Nearly Low Rank Tensors and Their Approximations

TL;DR

This paper introduces a new method for low rank tensor approximation that leverages linear relations, polynomial zeros, and least squares, especially effective when the original tensor is near a low rank tensor.

Contribution

It proposes a novel three-stage approach for low rank tensor approximation applicable to large-scale tensors, improving efficiency and accuracy.

Findings

Effective approximation when tensors are close to low rank

Applicable to symmetric and nonsymmetric tensors

Enhances low rank tensor decomposition methods

Abstract

The low rank tensor approximation problem (LRTAP) is to find a tensor whose rank is small and that is close to a given one. This paper studies the LRTAP when the tensor to be approximated is close to a low rank one. Both symmetric and nonsymmetric tensors are discussed. We propose a new approach for solving the LRTAP. It consists of three major stages: i) Find a set of linear relations that are approximately satisfied by the tensor; such linear relations can be expressed by polynomials and can be found by solving linear least squares. ii) Compute a set of points that are approximately common zeros of the obtained polynomials; they can be found by computing Schur decompositions. iii) Construct a low rank approximating tensor from the obtained points; this can be done by solving linear least squares. Our main conclusion is that if the given tensor is sufficiently close to a low rank one, then the computed tensor is a good enough low rank approximation. This approach can also be applied to efficiently compute low rank tensor decompositions, especially for large scale tensors.