TY - GEN

T1 - Kronecker powers of tensors and Strassen’s laser method

AU - Conner, Austin

AU - Landsberg, Joseph M.

AU - Gesmundo, Fulvio

AU - Ventura, Emanuele

PY - 2020

Y1 - 2020

N2 - We answer a question, posed implicitly in [18, §11], [11, Rem. 15.44] and explicitly in [9, Problem 9.8], showing the border rank of the Kronecker square of the little Coppersmith-Winograd tensor is the square of the border rank of the tensor for all q > 2, a negative result for complexity theory. We further show that when q > 4, the analogous result holds for the Kronecker cube. In the positive direction, we enlarge the list of explicit tensors potentially useful for the laser method. We observe that a well-known tensor, the 3×3 determinant polynomial regarded as a tensor, det3 ∈ C9 C9 C9, could potentially be used in the laser method to prove the exponent of matrix multiplication is two. Because of this, we prove new upper bounds on its Waring rank and rank (both 18), border rank and Waring border rank (both 17), which, in addition to being promising for the laser method, are of interest in their own right. We discuss “skew” cousins of the little Coppersmith-Winograd tensor and indicate why they may be useful for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3 C3 C3

AB - We answer a question, posed implicitly in [18, §11], [11, Rem. 15.44] and explicitly in [9, Problem 9.8], showing the border rank of the Kronecker square of the little Coppersmith-Winograd tensor is the square of the border rank of the tensor for all q > 2, a negative result for complexity theory. We further show that when q > 4, the analogous result holds for the Kronecker cube. In the positive direction, we enlarge the list of explicit tensors potentially useful for the laser method. We observe that a well-known tensor, the 3×3 determinant polynomial regarded as a tensor, det3 ∈ C9 C9 C9, could potentially be used in the laser method to prove the exponent of matrix multiplication is two. Because of this, we prove new upper bounds on its Waring rank and rank (both 18), border rank and Waring border rank (both 17), which, in addition to being promising for the laser method, are of interest in their own right. We discuss “skew” cousins of the little Coppersmith-Winograd tensor and indicate why they may be useful for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3 C3 C3

KW - Asymptotic rank

KW - Laser method

KW - Matrix multiplication complexity

KW - Tensor rank

UR - http://www.scopus.com/inward/record.url?scp=85078023161&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2020.10

DO - 10.4230/LIPIcs.ITCS.2020.10

M3 - Article in proceedings

AN - SCOPUS:85078023161

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

A2 - Vidick, Thomas

PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik

T2 - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

Y2 - 12 January 2020 through 14 January 2020

ER -