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

这一合作研究项目旨在开发一个基于张量表示的新的超图信号处理框架，以利用复杂关系中数据的多路交互作用。简单图只能对数据之间的成对关系进行建模，这使得它们不能应用于具有更高阶关系的网络建模。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潜在多维关系。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--从机器人和自动驾驶导航到遥感和网络物理系统。例如，遥感中的点云三维成像是一项新兴的关键技术，可以在其中应用该项目正在开发的工具。与该项目的科学目标相一致，研究团队将开发关于图形和超图形信号处理的教学模块，以将这一新兴领域介绍给本国机构的广泛学生。这项研究工作从根本上背离了以前依赖对称正则多进张量分解的工作。相反，理论基础是基于最近在张量代数中引入的t-积运算，该运算允许类似于矩阵因式分解和特征分解的张量因式分解。采用t-特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，从该公式派生的正交特征基允许无损的傅立叶分解和计算效率。因此，新的框架将允许推广传统的图形信号处理技术，同时保持由超图表示的复杂系统的维度特征。为此，将引入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。还将定义超图傅立叶空间，随后将定义带宽受限信号、采样和学习的概念。新框架的好处将在光谱聚类、去噪和分类等应用中得到展示。这一奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.