Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time

Abstract. We consider Tucker-like approximations with an r × r × r core tensor for threedimensional n×n×n arrays in the case of r ¿ n and possibly very large n (up to 104−106). As the approximation contains only O(rn + r3) parameters, it is natural to ask if it can be computed using only a small amount of entries of the given array. A similar question for matrices (two-dimensional tensors) was asked and positively answered in [14]. In the present paper we extend the positive answer to the case of three-dimensional tensors. More specifically, it is shown that if the tensor admits a good Tucker approximation for some (small) rank r, then this approximation can be computed using only O(nr) entries. Moreover, in many cases it can be computed with O(nr3) complexity.