A Krylov-Schur like method for computing the best rank-$(r_1,r_2,r_3)$ approximation of large and sparse tensors

TL;DR

This paper introduces a Krylov-Schur like method for efficiently computing the best low multilinear rank approximation of large, sparse tensors, demonstrating faster convergence than existing methods.

Contribution

It generalizes the Krylov-Schur method to tensors, enabling efficient, memory-conscious approximation of large sparse tensors with proven convergence under certain conditions.

Findings

Faster convergence than higher order orthogonal iteration

Robust performance on synthetic and application-based sparse tensors

Memory-efficient tensor computations using block Krylov methods

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. We describe a generalization of this method to tensors, 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.