On Comon’s and Strassen’s Conjectures
Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known resul...
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| Format: | Article |
| Language: | English |
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MDPI AG
2018-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/6/11/217 |
| _version_ | 1818913706401071104 |
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| author | Alex Casarotti Alex Massarenti Massimiliano Mella |
| author_facet | Alex Casarotti Alex Massarenti Massimiliano Mella |
| author_sort | Alex Casarotti |
| collection | DOAJ |
| description | Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties. |
| first_indexed | 2024-12-19T23:34:45Z |
| format | Article |
| id | doaj.art-f2e5db08d16e464faae8123d833a1a38 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-19T23:34:45Z |
| publishDate | 2018-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f2e5db08d16e464faae8123d833a1a382022-12-21T20:01:39ZengMDPI AGMathematics2227-73902018-10-0161121710.3390/math6110217math6110217On Comon’s and Strassen’s ConjecturesAlex Casarotti0Alex Massarenti1Massimiliano Mella2Department of Mathematics and Informatics, University of Ferrara, 44121 Ferrara, ItalyDepartment of Mathematics and Informatics, University of Ferrara, 44121 Ferrara, ItalyDepartment of Mathematics and Informatics, University of Ferrara, 44121 Ferrara, ItalyComon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.https://www.mdpi.com/2227-7390/6/11/217Strassen’s conjectureComon’s conjecturetensor decompositionWaring decomposition |
| spellingShingle | Alex Casarotti Alex Massarenti Massimiliano Mella On Comon’s and Strassen’s Conjectures Mathematics Strassen’s conjecture Comon’s conjecture tensor decomposition Waring decomposition |
| title | On Comon’s and Strassen’s Conjectures |
| title_full | On Comon’s and Strassen’s Conjectures |
| title_fullStr | On Comon’s and Strassen’s Conjectures |
| title_full_unstemmed | On Comon’s and Strassen’s Conjectures |
| title_short | On Comon’s and Strassen’s Conjectures |
| title_sort | on comon s and strassen s conjectures |
| topic | Strassen’s conjecture Comon’s conjecture tensor decomposition Waring decomposition |
| url | https://www.mdpi.com/2227-7390/6/11/217 |
| work_keys_str_mv | AT alexcasarotti oncomonsandstrassensconjectures AT alexmassarenti oncomonsandstrassensconjectures AT massimilianomella oncomonsandstrassensconjectures |