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。对于包括独立同分布（i.i.d.）高斯和旋转不变矩阵在内的结构化随机测量系综，AMP的性能可通过称为状态演化（SE）的标量递归来表征。通常假定其具有伪Lipschitz（多项式）光滑性。在本研究中，我们将AMP的状态演化方法扩展至一类具有独立（不必同分布）元素的新测量矩阵。同时，我们将其推广至一类被称为受控函数的一般函数族——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Lipschitz函数不同，受控函数呈指数增长。通过利用Lindeberg-Feller定理解决了假设测量系综缺乏结构性的问题；针对假设受控函数光滑性缺失的问题，我们提出了一种基于AMP实例经验统计量的条件化技巧。最终成果使得状态演化方法可应用于更广泛的测量系综类别及一类全新的函数族。