On computing the minimum singular value of a tensor sum

Recently, the Lanczos bidiagonalization method over tensor space has been proposed for computing the maximum and minimum singular values of a tensor sum T. The method over tensor space is practical in memory and has a simple implementation due to recent developments in tensor computations; however, there is still room for improvement in the convergence to the minimum singular value. This study reconstructed an invert Lanczos bidiagonalization method from vector space to tensor space. The resulting algorithm requires solving linear systems at each iteration step. Using standard direct methods, such as the LU decomposition for solving the linear systems requires a huge memory of O(n(6)). Therefore, this paper proposes a tensor-structure-preserving direct methods of T whose memory requirements are of O(n(3)), which is equivalent to the order of iterative methods. Numerical examples indicate that the number of iterations tends to be much smaller than that of the conventional method.