Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-09-03 DOI:10.1137/23m1592547

Joshua Pickard, Can Chen, Cooper Stansbury, Amit Surana, Anthony M. Bloch, Indika Rajapakse

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引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1621-1642, September 2024.
Abstract. Hypergraphs and tensors extend classic graph and matrix theories to account for multiway relationships, which are ubiquitous in engineering, biological, and social systems. While the Kronecker product is a potent tool for analyzing the coupling of systems in a graph or matrix context, its utility in studying multiway interactions, such as those represented by tensors and hypergraphs, remains elusive. In this article, we present a comprehensive exploration of algebraic, structural, and spectral properties of the tensor Kronecker product. We express Tucker and tensor train decompositions and various tensor eigenvalues in terms of the tensor Kronecker product. Additionally, we utilize the tensor Kronecker product to form Kronecker hypergraphs, which are tensor-based hypergraph products, and investigate the structure and stability of polynomial dynamics on Kronecker hypergraphs. Finally, we provide numerical examples to demonstrate the utility of the tensor Kronecker product in computing Z-eigenvalues, performing various tensor decompositions, and determining the stability of polynomial systems.

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张量和超图的克朗克积：结构与动力学

SIAM 矩阵分析与应用期刊》，第 45 卷第 3 期，第 1621-1642 页，2024 年 9 月。摘要超图和张量扩展了经典的图和矩阵理论，以解释工程、生物和社会系统中无处不在的多向关系。虽然克罗内克乘积是分析图或矩阵背景下系统耦合的有效工具，但它在研究多向相互作用（如张量和超图所代表的相互作用）方面的实用性仍然难以捉摸。在本文中，我们全面探讨了张量克罗内克乘的代数、结构和谱特性。我们用张量 Kronecker 积来表达塔克和张量列车分解以及各种张量特征值。此外，我们还利用张量克罗内克积形成了克罗内克超图（即基于张量的超图积），并研究了克罗内克超图上多项式动力学的结构和稳定性。最后，我们提供了数值示例，以证明张量克罗内克积在计算 Z 特征值、执行各种张量分解和确定多项式系统稳定性方面的实用性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.