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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Barriers for Fast Matrix Multiplication from Irreversibility
Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent omega, is a central problem in algebraic complexity theory. The best upper bounds on omega, leading to the stateoftheart omega <= 2.37.., have been obtained via the laser method of Strassen and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on omega. We introduce a new and more general barrier, providing stronger limitations than in previous work. Concretely, we introduce the notion of "irreversibility" of a tensor and we prove (in some precise sense) that any approach that uses an irreversible tensor in an intermediate step (e.g., as a starting tensor in the laser method) cannot give omega = 2. In quantitative terms, we prove that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor. The quantum functionals and Strassen support functionals give (so far, the best) lower bounds on irreversibility. We provide lower bounds on the irreversibility of key intermediate tensors, including the small and big Coppersmith  Winograd tensors, that improve limitations shown in previous work. Finally, we discuss barriers on the grouptheoretic approach in terms of "monomial" irreversibility.
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 Award ID(s):
 1638352
 NSFPAR ID:
 10107903
 Date Published:
 Journal Name:
 34th Computational Complexity Conference (CCC 2019)
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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