# Bad and good news for Strassen's laser method: Border rank of the 3x3 permanent and strict submultiplicativity

@article{Conner2020BadAG, title={Bad and good news for Strassen's laser method: Border rank of the 3x3 permanent and strict submultiplicativity}, author={Austin Conner and Hang Huang and Joseph Landsberg}, journal={ArXiv}, year={2020}, volume={abs/2009.11391} }

We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent $\omega$ of matrix multiplication. The Kronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the $3\times 3$ permanent, and could potentially be used to show $\omega=2$. We prove the negative result for complexity theory that its border rank is $16$, resolving a longstanding problem. Regarding its $q=4$ skew cousin in $ C^5\otimes C^5\otimes C^5$, which could…

## 3 Citations

ON THE STRUCTURE TENSOR OF sln

- Mathematics
- 2021

The structure tensor of sln, denoted Tsln , 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…

Technical Report Column

- ChemistrySIGACT News
- 2020

This report presents a meta-modelling system that automates the very labor-intensive and therefore time-heavy and therefore expensive and expensive process of manually cataloging and cataloging all the components of a smart phone.

On the geometry of geometric rank

- Mathematics, Computer ScienceArXiv
- 2020

This work makes a geometric study of the Geometric Rank of tensors recently introduced by Kopparty et al, showing that upper bounds on geometric rank imply lower bounds on tensor rank.

## References

SHOWING 1-10 OF 49 REFERENCES

Kronecker powers of tensors and Strassen's laser method

- MathematicsITCS
- 2020

It is observed that a well-known tensor could potentially be used in the laser method to prove the exponent of matrix multiplication is two and new upper bounds are proved on its Waring rank and rank, border rank and Waring border rank, which are of interest in their own right.

Further Limitations of the Known Approaches for Matrix Multiplication

- Computer Science, MathematicsITCS
- 2018

A unifying framework is provided, showing that all known matrix multiplication running times since 1986 can be achieved from a single very natural tensor - the structural tensor $T_q$ of addition modulo an integer $q$.

About the maximal rank of 3-tensors over the real and the complex number field

- Computer Science, Mathematics
- 2010

This paper considers the maximal rank problem of 3-tensors and extends Atkinson and Stephens’ and Atkinson and Lloyd’s results over the real number field.

New lower bounds for matrix multiplication and the 3x3 determinant

- MathematicsArXiv
- 2019

The two key ingredients are: (i) the use of a multi-graded ideal associated to a border rank $r$ decomposition of any tensor, and (ii) the exploitation of the large symmetry group of $T$ to restrict to $B_T$-invariant ideals, where $B-T$ is a maximal solvable subgroup of the symmetry groupof $T$.

Multigraded Apolarity.

- Mathematics
- 2016

We generalize methods to compute various kinds of rank to the case of a toric variety $X$ embedded into projective space using a very ample line bundle $\mathcal{L}$. We find an upper bound on the…

Limits on the Universal method for matrix multiplication

- Computer ScienceComputational Complexity Conference
- 2019

This work proves that the Universal method applied to any Coppersmith-Winograd tensor CWq cannot yield a bound on ω, the exponent of matrix multiplication, better than 2.16805.

Fast Matrix Multiplication: Limitations of the Coppersmith-Winograd Method

- Computer ScienceSTOC
- 2015

A new framework is described extending the original laser method, which is the method underlying the algorithms by Coppersmith and Winograd, Stothers, Vassilevska-Williams and Le Gall, and is the first to explain why taking tensor powers of the Coppermith-Winograd identity results in faster algorithms.

On Degeneration of Tensors and Algebras

- Mathematics, Computer ScienceMFCS
- 2016

This work describes the smoothable algebra associated to the Coppersmith-Winograd tensor and proves a lower bound for the border rank of the tensor used in the "easy construction" of Coppermith and Winograd.

Barriers for fast matrix multiplication from irreversibility

- Computer ScienceComputational Complexity Conference
- 2019

The notion of "irreversibility" of a tensor is introduced and it is proved that any approach that uses an irreversible tensor in an intermediate step cannot give ω = 2.37, proving that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor.

New Lower Bounds for the Border Rank of Matrix Multiplication

- Mathematics, Computer ScienceTheory Comput.
- 2015

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the…