Minimality of Tree Tensor Network Ranks
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866908532970881024 |
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| author | Jovcheva, Jana Seynnaeve, Tim Vannieuwenhoven, Nick |
| author_facet | Jovcheva, Jana Seynnaeve, Tim Vannieuwenhoven, Nick |
| contents | For a given tree tensor network $G$, we call a tuple of bond dimensions minimal if there exists a tensor $T$ that can be represented by this network but not on the same tree topology with strictly smaller bond dimensions. We establish necessary and sufficient conditions on the bond dimensions of a tree tensor network to be minimal, generalizing a characterization of Carlini and Kleppe about existence of tensors with a given multilinear rank. We also show that in a minimal tree tensor network, the non-minimal tensors form a Zariski closed subset, so minimality is a generic property in this sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09463 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimality of Tree Tensor Network Ranks Jovcheva, Jana Seynnaeve, Tim Vannieuwenhoven, Nick Numerical Analysis 15A69 (Primary) 65F99 (Secondary) For a given tree tensor network $G$, we call a tuple of bond dimensions minimal if there exists a tensor $T$ that can be represented by this network but not on the same tree topology with strictly smaller bond dimensions. We establish necessary and sufficient conditions on the bond dimensions of a tree tensor network to be minimal, generalizing a characterization of Carlini and Kleppe about existence of tensors with a given multilinear rank. We also show that in a minimal tree tensor network, the non-minimal tensors form a Zariski closed subset, so minimality is a generic property in this sense. |
| title | Minimality of Tree Tensor Network Ranks |
| topic | Numerical Analysis 15A69 (Primary) 65F99 (Secondary) |
| url | https://arxiv.org/abs/2509.09463 |