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Bad and Good News for Strassen’s Laser Method: Border Rank of \(\mathrm{Perm}_3\) and Strict Submultiplicativity

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Abstract

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 \({\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5\), which could potentially be used to prove \(\le 2.11\), we show the border rank of its Kronecker square is at most 42, a remarkable sub-multiplicativity result, as the square of its border rank is 64. We also determine moduli spaces VSP for the small Coppersmith–Winograd tensors.

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References

  1. J. Alman and V. Vassilevska Williams, Limits on All Known (and Some Unknown) Approaches to Matrix Multiplication, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (2018).

  2. Josh Alman, Limits on the universal method for matrix multiplication, 34th Computational Complexity Conference, LIPIcs. Leibniz Int. Proc. Inform., vol. 137, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019, pp. Art. No. 12, 24.

  3. Josh Alman and Virginia Vassilevska Williams, A refined laser method and faster matrix multiplication, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), [Society for Industrial and Applied Mathematics (SIAM)], Philadelphia, PA, 2021, pp. 522–539.

  4. Josh Alman and Virginia Vassilevska Williams, Further limitations of the known approaches for matrix multiplication, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, 2018, pp. 25:1–25:15.

  5. Andris Ambainis, Yuval Filmus, and François Le Gall, Fast matrix multiplication: limitations of the Coppersmith-Winograd method (extended abstract), STOC’15—Proceedings of the 2015 ACM Symposium on Theory of Computing, ACM, New York, 2015, pp. 585–593.

  6. Dario Bini, Grazia Lotti, and Francesco Romani, Approximate solutions for the bilinear form computational problem, SIAM J. Comput. 9 (1980), no. 4, 692–697.

    Article  MathSciNet  MATH  Google Scholar 

  7. Markus Bläser, Fast matrix multiplication, Graduate Surveys, no. 5, Theory of Computing Library, 2013.

  8. Markus Bläser and Vladimir Lysikov, On degeneration of tensors and algebras, 41st International Symposium on Mathematical Foundations of Computer Science, LIPIcs. Leibniz Int. Proc. Inform., vol. 58, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016, pp. Art. No. 19, 11.

  9. Jerome Brachat, Pierre Comon, Bernard Mourrain, and Elias Tsigaridas, Symmetric tensor decomposition, Linear Algebra Appl. 433 (2010), no. 11-12, 1851–1872.

    Article  MathSciNet  MATH  Google Scholar 

  10. Weronika Buczyńska and Jarosław Buczyński, Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes, J. Algebraic Geom. 23 (2014), no. 1, 63–90.

    Article  MathSciNet  MATH  Google Scholar 

  11. Weronika Buczyńska and Jarosław Buczyński, Apolarity, border rank, and multigraded Hilbert scheme, Duke Math. J. 170 (2021), no. 16, 3659–3702.

    Article  MathSciNet  MATH  Google Scholar 

  12. Jarosław Buczyński and J. M. Landsberg, On the third secant variety, J. Algebraic Combin. 40 (2014), no. 2, 475–502.

  13. Peter Bürgisser, Michael Clausen, and M. Amin Shokrollahi, Algebraic complexity theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 315, Springer-Verlag, Berlin, 1997, With the collaboration of Thomas Lickteig.

  14. Luca Chiantini, Jonathan D. Hauenstein, Christian Ikenmeyer, Joseph M. Landsberg, and Giorgio Ottaviani, Polynomials and the exponent of matrix multiplication, Bull. Lond. Math. Soc. 50 (2018), no. 3, 369–389.

    Article  MathSciNet  MATH  Google Scholar 

  15. Matthias Christandl, Fulvio Gesmundo, and Asger Kjærulff Jensen, Border rank is not multiplicative under the tensor product, SIAM J. Appl. Algebra Geom. 3 (2019), no. 2, 231–255.

  16. Matthias Christandl, Péter Vrana, and Jeroen Zuiddam, Barriers for fast matrix multiplication from irreversibility, 34th Computational Complexity Conference, LIPIcs. Leibniz Int. Proc. Inform., vol. 137, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019, pp. Art. No. 26, 17.

