A Normal Form Algorithm for Tensor Rank Decomposition
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system of polynomial equations allows us to leverage recent numerica...
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| Published in | ACM transactions on mathematical software Vol. 48; no. 4; pp. 1 - 35 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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New York, NY
ACM
19.12.2022
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| Online Access | Get full text |
| ISSN | 0098-3500 1557-7295 1557-7295 |
| DOI | 10.1145/3555369 |
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| Abstract | We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We characterize the complexity of our algorithm in terms of an algebraic property of this polynomial system—the multigraded regularity. We prove effective bounds for many tensor formats and ranks, which are of independent interest for overconstrained polynomial system solving. Moreover, we conjecture a general formula for the multigraded regularity, yielding a (parameterized) polynomial time complexity for the tensor rank decomposition problem in the considered setting. Our numerical experiments show that our algorithm can outperform state-of-the-art numerical algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption. |
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| AbstractList | We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We characterize the complexity of our algorithm in terms of an algebraic property of this polynomial system—the multigraded regularity. We prove effective bounds for many tensor formats and ranks, which are of independent interest for overconstrained polynomial system solving. Moreover, we conjecture a general formula for the multigraded regularity, yielding a (parameterized) polynomial time complexity for the tensor rank decomposition problem in the considered setting. Our numerical experiments show that our algorithm can outperform state-of-the-art numerical algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption. |
| ArticleNumber | 38 |
| Author | Vannieuwenhoven, Nick Telen, Simon |
| Author_xml | – sequence: 1 givenname: Simon orcidid: 0000-0002-3459-5845 surname: Telen fullname: Telen, Simon email: simon.telen@mis.mpg.de organization: Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany – sequence: 2 givenname: Nick orcidid: 0000-0001-5692-4163 surname: Vannieuwenhoven fullname: Vannieuwenhoven, Nick email: nick.vannieuwenhoven@kuleuven.be organization: KU Leuven, Heverlee, Belgium |
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| CitedBy_id | crossref_primary_10_1016_j_jalgebra_2024_11_017 crossref_primary_10_1137_24M1653793 |
| Cites_doi | 10.1007/s10208-017-9372-x 10.1021/ac00289a052 10.1109/TSP.2017.2690524 10.1137/130916084 10.1016/j.jsc.2012.05.012 10.1137/040608830 10.1145/2512329 10.3792/chmm/1263317740 10.1007/978-1-4613-9425-9 10.1007/s10208-015-9291-7 10.1007/BF02289464 10.1137/18M1200531 10.1137/110829180 10.1137/16M1090132 10.1007/978-3-319-26765-4 10.1137/0614071 10.1145/1993886.1993898 10.1137/17M1162433 10.1137/S0895479896305696 10.1090/S0025-5718-1977-0428694-0 10.1016/j.laa.2018.07.004 10.1090/gsm/124 10.1002/cem.1180040105 10.1016/j.laa.2016.10.019 10.1002/1099-128X(200005/06)14:3<229::AID-CEM587>3.0.CO;2-N 10.1007/s10231-013-0352-8 10.1007/BFb0093426 10.1016/j.jpaa.2020.106367 10.1137/07070111x 10.1137/060655894 10.1002/sapm192761164 10.1137/110836067 10.1016/0024-3795(77)90069-6 10.1137/140961389 10.1007/978-3-662-03338-8 10.1137/120877258 10.1016/j.matpur.2020.07.003 10.1112/S0024610706022630 10.1016/j.jpaa.2010.11.010 10.1016/j.laa.2010.06.046 |
| ContentType | Journal Article |
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| Keywords | canonical polyadic decomposition polynomial systems Tensor rank decomposition normal form algorithms |
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| Title | A Normal Form Algorithm for Tensor Rank Decomposition |
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