Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics
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arXiv
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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916199859748864 |
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| author | Pickard, Joshua Chen, Can Stansbury, Cooper Surana, Amit Bloch, Anthony Rajapakse, Indika |
| author_facet | Pickard, Joshua Chen, Can Stansbury, Cooper Surana, Amit Bloch, Anthony Rajapakse, Indika |
| contents | Hypergraphs and tensors extend classic graph and matrix theory 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 graph or matrix contexts, its effectiveness in capturing multiway interactions 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, a tensor-based hypergraph product, 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-eigenvectors, various tensor decompositions, and determining the stability of polynomial systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03875 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics Pickard, Joshua Chen, Can Stansbury, Cooper Surana, Amit Bloch, Anthony Rajapakse, Indika Dynamical Systems Spectral Theory Hypergraphs and tensors extend classic graph and matrix theory 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 graph or matrix contexts, its effectiveness in capturing multiway interactions 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, a tensor-based hypergraph product, 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-eigenvectors, various tensor decompositions, and determining the stability of polynomial systems. |
| title | Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics |
| topic | Dynamical Systems Spectral Theory |
| url | https://arxiv.org/abs/2305.03875 |