Message-Passing Strategies for High-Dimensional Inference

本研究集中在设计和分析矩阵恢复算法的问题，使用突破性的“近似消息传递”（AMP）的框架，最近提出的压缩感知的背景下。 该项目中考虑的矩阵恢复问题包括矩阵完备化、鲁棒PCA、字典学习、矩阵不确定性下的压缩感知和多项式分类，其中目标是从其乘积的噪声观测中恢复一对矩阵，其中观测可能被元素非线性扭曲，并且其中矩阵可能包括诸如组稀疏性、非负性、以及每行/向量范数约束。 这些问题经常出现在工程、科学和医学的许多子学科中，并且数据量不断增加。 因此，我们迫切需要一种精确、鲁棒、计算效率高的算法。本研究主要包括：1）线性推理的AMP方法的扩展（如在压缩感测中）到双线性推理，2）由通信中的涡轮解码启发的消息传递策略的集成，其促进了复杂结构先验的利用，3）自动调整信号先验和观测先验建模参数的概率贝叶斯方法的开发，4）假设模型中不确定性的显式处理，以及5）用于严格性能分析的AMP状态演化框架的开发。

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.