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 existin...
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| Format: | Article |
| Language: | English |
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SciPost
2025-03-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.18.3.104 |
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| Summary: | 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. |
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| ISSN: | 2542-4653 |