  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.
