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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2309.15105 |
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| _version_ | 1866909996107694080 |
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| author | Kozhasov, Khazhgali Muniz, Alan Qi, Yang Sodomaco, Luca |
| author_facet | Kozhasov, Khazhgali Muniz, Alan Qi, Yang Sodomaco, Luca |
| contents | We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15105 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the minimal algebraic complexity of the rank-one approximation problem for general inner products Kozhasov, Khazhgali Muniz, Alan Qi, Yang Sodomaco, Luca Algebraic Geometry Optimization and Control We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry. |
| title | On the minimal algebraic complexity of the rank-one approximation problem for general inner products |
| topic | Algebraic Geometry Optimization and Control |
| url | https://arxiv.org/abs/2309.15105 |