Analysis of faster matrix multiplication

Research Article
Open access

Analysis of faster matrix multiplication

Jie Liu 1*
  • 1 University of California, Berkeley    
  • *corresponding author jasperl@berkeley.edu
Published on 17 November 2023 | https://doi.org/10.54254/2753-8818/11/20230394
TNS Vol.11
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-133-9
ISBN (Online): 978-1-83558-134-6

Abstract

Since the definition of matrices in 1855, matrix multiplication has played a crucial role in a wide range of fields. Over the years, numerous researchers have dedicated their efforts to improving the time complexity of this fundamental operation. This paper aims to delve into the historical development of matrix multiplication algorithms and methodologies employed to achieve these significant advancements in time complexity. By employing various approaches, researchers have been able to improve the time complexity of matrix multiplication, leading to a significant reduction from O () to O (). Across nearly two centuries, this progress is contributed by a lot of extraordinary scientists and researchers. This paper explores the practical implications of these improvements across various domains, such as computer science, physics, economics, and more. The development of more efficient matrix multiplication algorithms has enabled researchers and practitioners to tackle complex problems and explore new frontiers. In the future, with the rapid growth of machine learning techniques, matrix multiplication will continue to evolve and improve.

Keywords:

matrix, multiplication, time complexity, strassen, machine learning.

Liu,J. (2023). Analysis of faster matrix multiplication. Theoretical and Natural Science,11,142-146.
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References

[1]. Le Gall, François (2014), "Algebraic complexity theory and matrix multiplication", in Ka-tsusuke Nabeshima (ed.), Proceedings of the 39th International Symposium on Symbo-lic and Algebraic Computation - ISSAC '14, pp. 296–303, arXiv:1401.7714, Bibcode:2-014arXiv1401.7714L, doi:10.1145/2608628.2627493, ISBN 978-1-4503-2501-1, S2CID 25-97483

[2]. Zhang, Y., & Yu, H. (2022). Faster Matrix Multiplication via Partitioned Computing. arXiv preprint arXiv:2210.10173. Retrieved from https://arxiv.org/pdf/2210.10173.pdf

[3]. V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354-356, 1969.

[4]. Steven Huss-Lederman, Elaine M. Jacobson, Anna Tsao, Thomas Turnbull, and Jeremy R. Johnson. 1996. Implementation of Strassen's algorithm for matrix multiplication. In Proceedings of the 1996 ACM/IEEE conference on Supercomputing (Supercomputing '96). IEEE Computer Society, USA, 32–es. https://doi.org/10.1145/369028.369096

[5]. Virginia Vassilevska Williams (2012). "Multiplying Matrices Faster than Coppersmith-Winograd". In Howard J. Karloff; Toniann Pitassi (eds.). Proc. 44th Symposium on Theory of Computing (STOC). ACM. pp. 887–898. doi:10.1145/2213977.2214056. S2CID 14350287.

[6]. Williams, Virginia Vassilevska. Multiplying matrices in O()time (PDF) (Technical Report). Stanford University.

[7]. D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symbolic Computation, 9(3):251–280, 1990.

[8]. D. Coppersmith and S. Winograd. On the asymptotic complexity of matrix multiplication. In Proc. SFCS, pages 82–90, 1981.

[9]. Hardesty, L. (n.d.). Explained: Matrices. MIT News | Massachusetts Institute of Technology. Retrieved April 25, 2023, from https://news.mit.edu/2013/explained-matrices-1206

[10]. Alhussein Fawzi, Matej Balog, Aja Huang, Thomas Hubert, Bernardino Romera-Paredes, Mohammadamin Barekatain, Alexander Novikov, Francisco J. R. Ruiz, Julian Schrittwieser, Grzegorz Swirszcz, David Silver, Demis Hassabis, and Pushmeet Kohli. Discovering faster matrix multiplication algorithms with reinforcement learning. Nature, 610(7930):47–53, 2022. doi:10.1038/s41586-022-05172-4

