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KRONECKER PRODUCT OF TENSORS AND HYPERGRAPHS: STRUCTURE AND DYNAMICS.

Joshua Pickard1, Can Chen2, Cooper Stansbury1

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SIAM Journal on Matrix Analysis and Applications : a Publication of the Society for Industrial and Applied Mathematics
|June 3, 2025
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Summary

This study introduces the tensor Kronecker product, a new tool for analyzing multiway data. It enables new methods for tensor decomposition, eigenvalue computation, and understanding complex system dynamics.

Keywords:
05C6515A69block tensorshypergraph productsmultilinear systemtensor Kronecker producttensor decompositiontensor eigenvalues

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Area of Science:

  • Multidisciplinary Mathematics
  • Data Science
  • Network Theory

Background:

  • Hypergraphs and tensors generalize graph and matrix theory for multiway relationships.
  • The Kronecker product is valuable for coupled systems but its tensor/hypergraph application is unclear.

Purpose of the Study:

  • To comprehensively explore the algebraic, structural, and spectral properties of the tensor Kronecker product.
  • To extend the utility of Kronecker products to tensor and hypergraph analysis.
  • To investigate applications in tensor decompositions, eigenvalues, and dynamical systems.

Main Methods:

  • Algebraic manipulation of tensor operations.
  • Structural analysis of Kronecker hypergraphs.
  • Spectral analysis of tensor eigenvalues and polynomial dynamics.

Main Results:

  • Tucker and tensor train decompositions expressed via tensor Kronecker product.
  • Tensor eigenvalues related to the tensor Kronecker product.
  • Kronecker hypergraphs defined and their properties investigated.
  • Stability analysis of polynomial dynamics on Kronecker hypergraphs.

Conclusions:

  • The tensor Kronecker product is a powerful tool for multiway data analysis.
  • It facilitates computation of tensor decompositions and eigenvalues.
  • It provides a framework for analyzing complex dynamics in systems represented by tensors and hypergraphs.