SciPost Submission Page
Learning tensor networks with tensor cross interpolation: new algorithms and libraries
by Yuriel Núñez Fernández, Marc K. Ritter, Matthieu Jeannin, Jheng-Wei Li, Thomas Kloss, Thibaud Louvet, Satoshi Terasaki, Olivier Parcollet, Jan von Delft, Hiroshi Shinaoka, Xavier Waintal
Submission summary
| Authors (as registered SciPost users): | Yuriel Núñez Fernández · Xavier Waintal |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2407.02454v3 (pdf) |
| Code repository: | http://tensor4all.org |
| Date submitted: | 2025-01-06 16:04 |
| Submitted by: | Núñez Fernández, Yuriel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
List of changes
Related to Report #1:
1. We have rewritten the explanation given in section 3.2.4 (page 11) to show the derivation of Eq. (24) from Eq. (15) explicitly.
2. We have added an example for a fully nested configuration of pivot lists, with examples for pivots that would break nesting conditions (pages 17, 18).
3. We have added a remark explaining the terminology above Eq. (33) (page 16).
4. We have added an explaining sentence after Eq. (43) (page 21).
5. We have corrected the sentence accordingly (page 22).
6. We have added a paragraph detailing this motivation to section 4.5 (page 23). We give explicit examples where the conversion is advantageous.
7. We have removed the constant factors of 3 from the asymptotic complexities (now Table 2, page 27).
8. We now use two different color schemes $\beta$ and $\mathcal{L}$ in Fig.~5, as suggested by the referee.
Related to Report #2:
1. About failure mode of tci
- We have added a paragraph that describes more examples for QTT ranks of simple analytic functions in multiple dimensions to section 6.1 (page 35).
2. We have added a footnote about the approach of Ref. 64 in the main text (page 44).
3. About typos:
- We have changed the sentence accordingly (page 17).
- We have changed the main text accordingly (page 31).
- We now cite both references in all places where this statement is made (pages 35, 36, 58).
The following entries in the bibliography were changed:
* Sozykin, Chertkov, et al., 2022: removed duplicate arXiv number.
* Jolly, Núñez Fernández, and Waintal, 2023: corrected second author's name (Núñez Fernández) and removed duplicate arXiv number.
* Sakurai, Takahashi, and Miyamoto, 2024: added digital object identifier and removed duplicate arXiv number.
* Takahashi, Sakurai, and Shinaoka, 2024: added digital object identifier and removed duplicate arXiv number.
* Ishida, Okada, Hoshino, and Shinaoka, 2024: added digital object identifier and removed duplicate arXiv number.
* \'Sroda, Inayoshi, Shinaoka, and Werner, 2024: new reference.
* Gourianov, Lubasch, et al., 2022: removed duplicate arXiv number.
* Peddinti, Pisoni, et al., 2023: removed duplicate arXiv number.
* Sakaue, Shinaoka, and Sakurai, 2024: added digital object identifier and removed duplicate arXiv number.
* Golub and van Loan, 1996: corrected ISBN.
* Poole and Neal, 2000: added digital object identifier.
* Verstraete and Cirac, 2004: removed duplicate arXiv number.
* Woolfe, Hill, and Hollenberg, 2014: removed duplicate arXiv number.
* Removed entry https://gitlab.com/tensors4fields, as all material there has been moved to tensor4all.org. All references have been updated accordingly.
* Chen and Lindsey, 2024: removed duplicate arXiv number.
Current status:
Reports on this Submission
Report #1 by Christian Mendl (Referee 2) on 2025-1-7 (Invited Report)
Report
The comments and suggestions from the first review report have been well-addressed in the revised version. I recommend publication.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)