[11]. Benj Edwards - Oct 13, 2022 8:46 pm U. T. C., & Scintillant Seniorius Lurkius et Sub-scriptor jump to post. (2022, October 13). Deepmind breaks 50-year math record usi-ng AI; New record falls a week later. Ars Technica. Retrieved April 20, 2023, fromhttps://arstechnica.com/information-technology/2022/10/deepmind-breaks-50-year-math-record-using-ai-new-record-falls-a-week-later/

Cite this article

Liu,J. (2023). Analysis of faster matrix multiplication. Theoretical and Natural Science,11,142-146.

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The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation

ISBN：978-1-83558-133-9(Print) / 978-1-83558-134-6(Online)
Editor：Roman Bauer
Conference website: https://www.confmpcs.org/
Conference date: 12 August 2023
Series: Theoretical and Natural Science
Volume number: Vol.11
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. Le Gall, François (2014), "Algebraic complexity theory and matrix multiplication", in Ka-tsusuke Nabeshima (ed.), Proceedings of the 39th International Symposium on Symbo-lic and Algebraic Computation - ISSAC '14, pp. 296–303, arXiv:1401.7714, Bibcode:2-014arXiv1401.7714L, doi:10.1145/2608628.2627493, ISBN 978-1-4503-2501-1, S2CID 25-97483

[2]. Zhang, Y., & Yu, H. (2022). Faster Matrix Multiplication via Partitioned Computing. arXiv preprint arXiv:2210.10173. Retrieved from https://arxiv.org/pdf/2210.10173.pdf

[3]. V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354-356, 1969.

[4]. Steven Huss-Lederman, Elaine M. Jacobson, Anna Tsao, Thomas Turnbull, and Jeremy R. Johnson. 1996. Implementation of Strassen's algorithm for matrix multiplication. In Proceedings of the 1996 ACM/IEEE conference on Supercomputing (Supercomputing '96). IEEE Computer Society, USA, 32–es. https://doi.org/10.1145/369028.369096

[5]. Virginia Vassilevska Williams (2012). "Multiplying Matrices Faster than Coppersmith-Winograd". In Howard J. Karloff; Toniann Pitassi (eds.). Proc. 44th Symposium on Theory of Computing (STOC). ACM. pp. 887–898. doi:10.1145/2213977.2214056. S2CID 14350287.

[6]. Williams, Virginia Vassilevska. Multiplying matrices in O()time (PDF) (Technical Report). Stanford University.

[7]. D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symbolic Computation, 9(3):251–280, 1990.

[8]. D. Coppersmith and S. Winograd. On the asymptotic complexity of matrix multiplication. In Proc. SFCS, pages 82–90, 1981.

[9]. Hardesty, L. (n.d.). Explained: Matrices. MIT News | Massachusetts Institute of Technology. Retrieved April 25, 2023, from https://news.mit.edu/2013/explained-matrices-1206

[10]. Alhussein Fawzi, Matej Balog, Aja Huang, Thomas Hubert, Bernardino Romera-Paredes, Mohammadamin Barekatain, Alexander Novikov, Francisco J. R. Ruiz, Julian Schrittwieser, Grzegorz Swirszcz, David Silver, Demis Hassabis, and Pushmeet Kohli. Discovering faster matrix multiplication algorithms with reinforcement learning. Nature, 610(7930):47–53, 2022. doi:10.1038/s41586-022-05172-4

[11]. Benj Edwards - Oct 13, 2022 8:46 pm U. T. C., & Scintillant Seniorius Lurkius et Sub-scriptor jump to post. (2022, October 13). Deepmind breaks 50-year math record usi-ng AI; New record falls a week later. Ars Technica. Retrieved April 20, 2023, fromhttps://arstechnica.com/information-technology/2022/10/deepmind-breaks-50-year-math-record-using-ai-new-record-falls-a-week-later/

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