Signal Processing Over Networks: Graph-Based Methods for Data Analysis

对于当今的工程师和数据科学家来说，利用数据造福人类是一个巨大的挑战（例如，通过环境监测、改善公共卫生和可持续性）。该研究项目旨在开发新的信号处理理论和方法，以解决与处理和学习非结构化或不规则采样数据相关的挑战。现有的信号处理技术非常适合于具有规则、有序域的时间序列和图像等数据。许多当代应用程序产生大量、复杂和非结构化的数据。图提供了一种有原则的形式来捕捉变量和实体之间的复杂关系。在某些应用中，图表可以捕捉信号背后的物理结构（例如，道路网络链路上的交通强度，智能电网节点上的需求信号，或社会网络中人们意见的衡量标准）。当信号在空间和/或时间上不规则采样时（例如，在大脑表面采样的EEG和/或MRI信号，或在一个区域上不规则分布的传感器网络），图也可以作为一种有用的手段来编码我们期望产生相似值的数据采样位置。在其他应用程序中，图不是直接明显的，它可以编码实体之间的逻辑关系（例如，相关关系、因果关系或其他影响关系）。在这种情况下，为了更好地理解复杂系统的结构、组织和功能，从观察中推断出一个图通常是很有趣的。******本研究计划将对新兴的图信号处理领域作出贡献。拟议方案的具体目标是按照三个重点组织的。(1)我们将开发新的理论和方法，在假设信号在图上平滑的模型下从信号推断图。在不能在每个顶点都观察到信号的情况下，我们将推断图的结构或统计性质，这些性质对于抽样等其他应用可能仍然有用。(2)我们将开发新的理论和方法来逼近和压缩图信号。虽然图信号的采样和滤波理论变得越来越成熟，但对于信号产生过程和图结构的哪些条件是使结果信号平滑或以其他方式简约可表示所必需的，人们知之甚少。理论结果将有助于量化用于表示（即，近似或压缩）图形信号的系数数量与所产生的结果误差之间的权衡，给定一系列图形和图形信号。(3)我们将开发高维非线性/非高斯状态空间模型的跟踪方法，以及利用图结构提高计算效率的图信号采样/滤波方法。

Using data to benefit humanity is a grand challenge for today's engineers and data scientists (e.g., via environmental monitoring, improving public health, and sustainability). This research program aims to develop novel signal processing theory and methods to address challenges associated with processing and learning from unstructured or irregularly-sampled data. Existing signal processing techniques are well-suited for data such as time series and images which have a regular, well-ordered domain. Many contemporary applications produce data that is massive, complex, and unstructured. Graphs provide a principled formalism to capture complex relationships among variables and entities. In some applications the graph may capture a physical structure underlying the signal (e.g., traffic intensities on links of a road network, demand signals at nodes of the smart grid, or a measure of people's opinions in a social network). Graphs may also serve as a useful means to encode which data sampling locations we expect to produce similar values when signals are sampled irregularly in space and/or time (e.g., EEG and/or MRI signals sampled on the surface of the brain, or a network of sensors spread irregularly over a region). In other applications, where the graph is not directly apparent, it may encode logical relationships among entities (e.g., correlative, causal, or otherwise influential relationships). In this case it is often of interest to infer a graph from the observations in order to better understand the structure, organization, and function of a complex system.******This research program will make contributions to the burgeoning field of graph signal processing. The specific objectives of the proposed program are organized along three thrusts. (1) We will develop novel theory and methods for inferring graphs from signals under models where the signals are assumed to be smooth over the graph. In cases where signals cannot be observed at every vertex, we will infer structural or statistical properties of the graph that may still be useful for other applications like sampling. (2) We will develop novel theory and methods for approximating and compressing graph signals. While the theories of sampling and filtering graph signals are becoming more mature, little is known about what conditions on the signal-generating process and the graph structure are necessary for the resulting signal to be smooth or otherwise parsimoniously representable. The theoretical results will facilitate quantifying the tradeoff between the number of coefficients used to represent (i.e., approximate or compress) a graph signal and the resulting error incurred, given a family of graphs and graph signals. (3) We will develop methods for tracking in high-dimensional non-linear/non-Gaussian state-space models and sampling/filtering graph signals that exploit graph structure to improve computational efficiency.