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.

翻译：在张量空间框架下，我们考虑利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。具体而言，我们通过实验研究了六种正交化方法的正交性损失，即经典和修正的Gram-Schmidt分别带有重正交化（CGS2, MGS2）与不带重正交化（CGS, MGS）的变体、Gram方法以及Householder变换。为克服维度灾难，我们采用张量列（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通过TT舍入方法按指定精度执行的重压缩步骤。在描述各算法的结构与性质后，通过数值实验展示其正交性损失特性。经典矩阵计算舍入分析经过数十年发展得出的理论界似乎得以保持，仅将单位舍入误差替换为TT舍入精度。本研究最终以存储需求及计算复杂度（基于TT舍入次数度量，该操作为最昂贵计算步骤）为指标，完成了对各正交化核的计算分析。