关于张量列格式中的若干正交化方案
On some orthogonalization schemes in Tensor Train format
AI总结：
本文在张量列格式下实验比较六种正交化方法的正交性损失，并引入TT-rounding再压缩步骤，验证经典舍入误差界在低秩张量场景下仍以TT-rounding精度形式成立。
摘要：
在张量空间框架下，我们考虑使用正交化核从一组线性无关张量生成张量子空间的正交基。特别地，我们通过实验研究了六种正交化方法的正交性损失，即带重正交化（CGS2、MGS2）和不带重正交化（CGS、MGS）的经典Gram-Schmidt方法与修正Gram-Schmidt方法、Gram方法以及Householder变换。为克服维数灾难，我们使用张量列（TT）形式对张量进行低秩近似表示。此外，我们在标准算法流程中引入再压缩步骤，通过TT-rounding方法在指定精度下进行。在描述算法的结构和性质之后，我们通过数值实验展示了它们的正交性损失。几十年来从经典矩阵计算舍入误差分析中获得的理论界似乎得以保持，其中单位舍入被TT-rounding精度所替代。针对每个正交化核的计算分析，包括内存需求和以TT-rounding次数衡量的计算复杂度（TT-rounding恰好是计算开销最大的操作），完善了本研究。
英文摘要：
In the framework of tensor spaces, we consider orthogonalization kernels to generate an orthogonal basis of a tensor subspace from a set of linearly independent tensors. In particular, we experimentally study the loss of orthogonality of six orthogonalization methods, namely Classical and Modified Gram-Schmidt with (CGS2, MGS2) and without (CGS, MGS) re-orthogonalization, the Gram approach, and the Householder transformation. To overcome the curse of dimensionality, we represent tensors with a low-rank approximation using the Tensor Train (TT) formalism. In addition, we introduce recompression steps in the standard algorithm outline through the TT-rounding method at a prescribed accuracy. After describing the structure and properties of the algorithms, we illustrate their loss of orthogonality with numerical experiments. The theoretical bounds from the classical matrix computation round-off analysis, obtained over several decades, seem to be maintained, with the unit round-off replaced by the TT-rounding accuracy. The computational analysis for each orthogonalization kernel in terms of the memory requirements and the computational complexity measured as a function of the number of TT-rounding, which happens to be the most computationally expensive operation, completes the study.
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。
