Learning tensor networks with tensor cross interpolation: new algorithms and libraries
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917967473672192 |
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| author | Fernández, Yuriel Núñez Ritter, Marc K. Jeannin, Matthieu Li, Jheng-Wei Kloss, Thomas Louvet, Thibaud Terasaki, Satoshi Parcollet, Olivier von Delft, Jan Shinaoka, Hiroshi Waintal, Xavier |
| author_facet | Fernández, Yuriel Núñez Ritter, Marc K. Jeannin, Matthieu Li, Jheng-Wei Kloss, Thomas Louvet, Thibaud Terasaki, Satoshi Parcollet, Olivier von Delft, Jan Shinaoka, Hiroshi Waintal, Xavier |
| contents | The tensor cross interpolation (TCI) algorithm is a rank-revealing algorithm for decomposing low-rank, high-dimensional tensors into tensor trains/matrix product states (MPS). TCI learns a compact MPS representation of the entire object from a tiny training data set. Once obtained, the large existing MPS toolbox provides exponentially fast algorithms for performing a large set of operations. We discuss several improvements and variants of TCI. In particular, we show that replacing the cross interpolation by the partially rank-revealing LU decomposition yields a more stable and more flexible algorithm than the original algorithm. We also present two open source libraries, xfac in Python/C++ and TensorCrossInterpolation.jl in Julia, that implement these improved algorithms, and illustrate them on several applications. These include sign-problem-free integration in large dimension, the superhigh-resolution quantics representation of functions, the solution of partial differential equations, the superfast Fourier transform, the computation of partition functions, and the construction of matrix product operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02454 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning tensor networks with tensor cross interpolation: new algorithms and libraries Fernández, Yuriel Núñez Ritter, Marc K. Jeannin, Matthieu Li, Jheng-Wei Kloss, Thomas Louvet, Thibaud Terasaki, Satoshi Parcollet, Olivier von Delft, Jan Shinaoka, Hiroshi Waintal, Xavier Computational Physics Strongly Correlated Electrons The tensor cross interpolation (TCI) algorithm is a rank-revealing algorithm for decomposing low-rank, high-dimensional tensors into tensor trains/matrix product states (MPS). TCI learns a compact MPS representation of the entire object from a tiny training data set. Once obtained, the large existing MPS toolbox provides exponentially fast algorithms for performing a large set of operations. We discuss several improvements and variants of TCI. In particular, we show that replacing the cross interpolation by the partially rank-revealing LU decomposition yields a more stable and more flexible algorithm than the original algorithm. We also present two open source libraries, xfac in Python/C++ and TensorCrossInterpolation.jl in Julia, that implement these improved algorithms, and illustrate them on several applications. These include sign-problem-free integration in large dimension, the superhigh-resolution quantics representation of functions, the solution of partial differential equations, the superfast Fourier transform, the computation of partition functions, and the construction of matrix product operators. |
| title | Learning tensor networks with tensor cross interpolation: new algorithms and libraries |
| topic | Computational Physics Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2407.02454 |