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Numerische Mathematik

, Volume 143, Issue 4, pp 905–922 | Cite as

Grothendieck constant is norm of Strassen matrix multiplication tensor

  • Shmuel Friedland
  • Lek-Heng LimEmail author
  • Jinjie Zhang
Article

Abstract

We show that two important quantities from two disparate areas of complexity theory—Strassen’s exponent of matrix multiplication \(\omega \) and Grothendieck’s constant \(K_G\) — are different measures of size for the same underlying object: the matrix multiplication tensor, i.e., the 3-tensor or bilinear operator \(\mu _{l,m,n} : {\mathbb {F}}^{l \times m} \times {\mathbb {F}}^{m \times n} \rightarrow {\mathbb {F}}^{l \times n}\), \((A,B) \mapsto AB\) defined by matrix-matrix product over \({\mathbb {F}} = {\mathbb {R}}\) or \({\mathbb {C}}\). It is well-known that Strassen’s exponent of matrix multiplication is the greatest lower bound on (the log of) the tensor rank of \(\mu _{l,m,n}\). We will show that Grothendieck’s constant is the least upper bound on a tensor norm of \(\mu _{l,m,n}\), taken over all \(l, m, n \in {\mathbb {N}}\). Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck’s inequality as a norm inequality
$$\begin{aligned} ||\mu _{l,m,n}||_{1,2,\infty } =\max _{X,Y,M\ne 0}\frac{|{{\,\mathrm{tr}\,}}(XMY)|}{||X||_{1,2}||Y||_{2,\infty }||M||_{\infty ,1}}\leqslant K_G. \end{aligned}$$
We prove that Grothendieck’s inequality is unique in the sense that if we generalize the \((1,2,\infty )\)-norm to arbitrary \(p,q, r \in [1, \infty ]\),
$$\begin{aligned} ||\mu _{l,m,n}||_{p,q,r}=\max _{X,Y,M\ne 0}\frac{|{{\,\mathrm{tr}\,}}(XMY)|}{||X||_{p,q}||Y||_{q,r}||M||_{r,p}}, \end{aligned}$$
then \((p,q,r )=(1,2,\infty )\) is, up to cyclic permutations, the only choice for which \(||\mu _{l,m,n}||_{p,q,r}\) is uniformly bounded by a constant independent of lmn.

Mathematics Subject Classification

15A60 46B28 46B85 47A07 65Y20 68Q17 68Q25 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of IllinoisChicagoUSA
  2. 2.Computational and Applied Mathematics Initiative, Department of StatisticsUniversity of ChicagoChicagoUSA
  3. 3.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

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