Bad and Good News for Strassen’s Laser Method: Border Rank of $$\mathrm{Perm}_3$$ and Strict Submultiplicativity

Conner, Austin; Huang, Hang; Landsberg, J. M.

doi:10.1007/s10208-022-09579-3

Cited by 3 publications

(2 citation statements)

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“…Since these tensors, and their Kronecker squares tend to have border rank below the cactus barrier, we expect to be able to effectively apply the method as is to determine the border rank at least for small Kronecker powers. After this paper was posted on arXiv, border apolarity was utilized to determine the border rank of 𝑇 2 𝑐𝑤,2 in [16] and the answer ended up being the known upper bound. We are developing techniques to write down usual border rank decompositions guided by the ideals produced by border apolarity to potentially determine new upper bounds for higher Kronecker powers of 𝑇 𝑐𝑤,2 and det 3 (or to show that the known bounds are sharp).…”

Section: Upper Bounds Especially For Tensors Relevant For Strassen's ...mentioning

confidence: 99%

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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Section: Upper Bounds Especially For Tensors Relevant For Strassen's ...mentioning

confidence: 99%

New lower bounds for matrix multiplication and

Conner

Harper

Landsberg

2023

Forum of Mathematics, Pi

View full text Add to dashboard Cite

show abstract

“…We stress however, that the ideals above are usually not saturated, hence there is important information lost when passing from I to V (I) ⊆ P n . The multigraded Hilbert scheme is recently applied very successfully in theory of tensors, for investigating the lower bounds on border rank [19,36].…”

Section: Problem XXIX Say That An Irreducible Component Z Of Hilbmentioning

confidence: 99%