Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

For a general third-order tensor $\mathcal{A}\in\mathbb{R}^{n\times n\times n}$ 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 $\mathcal{A}$. The algorithm works on $2\times2\times2$ 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.