Strassen's algorithm is not optimally accurate
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866916304491905024 |
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| author | Dumas, Jean-Guillaume Pernet, Clément Sedoglavic, Alexandre |
| author_facet | Dumas, Jean-Guillaume Pernet, Clément Sedoglavic, Alexandre |
| contents | We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity bound with the best currently known leading term (obtained via alternate basis sparsification). To build this algorithm, we consider matrix and tensor norms bounds governing the stability and accuracy of numerical matrix multiplication. First, we reduce those bounds by minimizing a growth factor along the unique orbit of Strassen's 2x2-matrix multiplication tensor decomposition. Second, we develop heuristics for minimizing the number of operations required to realize a given bilinear formula, while further improving its accuracy. Third, we perform an alternate basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strassen's algorithm is not optimally accurate Dumas, Jean-Guillaume Pernet, Clément Sedoglavic, Alexandre Numerical Analysis Symbolic Computation We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity bound with the best currently known leading term (obtained via alternate basis sparsification). To build this algorithm, we consider matrix and tensor norms bounds governing the stability and accuracy of numerical matrix multiplication. First, we reduce those bounds by minimizing a growth factor along the unique orbit of Strassen's 2x2-matrix multiplication tensor decomposition. Second, we develop heuristics for minimizing the number of operations required to realize a given bilinear formula, while further improving its accuracy. Third, we perform an alternate basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy. |
| title | Strassen's algorithm is not optimally accurate |
| topic | Numerical Analysis Symbolic Computation |
| url | https://arxiv.org/abs/2402.05630 |