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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 α via the big Coppersmith–Winograd tensors cannot exceed 0.625.
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New Bounds for Matrix Multiplication: from Alpha to Omega
The main contribution of this paper is a new improved variant of the laser method for designing matrix multiplication algorithms. Building upon the recent techniques of [Duan, Wu, Zhou, FOCS 2023], the new method introduces several new ingredients that not only yield an improved bound on the matrix multiplication exponent ω, but also improve the known bounds on rectangular matrix multiplication by [Le Gall and Urrutia, SODA 2018]. In particular, the new bound on ω is ω ≤ 2.371552 (improved from ω ≤ 2.371866). For the dual matrix multiplication exponent α defined as the largest α for which ω(1, α, 1) = 2, we obtain the improvement α ≥ 0.321334 (improved from α ≥ 0.31389). Similar improvements are obtained for various other exponents for multiplying rectangular matrices.
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- Award ID(s):
- 2330048
- PAR ID:
- 10524471
- Publisher / Repository:
- Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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