Message-Passing Strategies for High-Dimensional Inference

本研究的重点是利用最近在压缩感知背景下提出的突破性的“近似消息传递”（AMP）框架来设计和分析矩阵恢复算法。在这个项目中考虑的矩阵恢复问题包括矩阵补全、鲁棒PCA、字典学习、矩阵不确定性下的压缩感知和多项分类，其中的目标是从它们乘积的噪声观测中恢复一对矩阵，其中的观测可能被元素非线性扭曲，其中的矩阵可能包括诸如群稀疏性、非负性和逐行/向量范数约束等结构。这样的问题经常出现在工程、科学和医学的许多分支学科中，并且随着数据规模的不断增加。因此，非常需要一种精确、鲁棒和计算效率高的算法方法。特别是，该研究涉及1)将AMP方法从线性推理（如压缩感知）扩展到双线性推理，2)在通信中受涡轮解码启发的消息传递策略的集成，有助于利用复杂的结构先验，3)开发经验贝叶斯方法，自动调整参数建模信号先验以及观察先验。4)明确处理假设模型中的不确定性，5)利用AMP状态演化框架进行严格的性能分析。

This research centers on the problem of designing and analyzing matrix recovery algorithms using the breakthrough "approximate message-passing" (AMP) framework that was recently proposed in the context of compressive sensing. The matrix recovery problems considered in this project include matrix completion, robust PCA, dictionary learning, compressive sensing under matrix uncertainty, and multinomial classification, where the goal is to recover a pair of matrices from noisy observations of their product, where the observations may be warped by element-wise nonlinearities, and where the matrices may include structure such as group sparsity, non-negativity, and per-row/vector norm constraints. Such problems arise frequently across many subdisciplines of engineering, science, and medicine, and with ever increasing data sizes. Thus, there is a great need for an algorithmic approach that is accurate, robust, and computationally efficient.In particular, the research involves 1) the extension of AMP methods from linear inference (as in compressive sensing) to bilinear inference, 2) the integration of message passing strategies inspired by turbo-decoding in communications that facilitate the exploitation of sophisticated structural priors, 3) the development of empirical-Bayesian methods that automatically tune the parameters modeling the signal prior as well as the observation prior, 4) the explicit treatment of uncertainty in the assumed model, and 5) the exploitation of the AMP state-evolution framework for rigorous performance analysis.