The Tensor Rank of 5x5 Matrices Multiplication is Bounded by 98 andIts Border Rank by 89

Sedoglavic, Alexandre; Смирнов, А В

doi:10.1145/3452143.3465537

Cited by 6 publications

(2 citation statements)

References 26 publications

(32 reference statements)

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“…In all instances, we either matched the previously smallest known number of required multiplications or even found improvements. Our technique is based on investigating random paths in a certain graph, see [9,8,3,2] for other search techniques that have successfully been applied in the quest for reducing the number of multiplications for certain formats.…”

Section: Introductionmentioning

confidence: 99%

A Normal Form for Matrix Multiplication Schemes

Kauers

Moosbauer

2022

Algebraic Informatics

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Section: Introductionmentioning

confidence: 99%

A Normal Form for Matrix Multiplication Schemes

Kauers

Moosbauer

2022

Algebraic Informatics

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“…In fact, the search space is so large that even the optimal algorithm for multiplying two 3 × 3 matrices is still unknown. Nevertheless, in a longstanding research effort, matrix multiplication algorithms have been discovered by attacking this tensor decomposition problem using human search 2,15,16 , continuous optimization [17][18][19] and combinatorial search 20 . These approaches often rely on human-designed heuristics, which are probably suboptimal.…”

mentioning

confidence: 99%

Discovering faster matrix multiplication algorithms with reinforcement learning

Fawzi

Balog

Huang

et al. 2022

Nature

244

110

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Improving the efficiency of algorithms for fundamental computations can have a widespread impact, as it can affect the overall speed of a large amount of computations. Matrix multiplication is one such primitive task, occurring in many systems—from neural networks to scientific computing routines. The automatic discovery of algorithms using machine learning offers the prospect of reaching beyond human intuition and outperforming the current best human-designed algorithms. However, automating the algorithm discovery procedure is intricate, as the space of possible algorithms is enormous. Here we report a deep reinforcement learning approach based on AlphaZero1 for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. Our agent, AlphaTensor, is trained to play a single-player game where the objective is finding tensor decompositions within a finite factor space. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time, to our knowledge, since its discovery 50 years ago2. We further showcase the flexibility of AlphaTensor through different use-cases: algorithms with state-of-the-art complexity for structured matrix multiplication and improved practical efficiency by optimizing matrix multiplication for runtime on specific hardware. Our results highlight AlphaTensor’s ability to accelerate the process of algorithmic discovery on a range of problems, and to optimize for different criteria.

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