Tensor decomposition beyond uniqueness, with an application to the minrank problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911240663597056 |
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| author | Koiran, Pascal Oliveira, Rafael |
| author_facet | Koiran, Pascal Oliveira, Rafael |
| contents | We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call matrix-vector decomposition. Moreover, in the same settings in which our uniqueness theorem applies, we also design and analyze an efficient randomized algorithm to compute the unique minimum matrix-vector decomposition (and thus a tensor rank decomposition of minimum rank).
As an application of our uniqueness theorem and our efficient algorithm, we show how to compute all matrices of minimum rank (up to scalar multiples) in certain generic vector spaces of matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26587 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor decomposition beyond uniqueness, with an application to the minrank problem Koiran, Pascal Oliveira, Rafael Computational Complexity Discrete Mathematics Data Structures and Algorithms Rings and Algebras Representation Theory We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call matrix-vector decomposition. Moreover, in the same settings in which our uniqueness theorem applies, we also design and analyze an efficient randomized algorithm to compute the unique minimum matrix-vector decomposition (and thus a tensor rank decomposition of minimum rank). As an application of our uniqueness theorem and our efficient algorithm, we show how to compute all matrices of minimum rank (up to scalar multiples) in certain generic vector spaces of matrices. |
| title | Tensor decomposition beyond uniqueness, with an application to the minrank problem |
| topic | Computational Complexity Discrete Mathematics Data Structures and Algorithms Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2510.26587 |