Numerische Mathematik

, Volume 143, Issue 4, pp 905–922

# Grothendieck constant is norm of Strassen matrix multiplication tensor

• Shmuel Friedland
• Lek-Heng Lim
• 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

## References

1. 1.
Acín, A., Gisin, N., Toner, B.: Grothendieck’s constant and local models for noisy entangled quantum states. Phys. Rev. A 73(6), 0621055 (2006)
2. 2.
Alon, N., Berger, E.: The Grothendieck constant of random and pseudo-random graphs. Discrete Optim. 5(2), 323–327 (2008)
3. 3.
Alon, N., Makarychev, K., Makarychev, Y., Naor, A.: Quadratic forms on graphs. Invent. Math. 163(3), 499–522 (2006)
4. 4.
Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. SIAM J. Comput. 35(4), 787–803 (2006)
5. 5.
Arora, S., Berger, E., Hazan, E., Kindler, G., Safra, M.: On non-approximability for quadratic programs. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 206–215 (2005)Google Scholar
6. 6.
Braverman, M., Makarychev, K., Makarychev, Y., Naor, A.: The Grothendieck constant is strictly smaller than Krivine’s bound. Forum Math. Pi 1, e442 (2013)
7. 7.
Briët, J., Buhrman, H., Toner, B.: A generalized Grothendieck inequality and nonlocal correlations that require high entanglement. Commun. Math. Phys. 305(3), 827–843 (2011)
8. 8.
Briët, J., de Oliveira Filho, F.M., Vallentin, F.: The positive semidefinite Grothendieck problem with rank constraint. In: Automata, languages and programming. Part I, Lecture Notes in Comput. Sci., vol. 6198, pp. 31–42. Springer, Berlin (2010)
9. 9.
Briët, J., de Oliveira Filho, F.M., Vallentin, F.: Grothendieck inequalities for semidefinite programs with rank constraint. Theory Comput. 10, 77–105 (2014)
10. 10.
Bürgisser, P., Clausen, M., Shokrollahi, A.: Algebraic Complexity Theory, vol. 315. Grundlehren der Mathematischen Wissenschaften, Springer, Berlin (1997)
11. 11.
Charikar, M., Wirth, A.: Maximizing quadratic programs: extending Grothendieck’s inequality. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 54–60 (2004)Google Scholar
12. 12.
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symbolic Comput. 9(3), 251–280 (1990)
13. 13.
Davie, A.M.: Lower bound for $$k_g$$. Unpublished note (1984)Google Scholar
14. 14.
Davie, A.M.: Matrix norms related to Grothendieck’s inequality. In: Banach spaces (Columbia, Mo., 1984), Lecture Notes in Math., vol. 1166, pp. 22–26. Springer, Berlin (1985)
15. 15.
Diviánszky, P., Bene, E., Vértesi, T.: Qutrit witness from the Grothendieck constant of order four. Phys. Rev., A(96) (2017)Google Scholar
16. 16.
Dwork, C., Nikolov, A., Talwar, K.: Efficient algorithms for privately releasing marginals via convex relaxations. Discrete Comput. Geom. 53(3), 650–673 (2015)
17. 17.
Fishburn, P.C., Reeds, J.A.: Bell inequalities, Grothendieck’s constant, and root two. SIAM J. Discrete Math. 7(1), 48–56 (1994)
18. 18.
Friedland, S., Aliabadi, M.: Linear algebra and matrices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2018)Google Scholar
19. 19.
Friedland, S., Lim, L.-H.: Nuclear norm of higher-order tensors. Math. Comp. 87(311), 1255–1281 (2018)
20. 20.
Friedland, S., Lim, L.-H., Zhang, J.: An elementary proof of Grothendieck’s inequalty. Enseign. Math. 64(3/4), 327–351 (2018)Google Scholar
21. 21.
Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8, 1–79 (1953)
22. 22.
Haagerup, U.: A new upper bound for the complex Grothendieck constant. Israel J. Math. 60(2), 199–224 (1987)
23. 23.
Hendrickx, J.M., Olshevsky, A.: Matrix $$p$$-norms are NP-hard to approximate if $$p\ne 1,2,\infty$$. SIAM J. Matrix Anal. Appl. 31(5), 2802–2812 (2010)
24. 24.
Heydari, H.: Quantum correlation and Grothendieck’s constant. J. Phys. A 39(38), 11869–11875 (2006)
25. 25.
