Approximate Message Passing Algorithms and Networks

工程、科学和医学中最重要的问题是从高维测量中恢复信息信号的问题。这个问题表现在许多形式上，例如，从几个噪声傅立叶投影重建高质量的图像，确定患者数据中的哪些特征最有可能与给定的疾病相关联，或者分类图像中存在哪些对象。直到最近，信号恢复的主导方法还是算法。但如今，算法正越来越多地被深度神经网络(DNN)所取代，深度神经网络可以直接从数据中学习最优推理策略。本课题研究了算法和深度神经网络(DNN)在高维信号恢复中的应用，充分利用两者之间的联系，在算法方面研究了向量近似消息传递(VAMP)算法。与Donoho、Maleki和Montanari的原始AMP算法一样，VAMP算法具有低复杂度和标量状态演化的特点，严格而简洁地表征了其行为。然而，VAMP适用于比AMP大得多的一类问题。在DNN方面，本项目研究其架构灵感来自VAMP中的处理步骤的DNN。得到的DNN是高度可解释的，并且对于一些简单的应用来说，是统计最优的。该项目旨在开发基于VAMP的DNN设计框架，以处理更复杂的应用程序。

A problem of paramount importance in engineering, science, and medicine is that of recovering information signals from high-dimensional measurements. This problem manifests in many forms, e.g., reconstructing a high-quality image from a few noisy Fourier projections, determining which features in patient data are most likely associated with a given disease, or classifying which objects are present within an image. Until recently, the dominant approach to signal recovery was algorithmic. But nowadays, algorithms are increasingly being replaced by deep neural networks (DNNs), which can learn optimal inference strategies directly from the data. This project researches algorithmic as well as deep-neural-network (DNN) approaches to high-dimensional signal recovery, leveraging connections between them to make advances in both.On the algorithmic front, this project investigates the vector approximate message passing (VAMP) algorithm. Like the original AMP algorithm of Donoho, Maleki, and Montanari, the VAMP algorithm enjoys low complexity and a scalar state-evolution that rigorously and concisely characterizes its behavior. However, VAMP is applicable to a much larger class of problems than AMP. On the DNN front, this project investigates DNNs whose architecture is inspired by the processing steps within VAMP. The resulting DNNs are highly interpretable and, for some simple applications, statistical optimal. This project aims to develop this VAMP-based DNN design framework to work with more complex applications.