张量交叉插值学习张量网络：新算法与库
Learning tensor networks with tensor cross interpolation: new algorithms and libraries
AI总结：
本文改进张量交叉插值算法，用部分秩揭示LU分解提升稳定性，并发布两个开源库，应用于积分、函数表示、PDE求解、FFT、配分函数和MPO构造。
摘要：
张量交叉插值（TCI）算法是一种秩揭示算法，用于将低秩、高维张量分解为张量列/矩阵乘积态（MPS）。TCI从极小的训练数据集中学习整个对象的紧凑MPS表示。一旦获得，现有的大型MPS工具箱可提供指数级加速的算法来执行大量操作。我们讨论了TCI的若干改进和变体。特别地，我们展示了用部分秩揭示LU分解替代交叉插值，可得到比原始算法更稳定、更灵活的算法。我们还介绍了两个开源库，即Python/C++中的xfac和Julia中的TensorCrossInterpolation.jl，它们实现了这些改进算法，并在多个应用中进行了演示。这些应用包括无符号问题的大维积分、函数的高分辨率量子化表示、偏微分方程的求解、超快速傅里叶变换、配分函数的计算以及矩阵乘积算子的构造。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
