Learning tensor networks with tensor cross interpolation: New algorithms and libraries
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.
Authors
- Thibaud Louvet
- Thomas Kloss (ORCID: https://orcid.org/0000-0003-1120-209X)
- Jan von Delft (ORCID: https://orcid.org/0000-0002-8655-0999)
- Marc K. Ritter (ORCID: https://orcid.org/0000-0002-2960-5471)
- Olivier Parcollet (ORCID: https://orcid.org/0000-0002-0389-2660)
- Y. Núñez-Fernández (ORCID: https://orcid.org/0000-0002-0080-1903)
- Satoshi Terasaki
- Hiroshi Shinaoka (ORCID: https://orcid.org/0000-0002-7058-8765)
- Jheng-Wei Li (ORCID: https://orcid.org/0000-0002-2228-6762)
- Xavier Waintal (ORCID: https://orcid.org/0000-0003-3816-8290)
- Matthieu Jeannin (ORCID: https://orcid.org/0000-0003-2287-860X)
Institutions
- Institut polytechnique de Grenoble (FR)
- Centre National de la Recherche Scientifique (FR)
- Commissariat à l'Énergie Atomique et aux Énergies Alternatives (FR)
- Université Paris-Saclay (FR)
- Center for NanoScience (DE)
- CEA Paris-Saclay - Etablissement de Saclay (FR)
- Center for Theoretical Physics (PL)
- CEA Grenoble (FR)
- Institut de Recherche Interdisciplinaire de Grenoble (FR)
- Laboratoire de Photonique Quantique et Moléculaire (FR)
- CEA Paris-Saclay (FR)
- Munich Center for Quantum Science and Technology (DE)
- Institut Néel (FR)
- Flatiron Health (United States) (US)
- PHotonique ELectronique et Ingénierie QuantiqueS
- Flatiron Institute (US)
- Saitama University (JP)
- Université Grenoble Alpes (FR)
Publication Details
- Journal
- SciPost Physics
- Published
- 2025-03-20
- DOI
- https://doi.org/10.21468/scipostphys.18.3.104
- Citations
- 30
- Primary Topic
- Tensor decomposition and applications
- Type
- article
- Field-Weighted Citation Impact
- 46.01
Funders
- Deutsche Forschungsgemeinschaft
- Agence Nationale de la Recherche
- Forschungszentrum Jülich
- Kyushu University
- Commissariat à l'Énergie Atomique et aux Énergies Alternatives
- Japan Society for the Promotion of Science
- Japan Science and Technology Agency