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，推导了Tucker格式下张量的CUR型分解。这一新型采样技术使我们能够构造一种计算保结构分解的高效算法，显著降低了计算成本。针对大规模数据集，我们将随机采样技术与L-DEIM过程结合，以进一步提高效率。此外，我们提出了用于计算混合分解的随机算法，该算法能够产生可解释的分解，并且比张量CUR分解具有更小的近似误差。我们分析了所提算法相关的概率误差，并通过数值实验结果验证了方法的有效性。