Barriers for rectangular matrix multiplication

We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give optimal algorithms. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. We prove our result using the asymptotic spectrum of tensors. More precisely, we crucially make use of two families of real tensor parameters with special algebraic properties: the quantum functionals and the support functionals. In particular, we prove that any lower bound on the dual exponent of matrix multiplication $\alpha$ via the big Coppersmith-Winograd tensors cannot exceed 0.625.

[1]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[2]  Victor Y. Pan,et al.  Fast Rectangular Matrix Multiplication and Applications , 1998, J. Complex..

[3]  Matthias Christandl,et al.  Universal points in the asymptotic spectrum of tensors , 2017, STOC.

[4]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[5]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[6]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[7]  Yin Tat Lee,et al.  Solving Empirical Risk Minimization in the Current Matrix Multiplication Time , 2019, COLT.

[8]  Joshua A. Grochow,et al.  Which groups are amenable to proving exponent two for matrix multiplication? , 2017, ArXiv.

[9]  Matthias Christandl,et al.  Barriers for fast matrix multiplication from irreversibility , 2018, CCC.

[10]  Grazia Lotti,et al.  On the Asymptotic Complexity of Rectangular Matrix Multiplication , 1983, Theor. Comput. Sci..

[11]  Josh Alman,et al.  Further Limitations of the Known Approaches for Matrix Multiplication , 2017, ITCS.

[12]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[13]  Andris Ambainis,et al.  Fast Matrix Multiplication: Limitations of the Coppersmith-Winograd Method , 2014, STOC.

[14]  V. Strassen.  The asymptotic spectrum of tensors. , 1988 .

[15]  François Le Gall,et al.  Faster Algorithms for Rectangular Matrix Multiplication , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[16]  Joshua A. Grochow,et al.  On cap sets and the group-theoretic approach to matrix multiplication , 2016, ArXiv.

[17]  Josh Alman,et al.  Limits on the Universal method for matrix multiplication , 2018, CCC.

[18]  A. J. Stothers.  On the complexity of matrix multiplication , 2010 .

[19]  V. Strassen,et al.  Degeneration and complexity of bilinear maps: Some asymptotic spectra. , 1991 .

[20]  V. Pan,et al.  Fast rectangular matrix multiplication and some applications , 2008 .

[21]  Markus Bläser,et al.  Fast Matrix Multiplication , 2013, Theory Comput..

[22]  Don Coppersmith,et al.  Rectangular Matrix Multiplication Revisited , 1997, J. Complex..

[23]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[24]  Josh Alman,et al.  Limits on All Known (and Some Unknown) Approaches to Matrix Multiplication , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[25]  D. Coppersmiths.  RAPID MULTIPLICATION OF RECTANGULAR MATRICES * , 2014 .

[26]  François Le Gall,et al.  Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor , 2017, SODA.

[27]  Yin Tat Lee,et al.  Solving linear programs in the current matrix multiplication time , 2018, STOC.