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Matrix Multiplication and Number on the Forehead Communication

Authors Josh Alman, Jarosław Błasiok

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  • 23 pages

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Author Details

Josh Alman
  • Columbia University, New York, NY, USA
Jarosław Błasiok
  • Columbia University, New York, NY, USA


The authors thank Madhu Sudan and Toniann Pitassi for helpful discussions.

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Josh Alman and Jarosław Błasiok. Matrix Multiplication and Number on the Forehead Communication. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 16:1-16:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Three-player Number On the Forehead communication may be thought of as a three-player Number In the Hand promise model, in which each player is given the inputs that are supposedly on the other two players' heads, and promised that they are consistent with the inputs of the other players. The set of all allowed inputs under this promise may be thought of as an order-3 tensor. We surprisingly observe that this tensor is exactly the matrix multiplication tensor, which is widely studied in the design of fast matrix multiplication algorithms. Using this connection, we prove a number of results about both Number On the Forehead communication and matrix multiplication, each by using known results or techniques about the other. For example, we show how the Laser method, a key technique used to design the best matrix multiplication algorithms, can also be used to design communication protocols for a variety of problems. We also show how known lower bounds for Number On the Forehead communication can be used to bound properties of the matrix multiplication tensor such as its zeroing out subrank. Finally, we substantially generalize known methods based on slice-rank for studying communication, and show how they directly relate to the matrix multiplication exponent ω.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Number on the forehead
  • communication complexity
  • matrix multiplication


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  1. Josh Alman. Limits on the universal method for matrix multiplication. Theory Of Computing, 17(1):1-30, 2021. Google Scholar
  2. Josh Alman and Virginia Vassilevska Williams. Limits on all known (and some unknown) approaches to matrix multiplication. SIAM Journal on Computing, FOCS:18-285, 2021. Google Scholar
  3. Noga Alon, Ankur Moitra, and Benny Sudakov. Nearly complete graphs decomposable into large induced matchings and their applications. In Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, STOC '12, pages 1079-1090, New York, NY, USA, 2012. Association for Computing Machinery. URL:
  4. Noga Alon and Adi Shraibman. Algorithmic number on the forehead protocols yielding dense ruzsa-szemerédi graphs and hypergraphs, 2020. URL:
  5. Paul Beame, Matei David, Toniann Pitassi, and Philipp Woelfel. Separating deterministic from randomized multiparty communication complexity. Theory of Computing, 6(1):201-225, 2010. Google Scholar
  6. Felix A Behrend. On sets of integers which contain no three terms in arithmetical progression. Proceedings of the National Academy of Sciences, 32(12):331-332, 1946. Google Scholar
  7. Richard Beigel and Jun Tarui. On acc. Computational Complexity, 4:350-366, 1994. Google Scholar
  8. Dario Bini. Border rank of a p × q × 2 tensor and the optimal approximation of a pair of bilinear forms. In Automata, Languages and Programming: Seventh Colloquium Noordwijkerhout, the Netherlands July 14-18, 1980, pages 98-108. Springer, 2005. Google Scholar
  9. Markus Bläser. Fast matrix multiplication. Theory of Computing, pages 1-60, 2013. Google Scholar
  10. Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A Grochow, Eric Naslund, William F Sawin, and Chris Umans. On cap sets and the group-theoretic approach to matrix multiplication. Discrete Analysis, 3, 2017. Google Scholar
  11. Jonah Blasiak, Henry Cohn, Joshua A Grochow, Kevin Pratt, and Chris Umans. Matrix multiplication via matrix groups. arXiv preprint, 2022. URL:
  12. Ashok K Chandra, Merrick L Furst, and Richard J Lipton. Multi-party protocols. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 94-99, 1983. Google Scholar
  13. Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam. Symmetric subrank of tensors and applications, 2021. URL:
  14. Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam. Larger corner-free sets from combinatorial degenerations. In 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  15. Matthias Christandl, Péter Vrana, and Jeroen Zuiddam. Barriers for fast matrix multiplication from irreversibility. THEORY OF COMPUTING, 17(2):1-32, 2021. Google Scholar
  16. Lane H Clark, Roger C Entringer, Joseph E McCanna, and László A Székely. Extremal problems for local properties of graphs. Australas. J Comb., 4:25-32, 1991. Google Scholar
  17. Henry Cohn and Christopher Umans. A group-theoretic approach to fast matrix multiplication. In 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings., pages 438-449. IEEE, 2003. Google Scholar
  18. Austin Conner, Fulvio Gesmundo, Joseph M Landsberg, Emanuele Ventura, and Yao Wang. Towards a geometric approach to strassen’s asymptotic rank conjecture. Collectanea mathematica, 72:63-86, 2021. Google Scholar
  19. Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pages 1-6, 1987. Google Scholar
  20. Ernie Croot, Vsevolod F Lev, and Péter Pál Pach. Progression-free sets in are exponentially small. Annals of Mathematics, pages 331-337, 2017. Google Scholar
  21. Matei David and Toniann Pitassi. Separating nof communication complexity classes rp and np, 2008. URL:
  22. Jordan S. Ellenberg and Dion Gijswijt. On large subsets of f_qⁿ with no three-term arithmetic progression, 2016. URL:
  23. Jacob Fox. A new proof of the graph removal lemma. Annals of Mathematics, pages 561-579, 2011. Google Scholar
  24. J. Hastad and M. Goldmann. On the power of small-depth threshold circuits. In Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, pages 610-618 vol.2, 1990. URL:
  25. Swastik Kopparty, Guy Moshkovitz, and Jeroen Zuiddam. Geometric rank of tensors and subrank of matrix multiplication. arXiv preprint, 2020. URL:
  26. Nati Linial, Toniann Pitassi, and Adi Shraibman. On the communication complexity of high-dimensional permutations, 2019. Google Scholar
  27. Nati Linial and Adi Shraibman. Larger corner-free sets from better nof exactly-n protocols, 2021. URL:
  28. Imre Z Ruzsa and Endre Szemerédi. Triple systems with no six points carrying three triangles. Combinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai, 18(939-945):2, 1978. Google Scholar
  29. V Strassen. Relative bilinear complexity and matrix multiplication. Journal für die reine und angewandte Mathematik, 375:406-443, 1987. Google Scholar
  30. Volker Strassen. Vermeidung von divisionen. Journal für die reine und angewandte Mathematik, 264:184-202, 1973. Google Scholar
  31. Volker Strassen. The asymptotic spectrum of tensors and the exponent of matrix multiplication. In 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), pages 49-54. IEEE, 1986. Google Scholar
  32. Terence Tao. A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound. blog post, 2016. Google Scholar
  33. Jacobus Hendricus Van Lint and Richard Michael Wilson. A course in combinatorics. Cambridge university press, 2001. Google Scholar
  34. Avi Wigderson and Jeroen Zuiddam. Asymptotic spectra: Theory, applications and extensions, 2022. Google Scholar
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