TY - GEN

T1 - On degeneration of tensors and Algebras

AU - Bläser, Markus

AU - Lysikov, Vladimir

PY - 2016/8/1

Y1 - 2016/8/1

N2 - An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast algorithms for matrix multiplication, it seems to be important to understand those tensors whose border rank is as small as possible, so called tensors of minimal border rank. We investigate the connection between degenerations of associative algebras and degenerations of their structure tensors in the sense of Strassen. It allows us to describe an open subset of n × n × n tensors of minimal border rank in terms of smoothability of commutative algebras. We describe the smoothable algebra associated to the Coppersmith-Winograd tensor and prove a lower bound for the border rank of the tensor used in the "easy construction" of Coppersmith and Winograd.

AB - An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast algorithms for matrix multiplication, it seems to be important to understand those tensors whose border rank is as small as possible, so called tensors of minimal border rank. We investigate the connection between degenerations of associative algebras and degenerations of their structure tensors in the sense of Strassen. It allows us to describe an open subset of n × n × n tensors of minimal border rank in terms of smoothability of commutative algebras. We describe the smoothable algebra associated to the Coppersmith-Winograd tensor and prove a lower bound for the border rank of the tensor used in the "easy construction" of Coppersmith and Winograd.

KW - Bilinear complexity

KW - Border rank

KW - Commutative algebras

KW - Lower bounds

UR - http://www.scopus.com/inward/record.url?scp=85012910060&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2016.19

DO - 10.4230/LIPIcs.MFCS.2016.19

M3 - Article in proceedings

AN - SCOPUS:85012910060

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016

A2 - Muscholl, Anca

A2 - Faliszewski, Piotr

A2 - Niedermeier, Rolf

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

CY - Saarbrücken/Wadern

T2 - 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016

Y2 - 22 August 2016 through 26 August 2016

ER -