Hirsch, F., Quintino, M.T., Vértesi, T., Navascués, M., Brunner, N.: Better local hidden variable models for two-qubit werner states and an upper bound on the Grothendieck constant $$K_G(3)$$. Quantum 1, 3 (2017)
26. 26.
Hitchcock, F.L.: The expression of a tensor or a polyadic as a sum of products. J. Math. Phys. 6(1), 164–189 (1927)
27. 27.
Horadam, K.J.: Hadamard Matrices and their Applications. Princeton University Press, Princeton (2007)
28. 28.
Jameson, G.J.O.: Summing and Nuclear Norms in Banach Space Theory, vol. 8. London Mathematical Society Student Texts, Cambridge University Press, Cambridge (1987)
29. 29.
Khot, S., Naor, A.: Grothendieck-type inequalities in combinatorial optimization. Commun. Pure Appl. Math. 65(7), 992–1035 (2012)
30. 30.
Khot, S., Naor, A.: Sharp kernel clustering algorithms and their associated Grothendieck inequalities. Random Struct. Algorithms 42(3), 269–300 (2013)
31. 31.
Kindler, G., Naor, A., Schechtman, G.: The UGC hardness threshold of the $$L_p$$ Grothendieck problem. Math. Oper. Res. 35(2), 267–283 (2010)
32. 32.
Klaus, A.-L., Li, C.-K.: Isometries for the vector $$(p, q)$$ norm and the induced $$(p, q)$$ norm. Linear Multilinear Algebra 38(4), 315–332 (1995)
33. 33.
Krivine, J.-L.: Constantes de Grothendieck et fonctions de type positif sur les sphères. Adv. Math. 31(1), 16–30 (1979)
34. 34.
Landsberg, J.M.: Tensors: Geometry and Applications. Graduate Studies in Mathematics, American Mathematical Society, Providence (2012)
35. 35.
Lang, S.: Algebra, Graduate Texts in Mathematics, vol. 211, 3rd edn. Springer, New York (2002)Google Scholar
36. 36.
Le Gall, F.: Powers of tensors and fast matrix multiplication. In: ISSAC 2014—Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pp. 296–303. ACM, New York (2014)Google Scholar
37. 37.
Lim, L.-H.: Tensors and Hypermatrices. Handbook of Linear Algebra, vol. 211, 2nd edn. CRC Press, Boca Raton (2013)Google Scholar
38. 38.
Lindenstrauss, J., Pełczyński, A.: Absolutely summing operators in $$L_{p}$$-spaces and their applications. Studia Math. 29, 275–326 (1968)
39. 39.
Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. Random Struct. Algorithms 34(3), 368–394 (2009)
40. 40.
Pisier, G.: Factorization of linear operators and geometry of Banach spaces. In: CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, vol. 60 (1986)Google Scholar
41. 41.
Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. (N.S.) 49(2), 237–323 (2012)
42. 42.
Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? [extended abstract]. In: STOC’08, pp. 245–254. ACM, New York (2008)Google Scholar
43. 43.
Raghavendra, P., Steurer, D.: Towards computing the Grothendieck constant. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 525–534. SIAM, Philadelphia, PA (2009)Google Scholar
44. 44.
Regev, O.: Bell violations through independent bases games. Quantum Inf. Comput. 12(1–2), 9–20 (2012)
45. 45.
Regev, O., Toner, B.: Simulating quantum correlations with finite communication. SIAM J. Comput., 39(4):1562–1580(2009/10)
46. 46.
Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13(4), 354–356 (1969)
47. 47.
Strassen, V.: Vermeidung von Divisionen. J. Reine Angew. Math. 264, 184–202 (1973)
48. 48.
Strassen, V.: Rank and optimal computation of generic tensors. Linear Algebra Appl. 52(53), 645–685 (1983)
49. 49.
Strassen, V.: Relative bilinear complexity and matrix multiplication. J. Reine Angew. Math. 375(376), 406–443 (1987)
50. 50.
Tsirelson, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4(2), 93–100 (1980)
51. 51.
Williams, V.V.: Multiplying matrices faster than Coppersmith–Winograd [extended abstract]. In: STOC’12—Proceedings of the 2012 ACM Symposium on Theory of Computing, pp. 887–898. ACM, New York (2012)Google Scholar
52. 52.
Ye, K., Lim, L.-H.: Fast structured matrix computations: tensor rank and Cohn-Umans method. Found. Comput. Math. 18(1), 45–95 (2018)