  17. Ciro Ciliberto and Filip Cools, On Grassmann secant extremal varieties, Adv. Geom. 8 (2008), no. 3, 377–386.

    Article  MathSciNet  MATH  Google Scholar 

  18. Austin Conner, Fulvio Gesmundo, Joseph M. Landsberg, and Emanuele Ventura, Tensors with maximal symmetries, arXiv e-prints (2019), arXiv:1909.09518.

  19. Austin Conner, Fulvio Gesmundo, Joseph M. Landsberg, and Emanuele Ventura, Rank and border rank of Kronecker powers of tensors and Strassen’s laser method, Comput. Complexity 31 (2022), no. 1, Paper No. 1, 40.

  20. Austin Conner, Alicia Harper, and J.M. Landsberg, New lower bounds for matrix mulitplication and\(\operatorname{det}_3\), arXiv:1911.07981.

  21. D. Coppersmith and S. Winograd, On the asymptotic complexity of matrix multiplication, SIAM J. Comput. 11 (1982), no. 3, 472–492.

    Article  MathSciNet  MATH  Google Scholar 

  22. Don Coppersmith and Shmuel Winograd, Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9 (1990), no. 3, 251–280.

    Article  MathSciNet  MATH  Google Scholar 

  23. Harm Derksen, On the nuclear norm and the singular value decomposition of tensors, Found. Comput. Math. 16 (2016), no. 3, 779–811.

    Article  MathSciNet  MATH  Google Scholar 

  24. Klim Efremenko, Ankit Garg, Rafael Oliveira, and Avi Wigderson, Barriers for rank methods in arithmetic complexity, 9th Innovations in Theoretical Computer Science, LIPIcs. Leibniz Int. Proc. Inform., vol. 94, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018, pp. Art. No. 1, 19.

  25. Maciej Gałązka, Multigraded Apolarity, arXiv e-prints (2016), arXiv:1601.06211.

  26. Mark Haiman and Bernd Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), no. 4, 725–769.

    Article  MathSciNet  MATH  Google Scholar 

  27. Nathan Ilten and Zach Teitler, Product ranks of the\(3\times 3\)determinant and permanent, Canad. Math. Bull. 59 (2016), no. 2, 311–319.

    Article  MathSciNet  MATH  Google Scholar 

  28. Thomas A. Ivey and Joseph M. Landsberg, Cartan for beginners, Graduate Studies in Mathematics, vol. 175, American Mathematical Society, Providence, RI, 2016, Differential geometry via moving frames and exterior differential systems, Second edition [of MR2003610].

  29. J. M. Landsberg, The border rank of the multiplication of\(2\times 2\)matrices is seven, J. Amer. Math. Soc. 19 (2006), no. 2, 447–459.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. M. Landsberg, Geometry and complexity theory, Cambridge Studies in Advanced Mathematics, vol. 169, Cambridge University Press, Cambridge, 2017.

    Book  Google Scholar 

  31. J. M. Landsberg and Mateusz Michałek, Abelian tensors, J. Math. Pures Appl. (9) 108 (2017), no. 3, 333–371.

  32. J. M. Landsberg and Mateusz Michałek, On the geometry of border rank decompositions for matrix multiplication and other tensors with symmetry, SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 2–19.

  33. J. M. Landsberg and Giorgio Ottaviani, Equations for secant varieties of Veronese and other varieties, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 569–606.

  34. Joseph M. Landsberg and Mateusz Michałek, A\(2n^2-\log _2(n)-1\)lower bound for the border rank of matrix multiplication, Int. Math. Res. Not. IMRN (2018), no. 15, 4722–4733.

  35. Joseph M. Landsberg and Giorgio Ottaviani, New lower bounds for the border rank of matrix multiplication, Theory Comput. 11 (2015), 285–298.

    Google Scholar 

  36. François Le Gall, Powers of tensors and fast matrix multiplication, Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (New York, NY, USA), ISSAC ’14, ACM, 2014, pp. 296–303.

