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）的标量递归来描述。传统上假设伪利普希茨（多项式）光滑性。本文中，我们将AMP的SE扩展到一类新的具有独立（不必同分布）元素的测量矩阵。我们还将其扩展到一类称为受控函数的一般函数类，这类函数不受多项式光滑性约束；与具有多项式光滑性的伪利普希茨函数不同，受控函数呈指数增长。通过利用林德伯格-费勒定理来处理假定测量集合缺乏结构性的问题。针对假设的受控函数缺乏光滑性的问题，我们提出了一种基于AMP实例经验统计量的条件化技术。所得到的结果使得SE能够应用于更广泛的测量集合类和新的一类函数。