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.

该协作研究项目旨在开发基于张量表示的新的HyperGraph信号处理框架，以利用复杂关系中数据中的多路相互作用。简单的图形只能在数据之间建模成对关系，从而阻止其应用于具有高阶关系的网络。另一方面，HyperGraph信号处理技术更强大，因为它们可以解释数据节点之间的基本多层关系。 HyperGraph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许多应用 - 从机器人技术和自动驾驶导航到遥感和网络物理系统。例如，遥感中的Point-Cloud 3D成像是一种新兴和关键的技术，可以应用该项目中正在开发的工具。与该项目的科学目标一致，研究人员团队将开发有关图形和HyperGraph信号处理的教育模块，以将这个新兴领域介绍给其家庭机构的广泛学生。研究工作从根本上偏离了先前的工作，该工作依赖于对称的典型多辅助张量分解。取而代之的是，理论基础基于最近在张量代数中引入的T产品操作，该代数允许张量分解类似于矩阵因素化和特征性组成。采用T-Eigendecostosings的优点令人信服 - 它们保留了张量的内在结构和信号的高维质；最重要的是，从该公式中得出的正交本质可以进行无损失的傅立叶分解和计算有效的计算。因此，新框架将允许传统的图形信号处理技术的概括，同时保持由超图表代表的复杂系统的维度特征。为此，将引入新的HyperGraph信号处理框架的核心元素，包括转移操作员，卷积以及各种超图信号的定义。还将定义HyperGraph Fourier空间，其次是带有限制信号，采样和学习的概念。新框架的好处将在诸如光谱聚类，denoing和分类之类的应用中证明。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来提供支持。