Learning Latent Graphs from Stationary Signals

近年来，具有许多特征以及这些特征之间和/或跨时间和空间的复杂交互的数据已经变得无处不在。如何从如此庞大复杂的数据中提取有意义的信息是最紧迫的问题之一，具有重大的科学和社会影响。这项研究将产生新的工具，通过图形或基于网络的数据表示来建模和分析这些数据。 从这项研究中获得的分析和计算工具可以应用于许多领域，包括经济学，金融学，神经科学以及社会和生物科学中的各个子领域。该项目还将为新一代研究人员提供培训机会，使他们能够为统计、数据科学和相关领域的快速发展作出贡献。这项研究的结果将通过出版物，会议演示，讲座和开源软件传播。 该项目将开发一种新的谱图模型（SGM）框架，将多变量数据建模为图参考的平稳信号。SGM框架从信号处理的角度为网络推理提供了新的工具，并为具有复杂依赖性（包括可能的时间或空间依赖性）的多变量观测建模提供了新的工具。它还通过有效的基于图形的表示提供了协方差估计的新工具。SGM框架结合了四个关键方面-协方差的谱表示，谱图理论，半参数建模和稀疏参数化。它允许图形或参数的时间依赖性，并可用于建模独立和相关的多变量观测。SGM框架具有广泛的应用范围，通过该研究开发的方法将应用于包括大脑活动和国际贸易数据在内的各种数据。该奖项反映了NSF的法定使命，通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

In recent years, data with many features and complicated interactions among these features and/or across time and space have become ubiquitous. How to extract meaningful information from such large complex data is one of the most pressing questions with significant scientific and societal implications. This research will generate new tools for modeling and analyzing such data through graph- or network-based representations of the data. The analytical and computational tools derived from this research can be applied to many fields including economics, finance, neuroscience, and various sub-fields within the social and biological sciences. This project will also provide training opportunities for a new generation of researchers empowering them to contribute to the rapid development of statistics, data science and related fields. Results of this research will be disseminated through publications, conference presentations, lectures, and open source software. This project will develop a novel Spectral Graph Models (SGM) framework which models multivariate data as graph-referenced stationary signals. The SGM framework provides new tools for network inference from a signal processing perspective and for modeling multivariate observations with complicated dependencies, including possible temporal or spatial dependence. It also provides new tools for covariance estimation through efficient graph-based representations. The SGM framework combines four key aspects - spectral representation of covariances, spectral graph theory, semiparametric modeling, and sparse parameterization. It allows for temporal dependency in graphs or parameters and can be used to model both independent and dependent multivariate observations. The SGM framework has a wide range of applications, and methods developed through this research will be applied to various types of data including brain activity and international trade data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.