Collaborative Research: CIF: Small: Hypergraph Signal Processing and Networks via t-Product Decompositions

合作研究：CIF：小型：通过 t 产品分解的超图信号处理和网络

• 批准号: 2230162
• 负责人: Daniel Lau
• 金额: $ 25.41万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2230162&HistoricalAwards=false
• 关键词: Collaborative; Research; CIF; Small; Hypergraph

This collaborative research project aims to develop a new hypergraph signal-processing framework based on tensor representations to exploit multi-way interactions in data from complex relations. Simple graphs can model only pairwise relationships among data, which prevents their application to modeling networks with higher-order relationships. Hypergraph signal-processing techniques, on the other hand, are more powerful since they can account for the underlying polyadic relationships among data nodes. Hypergraph signal-processing tools can be used in different areas, including data science, communication networks, epidemiology, and sociology, and in numerous applications - from robotics and self-driving navigation to remote sensing and cyber-physical systems. Point-cloud 3D imaging in remote sensing, for instance, is an emerging and critical technology wherein the tools under development in this project can be applied. In concert with the scientific goals of the project, the team of researchers will develop educational modules on graph and hypergraph signal processing to introduce this emerging field to a broad set of students at their home institutions. The research effort radically departs from prior work that relied on symmetric canonical polyadic tensor decompositions. Instead, the theoretical underpinnings are based on the more recently introduced t-product operation in tensor algebra, which allows tensor factorizations that are analogous to matrix factorizations and eigendecompositions. The advantages of adopting t-eigendecompositions are compelling - they preserve the intrinsic structure of tensors and the high-dimensional nature of their signals; most importantly, the orthogonal eigenbasis derived from this formulation allows for a loss-free Fourier decomposition and computationally efficient calculations. The new framework will thus allow for the generalization of traditional graph signal-processing techniques while keeping the dimensionality characteristic of the complex systems represented by hypergraphs. To this end, core elements of the new hypergraph signal-processing framework will be introduced, including shifting operators, convolutions, and the definition of various hypergraph signals. The hypergraph Fourier space will also be defined, followed by the concepts of bandlimited signals, sampling, and learning. The benefits of the new framework will be demonstrated in applications such as spectral clustering, denoising, and classification.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.

该合作研究项目旨在开发一种基于张量表示的超图信号处理框架，以利用复杂关系中数据的多方式交互。简单图只能对数据之间的成对关系进行建模，这阻止了它们应用于对具有高阶关系的网络进行建模。另一方面，超图信号处理技术更强大，因为它们可以解释数据节点之间的底层多元关系。超图信号处理工具可用于不同领域，包括数据科学、通信网络、流行病学和社会学，以及从机器人技术和自动驾驶导航到遥感和网络物理系统等众多应用。例如，遥感中的点云三维成像是一项新兴的关键技术，本项目正在开发的工具可以应用于这项技术。与该项目的科学目标相一致，研究人员团队将开发关于图形和超图信号处理的教育模块，以向其所在机构的广大学生介绍这一新兴领域。这项研究工作从根本上背离了以前的工作，依赖于对称规范多元张量分解。相反，理论基础是基于最近在张量代数中引入的t-乘积运算，它允许类似于矩阵分解和特征分解的张量分解。采用t-本征分解的优点是引人注目的-它们保留了张量的内在结构及其信号的高维性质;最重要的是，从该公式导出的正交本征基允许无损失傅立叶分解和计算效率高的计算。因此，新的框架将允许传统的图信号处理技术的推广，同时保持由超图表示的复杂系统的维数特性。为此，将介绍新的超图信号处理框架的核心元素，包括移位运算符，卷积和各种超图信号的定义。超图傅立叶空间也将被定义，其次是带限信号，采样和学习的概念。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。