Decomposing tensors via rank-one approximations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912736869351424 |
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| author | Ribot, Alvaro Horobet, Emil Seigal, Anna Turatti, Ettore Teixeira |
| author_facet | Ribot, Alvaro Horobet, Emil Seigal, Anna Turatti, Ettore Teixeira |
| contents | Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_15935 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decomposing tensors via rank-one approximations Ribot, Alvaro Horobet, Emil Seigal, Anna Turatti, Ettore Teixeira Algebraic Geometry Spectral Theory 14N07, 15A18, 15A69 Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors. |
| title | Decomposing tensors via rank-one approximations |
| topic | Algebraic Geometry Spectral Theory 14N07, 15A18, 15A69 |
| url | https://arxiv.org/abs/2411.15935 |