This paper derives the CUR-type factorization for tensors in the Tucker format based on a new variant of the discrete empirical interpolation method known as L-DEIM. This novel sampling technique allows us to construct an efficient algorithm for computing the structure-preserving decomposition, which significantly reduces the computational cost. For large-scale datasets, we incorporate the random sampling technique with the L-DEIM procedure to further improve efficiency. Moreover, we propose randomized algorithms for computing a hybrid decomposition, which yield interpretable factorization and provide a smaller approximation error than the tensor CUR factorization. We provide comprehensive analysis of probabilistic errors associated with our proposed algorithms, and present numerical results that demonstrate the effectiveness of our methods.

翻译：本文基于离散经验插值方法的新变体L-DEIM，导出了Tucker格式张量的CUR型分解。这种新颖的采样技术使我们能够构建一个有效的算法来计算结构保持分解，从而显著降低了计算成本。对于大规模数据集，我们将随机采样技术与L-DEIM过程结合起来，以进一步提高效率。此外，我们提出了计算混合分解的随机算法，这些算法可以产生可解释的分解，并提供比张量CUR分解更小的近似误差。我们提供了与我们提出的算法相关的概率误差的全面分析，并呈现了证明我们方法有效性的数值结果。