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- W4294583887 abstract "We consider the techniques behind the current best algorithms for matrix multiplication. Our results are threefold. (1) We provide a unifying framework, showing that all known matrix multiplication running times since 1986 can be achieved from a single very natural tensor - the structural tensor $T_q$ of addition modulo an integer $q$. (2) We show that if one applies a generalization of the known techniques (arbitrary zeroing out of tensor powers to obtain independent matrix products in order to use the asymptotic sum inequality of Sch{o}nhage) to an arbitrary monomial degeneration of $T_q$, then there is an explicit lower bound, depending on $q$, on the bound on the matrix multiplication exponent $omega$ that one can achieve. We also show upper bounds on the value $alpha$ that one can achieve, where $alpha$ is such that $ntimes n^alpha times n$ matrix multiplication can be computed in $n^{2+o(1)}$ time. (3) We show that our lower bound on $omega$ approaches $2$ as $q$ goes to infinity. This suggests a promising approach to improving the bound on $omega$: for variable $q$, find a monomial degeneration of $T_q$ which, using the known techniques, produces an upper bound on $omega$ as a function of $q$. Then, take $q$ to infinity. It is not ruled out, and hence possible, that one can obtain $omega=2$ in this way." @default.
- W4294583887 created "2022-09-05" @default.
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- W4294583887 date "2017-12-19" @default.
- W4294583887 modified "2023-09-26" @default.
- W4294583887 title "Further limitations of the known approaches for matrix multiplication" @default.
- W4294583887 doi "https://doi.org/10.48550/arxiv.1712.07246" @default.
- W4294583887 hasPublicationYear "2017" @default.
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