  37. Kenneth Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math. 2 (1944), 164–168.

    Article  MathSciNet  MATH  Google Scholar 

  38. Thomas Lickteig, A note on border rank, Inform. Process. Lett. 18 (1984), no. 3, 173–178.

    Article  MathSciNet  MATH  Google Scholar 

  39. Donald W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math. 11 (1963), 431–441.

    Article  MathSciNet  Google Scholar 

  40. Victor Pan, How to multiply matrices faster, Lecture Notes in Computer Science, vol. 179, Springer-Verlag, Berlin, 1984.

  41. Kristian Ranestad and Frank-Olaf Schreyer, Varieties of sums of powers, J. Reine Angew. Math. 525 (2000), 147–181. (2001m:14009)

  42. A. Schönhage, Partial and total matrix multiplication, SIAM J. Comput. 10 (1981), no. 3, 434–455.

    Article  MathSciNet  MATH  Google Scholar 

  43. Alexandre Sedoglavic and Alexey V. Smirnov, The tensor rank of\(5\times 5\)matrices multiplication is bounded by 98 and its border rank by 89, ISSAC ’21—Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation, ACM, New York, [2021] \(\copyright \) 2021, pp. 345–351.

  44. Yaroslav Shitov, A counterexample to Comon’s conjecture, SIAM J. Appl. Algebra Geom. 2 (2018), no. 3, 428–443.

    Article  MathSciNet  MATH  Google Scholar 

  45. Yaroslav Shitov, The Waring rank of the\(3 \times 3\)permanent, SIAM J. Appl. Algebra Geom. 5 (2021), no. 4, 701–714.

    Article  MathSciNet  MATH  Google Scholar 

  46. A. V. Smirnov, The bilinear complexity and practical algorithms for matrix multiplication, Comput. Math. Math. Phys. 53 (2013), no. 12, 1781–1795.

    Article  MathSciNet  Google Scholar 

  47. A. V. Smirnov, The Approximate Bilinear Algorithm of Length 46 for Multiplication of 4 x 4 Matrices, ArXiv e-prints (2014).

  48. A. Stothers, On the complexity of matrix multiplication, PhD thesis, University of Edinburgh, 2010.

  49. V. Strassen, Rank and optimal computation of generic tensors, Linear Algebra Appl. 52/53 (1983), 645–685.

    Article  MathSciNet  MATH  Google Scholar 

  50. V. Strassen, Relative bilinear complexity and matrix multiplication, J. Reine Angew. Math. 375/376 (1987), 406–443. (88h:11026)

  51. V. Strassen, Algebra and complexity, First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 429–446.

  52. Volker Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354–356.

    Article  MathSciNet  MATH  Google Scholar 

  53. Toshio Sumi, Mitsuhiro Miyazaki, and Toshio Sakata, About the maximal rank of 3-tensors over the real and the complex number field, Ann. Inst. Statist. Math. 62 (2010), no. 4, 807–822.

    Article  MathSciNet  MATH  Google Scholar 

  54. Emil Toeplitz, Ueber ein Flächennetz zweiter Ordnung, Math. Ann. 11 (1877), no. 3, 434–463.

    Article  MathSciNet  MATH  Google Scholar 

  55. Virginia Williams, Breaking the coppersimith-winograd barrier, preprint.

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Acknowledgements

We thank J. Buczyński and J. Jelisiejew for very helpful conversations and J. Buczyński for extensive comments on a draft of this article. We also thank the anonymous referees for very careful readings of an earlier version and suggestions for significantly improving the exposition. In particular, we thank a referee for pointing out missed cases in the proof of Theorem 1.1 in an earlier version.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138-2901, USA

    Austin Conner

  2. Department of Mathematics and Statistics, 221 Parker Hall, Auburn University, Auburn, AL, 36849, USA

    Hang Huang

  3. Department of Mathematics, Texas A &M University, College Station, TX, 77843-3368, USA

    J. M. Landsberg

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  1. Austin Conner
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Correspondence to J. M. Landsberg.

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Communicated by Teresa Krick.

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Landsberg supported by NSF Grant AF-1814254, Conner supported by NSF Grant 2002149.

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Conner, A., Huang, H. & Landsberg, J.M. Bad and Good News for Strassen’s Laser Method: Border Rank of \(\mathrm{Perm}_3\) and Strict Submultiplicativity. Found Comput Math 23, 2049–2087 (2023). https://doi.org/10.1007/s10208-022-09579-3

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