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Faster Rectangular Matrix Multiplication by Combination Loss Analysis (2307.06535v2)

Published 13 Jul 2023 in cs.DS, cs.CC, and cs.SC

Abstract: Duan, Wu and Zhou (FOCS 2023) recently obtained the improved upper bound on the exponent of square matrix multiplication $\omega<2.3719$ by introducing a new approach to quantify and compensate the ``combination loss" in prior analyses of powers of the Coppersmith-Winograd tensor. In this paper we show how to use this new approach to improve the exponent of rectangular matrix multiplication as well. Our main technical contribution is showing how to combine this analysis of the combination loss and the analysis of the fourth power of the Coppersmith-Winograd tensor in the context of rectangular matrix multiplication developed by Le Gall and Urrutia (SODA 2018).

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References (41)
  1. Josh Alman. Limits on the universal method for matrix multiplication. Theory of Computing, 17:1–30, 2021.
  2. Further limitations of the known approaches for matrix multiplication. In Proceedings of the 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of LIPIcs, pages 25:1–25:15, 2018.
  3. Limits on all known (and some unknown) approaches to matrix multiplication. In Proceedings of the 59th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2018), pages 580–591, 2018.
  4. A refined laser method and faster matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pages 522–539, 2021.
  5. Fast algorithms for maximum subset matching and all-pairs shortest paths in graphs with a (not so) small vertex cover. In Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007), pages 175–186, 2007.
  6. Fast matrix multiplication: Limitations of the Coppersmith-Winograd method. In Proceedings of the 47th Symposium on Theory of Computing (STOC 2015), pages 585–593, 2015.
  7. O⁢(n2.7799)𝑂superscript𝑛2.7799O(n^{2.7799})italic_O ( italic_n start_POSTSUPERSCRIPT 2.7799 end_POSTSUPERSCRIPT ) complexity for n×n𝑛𝑛n\times nitalic_n × italic_n approximate matrix multiplication. Information Processing Letters, 8(5):234–235, 1979.
  8. Markus Bläser. Fast matrix multiplication. Theory of Computing, Graduate Surveys, 5:1–60, 2013.
  9. Algebraic complexity theory. Springer, 1997.
  10. Barriers for rectangular matrix multiplication. ArXiv:2003.03019, 2020.
  11. Barriers for fast matrix multiplication from irreversibility. Theory of Computing, 17:1–32, 2021.
  12. Don Coppersmith. Rapid multiplication of rectangular matrices. SIAM Journal on Computing, 11(3):467–471, 1982.
  13. Don Coppersmith. Rectangular matrix multiplication revisited. Journal of Complexity, 13(1):42–49, 1997.
  14. On the asymptotic complexity of matrix multiplication. SIAM Journal on Computing, 11(3):472–492, 1982.
  15. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9(3):251–280, 1990.
  16. Faster algorithms for finding lowest common ancestors in directed acyclic graphs. Theoretical Computer Science, 380(1-2):37–46, 2007.
  17. Improved bound for complexity of matrix multiplication. Proceedings of the Royal Society of Edinburgh, 143A:351–370, 2013.
  18. Fully dynamic transitive closure: Breaking through the o⁢(n2)𝑜superscript𝑛2o(n^{2})italic_o ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) barrier. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), pages 381–389, 2000.
  19. Faster matrix multiplication via asymmetric hashing. In Proceedings of the 64th Annual Symposium on Foundations of Computer Science (FOCS 2023), to appear, 2023. ArXiv:2210.10173.
  20. Fast rectangular matrix multiplication and applications. Journal of Complexity, 14(2):257–299, 1998.
  21. Fast rectangular matrix multiplication and some applications. Science in China Series A: Mathematics, 51(3):389–406, 2008.
  22. François Le Gall. Faster algorithms for rectangular matrix multiplication. In Proceedings of the 53rd Symposium on Foundations of Computer Science (FOCS 2012), pages 514–523, 2012.
  23. François Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC 2014), pages 296–303, 2014.
  24. François Le Gall and Florent Urrutia. Improved rectangular matrix multiplication using powers of the coppersmith-winograd tensor. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), pages 1029–1046, 2018.
  25. On the asymptotic complexity of rectangular matrix multiplication. Theoretical Computer Science, 23:171–185, 1983.
  26. Victor Y. Pan. Field extension and triangular aggregating, uniting and canceling for the acceleration of matrix multiplications. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science (FOCS 1979), pages 28–38, 1979.
  27. Victor Y. Pan. New combinations of methods for the acceleration of matrix multiplication. Computer and Mathematics with Applications, pages 73–125, 1981.
  28. All-pairs shortest paths with a sublinear additive error. ACM Transactions on Algorithms, 7(4):45, 2011.
  29. Francesco Romani. Some properties of disjoint sums of tensors related to matrix multiplication. SIAM Journal on Computing, 11(2):263–267, 1982.
  30. Fast dynamic transitive closure with lookahead. Algorithmica, 56(2):180–197, 2010.
  31. Arnold Schönhage. Partial and total matrix multiplication. SIAM Journal on Computing, 10(3):434–455, 1981.
  32. Andrew Stothers. On the Complexity of Matrix Multiplication. PhD thesis, University of Edinburgh, 2010.
  33. Volker Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354–356, 1969.
  34. Volker Strassen. The asymptotic spectrum of tensors and the exponent of matrix multiplication. In Proceedings of the 27th Annual Symposium on Foundations of Computer Science (FOCS 1986), pages 49–54, 1986.
  35. Volker Strassen. Relative bilinear complexity and matrix multiplication. Journal für die reine und angewandte Mathematik, 375-376:406–443, 1987.
  36. Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th Symposium on Theory of Computing (STOC 2012), pages 887–898, 2012.
  37. Raphael Yuster. Efficient algorithms on sets of permutations, dominance, and real-weighted APSP. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pages 950–957, 2009.
  38. Detecting short directed cycles using rectangular matrix multiplication and dynamic programming. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pages 254–260, 2004.
  39. Uri Zwick. All pairs lightest shortest paths. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC 1999), pages 61–69, 1999.
  40. Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM, 49(3):289–317, 2002.
  41. https://osf.io/hfs4y/?view_only=4035c6e72ba54083973c069e2e405a5f.
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Authors (1)
  1. François Le Gall (73 papers)
Citations (16)
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