Krylov-Type Methods for Tensor Computations

TL;DR

This paper introduces new Krylov-type algorithms for tensor computations, enabling efficient low-rank tensor approximations and decompositions, with proven convergence properties and demonstrated effectiveness on real and synthetic data.

Contribution

It presents novel Krylov methods for tensors, including minimal and maximal variants, with theoretical guarantees and improved approximation techniques.

Findings

Minimal Krylov recursion accurately extracts tensor subspaces.

Optimized minimal Krylov improves tensor approximation quality.

Numerical experiments confirm effectiveness on real and synthetic data.

Abstract

Several Krylov-type procedures are introduced that generalize matrix Krylov methods for tensor computations. They are denoted minimal Krylov recursion, maximal Krylov recursion, contracted tensor product Krylov recursion. It is proved that the for a given tensor with low rank, the minimal Krylov recursion extracts the correct subspaces associated to the tensor within certain number of iterations. An optimized minimal Krylov procedure is described that gives a better tensor approximation for a given multilinear rank than the standard minimal recursion. The maximal Krylov recursion naturally admits a Krylov factorization of the tensor. The tensor Krylov methods are intended for the computation of low-rank approximations of large and sparse tensors, but they are also useful for certain dense and structured tensors for computing their higher order singular value decompositions or obtaining starting points for the best low-rank computations of tensors. A set of numerical experiments, using real life and synthetic data sets, illustrate some of the properties of the tensor Krylov methods.