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 Ck ⊗ C3 ⊗ C3 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 C4 ⊗ C4 ⊗ C4. Some of our results are valid over an arbitrary base field. © 2020, Society for Industrial and Applied Mathematics
On Strassen's rank additivity for small three-way tensors / Buczynski, J.; Postinghel, E.; Rupniewski, F.. - In: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. - ISSN 0895-4798. - 2020, 41:1(2020), pp. 106-133. [10.1137/19M1243099]
On Strassen's rank additivity for small three-way tensors
Postinghel E.;
2020-01-01
Abstract
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 Ck ⊗ C3 ⊗ C3 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 C4 ⊗ C4 ⊗ C4. Some of our results are valid over an arbitrary base field. © 2020, Society for Industrial and Applied MathematicsFile | Dimensione | Formato | |
---|---|---|---|
Strassen.pdf
accesso aperto
Tipologia:
Post-print referato (Refereed author’s manuscript)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
440.59 kB
Formato
Adobe PDF
|
440.59 kB | Adobe PDF | Visualizza/Apri |
19m1243099.pdf
accesso aperto
Tipologia:
Versione editoriale (Publisher’s layout)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
485.03 kB
Formato
Adobe PDF
|
485.03 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione