Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for representing a matrix from a set of selected rows and columns-is an efficient method for constructing a tensor train decomposition of a tensor from few of its entries. While tensor train cross approximation has achieved remarkable performance in practical applications, its theoretical analysis, in particular regarding the error of the approximation, is so far lacking. To our knowledge, existing results only provide element-wise approximation accuracy guarantees, which lead to a very loose bound when extended to the entire tensor. In this paper, we bridge this gap by providing accuracy guarantees in terms of the entire tensor for both exact and noisy measurements. Our results illustrate how the choice of selected subtensors affects the quality of the cross approximation and that the approximation error caused by model error and/or measurement error may not grow exponentially with the order of the tensor. These results are verified by numerical experiments, and may have important implications for the usefulness of cross approximations for high-order tensors, such as those encountered in the description of quantum many-body states.

翻译：张量列分解因其对高维张量的简洁表示而克服了维度灾难，在机器学习和量子物理中得到广泛应用。交叉近似——最初用于从选定的行和列集合重构矩阵——是一种通过少量张量元素高效构建张量列分解的方法。尽管张量列交叉近似在实际应用中取得了显著成效，但其理论分析，尤其是近似误差方面的研究尚不充分。据我们所知，现有结果仅提供元素级的逼近精度保证，当扩展到整个张量时会产生非常宽松的误差界。本文通过提供针对精确测量和有噪声测量的整体张量精度保证，填补了这一空白。我们的结果揭示了所选子张量如何影响交叉近似的质量，并表明由模型误差和/或测量误差引起的近似误差不会随张量阶数呈指数增长。这些结果通过数值实验得到验证，并对高维张量（如描述量子多体态时遇到的张量）交叉近似的实用性具有重要意义。