Wedderburn rank reduction and Krylov subspace method for tensor approximation. Part 1: Tucker case

We propose algorithms for Tucker approximation of 3-tensor, that use it only through tensor-by-vector-by-vector multiplication subroutine. In matrix case, Krylov methods are methods of choice to approximate dominant column and row subspace of sparse or structured matrix given through matrix-by-vector subroutine. However, direct generalization to tensor case, proposed recently by Eldén and Savas, namely minimal Krylov recursion, does not have any convergence theory in background and in certain cases fails to compute an approximation with desired accuracy. Using Wedderburn rank reduction formula, we propose algorithm of matrix approximation that computes Krylov subspaces and allows generalization to tensor case. Several variants of proposed tensor algorithms differ by pivoting strategies, overall cost and quality of approximation. By convincing numerical experiments we show that proposed methods are faster and more accurate than minimal Krylov recursion.