For a general third-order tensor A ∈ R n×n×n the paper studies two closely related problems, the SVD-like tensor decomposition and the (approximate) tensor diagonalization. We develop the alternating least squares Jacobi-type algorithm that maximizes the squares of the diagonal entries of A. The algorithm works on 2 × 2 × 2 subtensors such that in each iteration the sum of the squares of two diagonal entries is maximized. We show how the rotation angles are calculated and prove the convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on the higher-order tensors.

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The paper introduces a hybrid approach to the CUR-type decomposition of tensors in the Tucker format. The idea of the hybrid algorithm is to write a tensor X as a product of a core tensor S, a matrix C obtained by extracting mode-k fibers of X , and matrices Uj , j = 1, . . . , k − 1, k + 1, . . . , d, chosen to minimize the approximation error. The approximation can easily be modified to preserve the fibers in more than one mode. The approximation error obtained this way is smaller than the one from the standard tensor CUR-type method. This difference increases as the tensor dimension increases. It also increases as the number of modes in which the original fibers are preserved decreases.