Probabilistic Tensors and Opportunistic Boolean Matrix Multiplication

Karppa, Matti; Kaski, Petteri

doi:10.1137/1.9781611975482.31

Cited by 7 publications

(21 citation statements)

References 52 publications

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“…Cohn and Umans [17] introduced the notion of the support rank of tensors, and showed that upper bounds on the support rank of matrix multiplication tensors can be used to design faster Boolean matrix multiplication algorithms. Recently, Karppa and Kaski [23] used "probabilistic tensors" as another way to design Boolean matrix multiplication algorithms.…”

Section: Other Related Workmentioning

confidence: 99%

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Alman¹

2021

Theory of Comput.

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We prove limitations on the known methods for designing matrix multiplication algorithms. Alman and Vassilevska Williams (FOCS'18) recently defined the Universal Method, which generalizes all the known approaches, including Strassen's Laser Method (J. reine angew. Math., 1987) and Cohn and Umans's Group Theoretic Method (FOCS'03). We prove concrete lower bounds on the algorithms one can design by applying the Universal Method to many different tensors. Our proofs use new tools to give upper bounds on the asymptotic slice rank of a wide range of tensors. Our main result is that the Universal Method applied to any Coppersmith-Winograd tensor CW q cannot yield a bound on ω, the exponent of matrix multiplication, better than 2.16805. It was previously known (Alman and Vassilevska Williams, FOCS'18) that the weaker "Galactic Method" applied to CW q could not achieve an exponent of 2.We also study the Laser Method (which is a special case of the Universal Method) and prove that it is "complete" for matrix multiplication algorithms: when it applies to a tensor T , it achieves ω = 2 if and only if it is possible for the Universal Method applied to T to achieve ω = 2. Hence, the Laser Method, which was originally used as an algorithmic tool,

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Section: Other Related Workmentioning

confidence: 99%

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Alman¹

2021

Theory of Comput.

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“…There is one critical difference between our analysis and that of [11]. In [11], the only relevant factor was the gross count of the number of terms computed by the pseudo-multiplication algorithm.…”

Section: Introductionmentioning

confidence: 98%

“…It seems difficult to make progress on better practical algorithms for matrix multiplication. In [11], Karppa & Kaski proposed a novel type of algorithm to break this impasse. Instead of reducing BMM directly to matrix multiplication, one reduces to to a randomized "broken" or "opportunistic" form of matrix multiplication.…”

Section: Introductionmentioning

confidence: 99%

“…In this note, we describe an alternative way to use the pseudo-multiplication algorithm of [11]. The idea is that, instead of executing multiple independent pseudo-multiplications, we combine them all into a single larger randomized pseudo-multiplication.…”

Section: Introductionmentioning

confidence: 99%

“…This avoids a number of tedious issues with implementing padding or choosing cut-off points for the recursion. We expect that our new algorithm will be more efficient in practice than Strassen's algorithm or the original algorithm of [11].…”

Section: Introductionmentioning

confidence: 99%

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Improved algorithms for Boolean matrix multiplication via opportunistic matrix multiplication

Harris¹

2021

Preprint

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Karppa & Kaski (2019) proposed a novel type of "broken" or "opportunistic" multiplication algorithm, based on a variant of Strassen's alkgorithm, and used this to develop new algorithms for Boolean matrix multiplication, among other tasks. For instance, their algorithm can compute Boolean matrix multiplication in O(n log 2 (6+6/7) log n) = O(n 2.778 ) time. While faster matrix multiplication algorithms exist asymptotically, in practice most such algorithms are infeasible for practical problems. Their opportunistic algorithm is a slight variant of Strassen's algorithm, so hopefully it should yield practical as well as asymptotic improvements to it.In this note, we describe a more efficient way to use the broken matrix multiplication algorithm to solve Boolean matrix multiplication. In brief, instead of running multiple iterations of the broken algorithm on the original input matrix, we form a new larger matrix by sampling and run a single iteration of the broken algorithm on it. The resulting algorithm has runtime O(n .763 ). We also describe an extension to witnessing Boolean matrix multiplication, as well as extensions to non-square matrices.The new algorithm is simple and has reasonable constants. We hope it may lead to improved practical algorithms

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Algorithms for Matrix Multiplication via Sampling and Opportunistic Matrix Multiplication

Harris

2024

Algorithmica

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Probabilistic Tensors and Opportunistic Boolean Matrix Multiplication

Cited by 7 publications

References 52 publications

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Improved algorithms for Boolean matrix multiplication via opportunistic matrix multiplication

Algorithms for Matrix Multiplication via Sampling and Opportunistic Matrix Multiplication

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