Generative modeling via tensor train sketching
In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear...
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| Veröffentlicht in: | Applied and computational harmonic analysis Jg. 67; H. C; S. 101575 |
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| Sprache: | Englisch |
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01.11.2023
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| Abstract | In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear systems for the individual tensor train cores. This approach can avoid the curse of dimensionality that threatens both the algorithmic and sample complexities of the recovery problem. Specifically, for Markov models under natural conditions, we prove that the tensor cores can be recovered with a sample complexity that scales logarithmically in the dimensionality. Finally, we illustrate the performance of the method with several numerical experiments. |
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| AbstractList | Not provided. In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear systems for the individual tensor train cores. This approach can avoid the curse of dimensionality that threatens both the algorithmic and sample complexities of the recovery problem. Specifically, for Markov models under natural conditions, we prove that the tensor cores can be recovered with a sample complexity that scales logarithmically in the dimensionality. Finally, we illustrate the performance of the method with several numerical experiments. |
| ArticleNumber | 101575 |
| Author | Lindsey, Michael Hur, YoonHaeng Khoo, Yuehaw Stoudenmire, E.M. Hoskins, Jeremy G. |
| Author_xml | – sequence: 1 givenname: YoonHaeng orcidid: 0000-0001-6308-8896 surname: Hur fullname: Hur, YoonHaeng email: yoonhaenghur@uchicago.edu organization: Department of Statistics, University of Chicago, United States of America – sequence: 2 givenname: Jeremy G. surname: Hoskins fullname: Hoskins, Jeremy G. email: jeremyhoskins@uchicago.edu organization: Department of Statistics, University of Chicago, United States of America – sequence: 3 givenname: Michael surname: Lindsey fullname: Lindsey, Michael email: michael.lindsey@cims.nyu.edu organization: Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, United States of America – sequence: 4 givenname: E.M. surname: Stoudenmire fullname: Stoudenmire, E.M. email: mstoudenmire@flatironinstitute.org organization: Center for Computational Quantum Physics, Flatiron Institute, United States of America – sequence: 5 givenname: Yuehaw surname: Khoo fullname: Khoo, Yuehaw email: ykhoo@uchicago.edu organization: Department of Statistics, University of Chicago, United States of America |
| BackLink | https://www.osti.gov/biblio/2576309$$D View this record in Osti.gov |
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| Cites_doi | 10.1137/090752286 10.1137/15M1036919 10.1111/j.1467-9868.2010.00769.x 10.1007/s10444-018-9622-8 10.1002/gamm.202100008 10.1073/pnas.0804869105 10.1017/9781316544938 10.1103/PhysRevLett.69.2863 10.1137/19M1257718 10.1137/090771806 10.1137/17M1154382 10.1007/BF01932678 10.4310/CMS.2010.v8.n1.a11 10.1007/s11222-019-09910-z 10.1103/PhysRevB.99.155131 10.1137/15M1010506 10.1016/j.laa.2009.07.024 10.1016/j.cma.2018.12.015 10.1561/2200000079 10.1137/20M1387158 |
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| Keywords | High-dimensional function approximation Generative modeling Tensor decompositions Tensor train Randomized algorithm |
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Eng. doi: 10.1016/j.cma.2018.12.015 – volume: 14 start-page: 566 issue: 5 year: 2021 ident: 10.1016/j.acha.2023.101575_br0050 article-title: Spectral methods for data science: a statistical perspective publication-title: Found. Trends Mach. Learn. doi: 10.1561/2200000079 – volume: 44 start-page: C25 issue: 1 year: 2022 ident: 10.1016/j.acha.2023.101575_br0070 article-title: Parallel algorithms for tensor train arithmetic publication-title: SIAM J. Sci. Comput. doi: 10.1137/20M1387158 |
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| SubjectTerms | Generative modeling High-dimensional function approximation Mathematics Randomized algorithm Tensor decompositions Tensor train |
| Title | Generative modeling via tensor train sketching |
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