CIF:Small:Learning Sparse Vector and Matrix Graphs from Time-Dependent Data

图是经常用于表达数据变量之间的依赖关系或相似性的数学结构。它们可以捕获看似不规则的高维数据中固有的复杂结构，使其成为信号处理，机器学习和数据科学中的宝贵工具。图形模型的应用包括金融、社交网络、环境网络、基因调控网络和功能性磁共振成像（fMRI）中的分类和探索性数据分析。然而，图表并不总是明确可用的。因此，给定数据，学习底层图结构是机器学习和信号处理应用的核心。在文献中，它通常是假设的时间数据包括多个独立实现的随机向量或矩阵的目标函数的选择，以优化以及在算法设计和分析。这一假设在实践中经常被违反。该项目明确考虑了时间依赖性数据，而不需要任何详细的参数化建模来捕获时间依赖性。据预计，更好的模型，结合短期和长期的记忆时间依赖将产生更准确的图形拓扑结构，因此，显着改善数据分析和学习任务。差分图估计的问题也在这个框架中解决，例如，在生物统计应用中，人们可能对健康和受损受试者的图形模型的差异感兴趣，或者在不同疾病状态下的模型，给定基因表达数据或fMRI信号。在这个项目中，考虑了三个主要研究方向：短期和长期依赖下的多变量依赖时间序列图学习、矩阵值依赖时间序列图学习和微分图学习。在所有三个推力的重点是稀疏图或稀疏微分图，在高维设置，其中的图形大小大于，或的顺序，数据样本大小。计算效率和准确，一般方法估计无向加权图的时间依赖的多元以及矩阵值的时间序列将进行调查。将考虑两类方法：频域方法，基于数据的离散傅立叶变换，在频域中产生近似独立的数据，允许利用基于复值信号处理的广泛的分析工具;以及基于时间延迟嵌入的时域方法，将问题转换为多属性图估计之一，其中随机向量，而不是标量，与每个图形节点相关联。将考虑问题的所有方面：算法设计和分析，稀疏参数估计的凸和非凸正则化函数下的优化，模型选择（惩罚参数的选择），理论性质的分析（如一致性和模型恢复），以及使用公共数据集应用于真实的数据。&该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Graphs are mathematical structures that are frequently used to express dependencies or similarities among data variables. They can capture complex structures inherent in seemingly irregular high-dimensional data, making them an invaluable tool in signal processing, machine learning, and data science. Applications of graphical models include classification and exploratory data analysis in finance, social networks, environmental networks, gene regulatory networks, and functional magnetic resonance imaging (fMRI). However, graphs are not always explicitly available. Therefore, given data, learning the underlying graph structure is central to applications in machine learning and signal processing. In the literature, it is typically assumed that the temporal data consists of multiple independent realizations of a random vector or matrix in the choice of the objective function to be optimized as well as in algorithm design and analysis. This assumption is often violated in practice. This project explicitly considers time-dependent data, without requiring any detailed parametric modeling to capture time dependencies. It is anticipated that better models incorporating short- and long-memory time dependence will yield more accurate graph topology, hence, significant improvements in data analysis and learning tasks. The problem of differential graph estimation is also addressed in this framework where, for example, in a bio-statistical application, one may be interested in the differences in the graphical models of healthy and impaired subjects, or models under different disease states, given gene-expression data or fMRI signals.In this project, three main research thrusts are considered: multivariate dependent time-series graph learning under both short- and long-range dependence, matrix-valued dependent time-series graph learning, and differential graph learning. The focus in all three thrusts is on sparse graphs or sparse differential graphs, under high-dimensional settings wherein the graph size is greater than, or of the order of, the data sample size. Computationally efficient and accurate, general approaches for estimation of undirected weighted graphs from time-dependent multivariate as well as matrix-valued time series will be investigated. Two classes of approaches will be considered: frequency-domain approaches based on the discrete Fourier transform of data which yields approximately independent data in the frequency domain, allowing a broad set of analysis tools based on complex-valued signal processing to be exploited; and time-domain approaches based on time-delay embedding, casting the problem as one of multi-attribute graph estimation wherein a random vector, instead of a scalar, is associated with each graph node. All aspects of the problem will be considered: algorithm design and analysis, optimization under both convex and non-convex regularizing functions for sparse parameter estimation, model selection (choice of penalty parameters), analysis of theoretical properties (such as consistency and model recovery), and application to real data using publicly available data sets.This project is jointly funded by the Communications & Information Foundations (CIF) and the Established Program to Stimulate Competitive Research (EPSCoR) programs.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.