This paper presents a unified framework for constructing Approximate Message Passing (AMP) algorithms for rotationally-invariant models. By employing a general iterative algorithm template and reducing it to long-memory Orthogonal AMP (OAMP), we systematically derive the correct Onsager terms of AMP algorithms. This approach allows us to rederive an AMP algorithm introduced by Fan and Opper et al., while shedding new light on the role of free cumulants of the spectral law. The free cumulants arise naturally from a recursive centering operation, potentially of independent interest beyond the scope of AMP. To illustrate the flexibility of our framework, we introduce two novel AMP variants and apply them to estimation in spiked models.

翻译：本文提出了一个用于构建旋转不变模型中近似消息传递（AMP）算法的统一框架。通过采用通用迭代算法模板并将其简化为长记忆正交AMP（OAMP），我们系统地推导出AMP算法的正确昂萨格项。该方法使我们能够重新推导出由Fan和Opper等人提出的AMP算法，同时为谱律自由累积量的作用提供了新的见解。自由累积量通过递归中心化操作自然产生，这一操作可能在AMP范畴之外也具有独立研究价值。为展示本框架的灵活性，我们引入了两种新型AMP变体，并将其应用于尖峰模型中的估计问题。