Canonical Polyadic Decomposition via the generalized Schur decomposition

TL;DR

This paper introduces a simplified and more accurate method for canonical polyadic decomposition (CPD) of tensors, replacing the generalized eigenvalue decomposition with a QZ decomposition, enhancing computational efficiency.

Contribution

It presents a novel simplification of the GEVD algorithm for CPD, replacing eigenvector computation with QZ decomposition for improved accuracy.

Findings

Improved accuracy over traditional GEVD-based CPD algorithms

Simplification reduces computational complexity

Applicable to low-rank tensor decompositions

Abstract

The canonical polyadic decomposition (CPD) is a fundamental tensor decomposition which expresses a tensor as a sum of rank one tensors. In stark contrast to the matrix case, with light assumptions, the CPD of a low rank tensor is (essentially) unique. The essential uniqueness of CPD makes this decomposition a powerful tool in many applications as it allows for extraction of component information from a signal of interest. One popular algorithm for algebraic computation of a CPD is the generalized eigenvalue decomposition (GEVD) which selects a matrix subpencil of a tensor, then computes the generalized eigenvectors of the pencil. In this article, we present a simplification of GEVD which improves the accuracy of the algorithm. Surprisingly, the generalized eigenvector computation in GEVD is in fact unnecessary and can be replaced by a QZ decomposition which factors a pair of matrices as a product of unitary and upper triangular matrices. Computing a QZ decomposition is a standard first step when computing generalized eigenvectors, so our algorithm can been seen as a direct simplification of GEVD.