Rank and border rank of Kronecker powers of tensors and Strassen's laser method

Conner, Austin; Gesmundo, Fulvio; Landsberg, J. M.; Ventura, Emanuele

doi:10.1007/s00037-021-00217-y

Cited by 9 publications

(6 citation statements)

References 31 publications

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“…The upper bound was proved in [15]. In [9], a lower bound of 15 for the Waring rank of det 3 was proven.…”

Section: Resultsmentioning

confidence: 96%

“…In [2,3,1,13], barriers to proving further upper bounds with the method were found for many tensors. In [15], we showed that the (unique up to scale) skew-symmetric tensor in C 3 ⊗C 3 ⊗C 3 , which we denote 𝑇 𝑠𝑘𝑒𝑤𝑐𝑤,2 , is not subject to these upper bound barriers and could potentially be used to prove the exponent of matrix multiplication is two via its Kronecker powers. Explicitly, if one were to prove that lim 𝑘 → ∞ R(𝑇 𝑘 𝑠𝑘𝑒𝑤𝑐𝑤,2 )…”

Section: Resultsmentioning

confidence: 99%

“…1 𝑘 equals 3, that would imply the exponent is two. One has R(𝑇 𝑠𝑘𝑒𝑤𝑐𝑤,2 ) = 5 and 𝑇 2 𝑠𝑘𝑒𝑤𝑐𝑤,2 = det 3 ; see [15]. Thus, the following result is important for matrix multiplication complexity upper bounds:…”

Section: Resultsmentioning

confidence: 99%

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New lower bounds for matrix multiplication and

Conner

Harper

Landsberg

2023

Forum of Mathematics, Pi

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Let $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ denote the matrix multiplication tensor (and write $M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ ), and let $\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ denote the determinant polynomial considered as a tensor. For a tensor T, let $\underline {\mathbf {R}}(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ , (ii) prove $\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ and $\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_{\langle 2\rangle }$ , (iii) prove $\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , (iv) prove $\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , improving the previous lower bound of $12$ , (v) prove $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ for all $\mathbf {n}\geq 25$ , where previously only $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ was known, as well as lower bounds for $4\leq \mathbf {n}\leq 25$ , and (vi) prove $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ for all $\mathbf {n} \ge 18$ , where previously only $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensor T and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable when T has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.

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“…The upper bound was proved in [15]. In [9], a lower bound of 15 for the Waring rank of det 3 was proven.…”

Section: Resultsmentioning

confidence: 96%

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

New lower bounds for matrix multiplication and

Conner

Harper

Landsberg

2023

Forum of Mathematics, Pi

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“…A numerical computer search has given a rank 20 decomposition of T sl 3 . The technique used was a combination of Newton's Method and Lenstra-Lenstra-Lovász Algorithm to find rational approximations [CLGV20]. This technique formulated the problem as a nonlinear optimization problem that was solved to machine precision and then subsequently modified using the Lenstra-Lenstra-Lovász Algorithm to generate a precise solution with algebraic numbers given the numerical solution.…”

Section: New Boundsmentioning

confidence: 99%

On the structure tensor of $\mathfrak{sl}_n$

Bari¹

2021

Preprint

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The structure tensor of sl n , denoted T sln , is the tensor arising from the Lie bracket bilinear operation on the set of traceless n × n matrices over C. This tensor is intimately related to the well studied matrix multiplication tensor. Studying the structure tensor of sl n may provide further insight into the complexity of matrix multiplication and the "hay in a haystack" problem of finding explicit sequences tensors with high rank or border rank. We aim to find new bounds on the rank and border rank of this structure tensor in the case of sl 3 and sl 4 . We additionally provide bounds in the case of the lie algebras so 4 and so 5 . The lower bounds on the border ranks were obtained via various recent techniques, namely Koszul flattenings, border substitution, and border apolarity. Upper bounds on the rank of T sl3 are obtained via numerical methods that allowed us to find an explicit rank decomposition.

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“…Fast forward in time, there were plenty and a variety of studies dedicated to explore the field under the scope of mathematical analysis, differential geometry and topology. In particular, Conner et al [2] have developed tensors with maximal symmetries, embedded in representation theory, computer science, computational complexity, algebraic geometry. In this paper we expand the notations expressed in previous studies by generalising the tensor concept.…”

Section: Introductionmentioning

confidence: 99%

Generalised Tensors

Pierros

2023

Preprint

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In this document we describe the novel concept of generalisation of tensor. We expand the concepts expressed in previous studies by considering the generalisation of the tensor concept. We introduce the generalised index of a tensor, an index describing several types of indices of a tensor. Furthermore, we describe the connection between category theory and tensor calculus. In particular, we explore the concepts of functor tensor, or functorial tensor, tensor with categories as elements. This work opens new avenues to mathematical analysis, tensor analysis, tensor theory, and category theory.

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Rank and border rank of Kronecker powers of tensors and Strassen's laser method

Abstract: We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor $$T_{cw,q}$$ T c w , q is the square of its border rank for $$q > 2$$ q > 2 and that the bord… Show more

Cited by 9 publications

References 31 publications

New lower bounds for matrix multiplication and

New lower bounds for matrix multiplication and

On the structure tensor of $\mathfrak{sl}_n$

Generalised Tensors

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