设测量集合缺乏结构性的问题，我们利用Lindeberg-Feller中心极限定理进行解决；针对假设控制函数缺乏光滑性的问题，我们提出基于AMP实例经验统计量的条件化技术。研究成果使得状态演化方法可适用于更广泛的测量集合类别及新函数类。