Approximate Message Passing (AMP) algorithms are a class of iterative procedures for computationally-efficient estimation in high-dimensional inference and estimation tasks. Due to the presence of an 'Onsager' correction term in its iterates, for $N \times M$ design matrices $\mathbf{A}$ with i.i.d. Gaussian entries, the asymptotic distribution of the estimate at any iteration of the algorithm can be exactly characterized in the large system limit as $M/N \rightarrow \delta \in (0, \infty)$ via a scalar recursion referred to as state evolution. In this paper, we show that appropriate functionals of the iterates, in fact, concentrate around their limiting values predicted by these asymptotic distributions with rates exponentially fast in $N$ for a large class of AMP-style algorithms, including those that are used when high-dimensional generalized linear regression models are assumed to be the data-generating process, like the generalized AMP algorithm, or those that are used when the measurement matrix is assumed to be right rotationally invariant instead of i.i.d. Gaussian, like vector AMP and generalized vector AMP. In practice, these more general AMP algorithms have many applications, for example in in communications or imaging, and this work provides the first study of finite sample behavior of such algorithms.

翻译：近似消息传递（AMP）算法是一类用于高维推断和估计任务中实现高效计算的迭代过程。由于迭代中引入了"昂萨格"修正项，对于具有独立同分布高斯元素的$N \times M$设计矩阵$\mathbf{A}$，在$M/N \rightarrow \delta \in (0, \infty)$的大系统极限下，算法任意迭代步估计量的渐近分布可通过标量递归（即状态演化）精确刻画。本文证明：对于一大类AMP类算法（包括假设数据生成过程为高维广义线性回归模型时使用的广义AMP算法，以及假设测量矩阵为右旋转不变（而非独立同分布高斯）时使用的向量AMP和广义向量AMP算法），其迭代量的适当泛函实际上以$N$的指数速率快速集中于这些渐近分布所预测的极限值。在实践中，这些更广义的AMP算法在通信或成像等领域具有广泛应用，而本研究首次提供了此类算法有限样本行为的理论分析。