On Strassen's rank additivity for small three-way tensors
We address the problem of the additivity of the tensor rank. That is, for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples...
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2020
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rr-article-92499142020-01-14T00:00:00Z On Strassen's rank additivity for small three-way tensors Jarosław Buczyński (7133282) Elisa Postinghel (2887796) Filip Rupniewski (7133288) Tensor rank Additivity of tensor rank Strassen's conjecture Slices of tensor Secant variety Border rank We address the problem of the additivity of the tensor rank. That is, for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. In this article we prove that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity holds. Or, if one of the tensors lives in ${\mathbb C}^k\otimes {\mathbb C}^3\otimes {\mathbb C}^3$ for any $k$, then the additivity also holds. More generally, if one of the tensors is concise and its rank is at most 2 more than the dimension of one of the linear spaces, then additivity holds. In addition we also treat some cases of the additivity of the border rank of such tensors. In particular, we show that the additivity of the border rank holds if the direct sum tensor is contained in ${\mathbb C}^4\otimes {\mathbb C}^4\otimes {\mathbb C}^4$. Some of our results are valid over an arbitrary base field.<br><br><br> 2020-01-14T00:00:00Z Text Journal contribution 2134/9249914.v1 https://figshare.com/articles/journal_contribution/On_Strassen_s_rank_additivity_for_small_three-way_tensors/9249914 All Rights Reserved |
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Tensor rank Additivity of tensor rank Strassen's conjecture Slices of tensor Secant variety Border rank |
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Tensor rank Additivity of tensor rank Strassen's conjecture Slices of tensor Secant variety Border rank Jarosław Buczyński Elisa Postinghel Filip Rupniewski On Strassen's rank additivity for small three-way tensors |
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We address the problem of the additivity of the tensor rank. That is, for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. In this article we prove that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity holds. Or, if one of the tensors lives in ${\mathbb C}^k\otimes {\mathbb C}^3\otimes {\mathbb C}^3$ for any $k$, then the additivity also holds. More generally, if one of the tensors is concise and its rank is at most 2 more than the dimension of one of the linear spaces, then additivity holds. In addition we also treat some cases of the additivity of the border rank of such tensors. In particular, we show that the additivity of the border rank holds if the direct sum tensor is contained in ${\mathbb C}^4\otimes {\mathbb C}^4\otimes {\mathbb C}^4$. Some of our results are valid over an arbitrary base field. |
| format |
Default Article |
| author |
Jarosław Buczyński Elisa Postinghel Filip Rupniewski |
| author_facet |
Jarosław Buczyński Elisa Postinghel Filip Rupniewski |
| author_sort |
Jarosław Buczyński (7133282) |
| title |
On Strassen's rank additivity for small three-way tensors |
| title_short |
On Strassen's rank additivity for small three-way tensors |
| title_full |
On Strassen's rank additivity for small three-way tensors |
| title_fullStr |
On Strassen's rank additivity for small three-way tensors |
| title_full_unstemmed |
On Strassen's rank additivity for small three-way tensors |
| title_sort |
on strassen's rank additivity for small three-way tensors |
| publishDate |
2020 |
| url |
https://hdl.handle.net/2134/9249914.v1 |
| _version_ |
1797731235302211584 |