Low-rank approximation of tensors

Abstract

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank k-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of the CUR-decomposition are most suitable. For d-mode tensors (formula presented), with d > 2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r1,…, rd)-approximation, which maximizes the l2 norm of the projection of T on (formula presented), where Ui is an ri - dimensional subspace RniOne of the most common methods is the alternating maximization method (AMM). It is obtained by maximizing on one subspace Ui, while keeping all other fixed, and alternating the procedure repeatedly for i = 1,…, d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point.We also discuss variants of CUR-approximation method for tensors. The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best.(r1,…, rd)-approximation.We compare numerically different approximation methods.