Approximate Message Passing Algorithms and Networks

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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.