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
Elden 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.