A Krylov-Schur-like method for computing the best rank-(r1,r2,r3) approximation of large and sparse tensors

Abstract

The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here block versions are introduced. For the computation of partial eigenvalue and singular value decompositions of matrices the Krylov-Schur (restarted Arnoldi) method is used. A generalization of this method to tensors is described, for computing the best low multilinear rank approximation of large and sparse tensors. In analogy to the matrix case, the large tensor is only accessed in multiplications between the tensor and blocks of vectors, thus avoiding excessive memory usage. It is proved that if the starting approximation is good enough, then the tensor Krylov-Schur method is convergent. Numerical examples are given for synthetic tensors and sparse tensors from applications, which demonstrate that for most large problems the Krylov-Schur method converges faster and more robustly than higher order orthogonal iteration.

Spectral partitioning of large and sparse 3‐tensors using low‐rank tensor approximation

“…The 3-slices of a normalized (1,2)-symmetric tensor all have the maximum eigenvalue equal to 1. In addition, for a tensor of the structure (20) we can expect that…”

“…For problems with sparse matrices, restarted Krylov methods are standard 33,34 . In References 20,35 we developed a block‐Krylov type method for tensors, which accesses the tensor only in tensor‐matrix multiplications, where the matrix consists of a relatively small number of columns. Thus the method is memory‐efficient.…”

“…Thus the method is memory‐efficient. In Reference 20 it was combined with the Krylov‐Schur approach for the computation of the best rank‐ approximation of a large and sparse tensors. For the problems solved in Section 7 convergence is very fast.…”

“…For the problems solved in Section 7 convergence is very fast. It is shown in Reference 20 that the block Krylov–Schur method is in general considerably faster than the Higher‐Order Orthogonal Iteration 8…”

“…The present paper is the second in a series of three on the analysis of large and sparse tensor data. In the first 20 we develop a Krylov–Schur like algorithm for efficiently computing best low multilinear rank approximations of large and sparse tensors. In the third paper 21 we use the methods to analyze large tensors from data science applications.…”