Symmetric Orthogonal Tensor Decomposition is Trivial

TL;DR

This paper shows that certain symmetric tensor decomposition problems can be simplified to orthogonal problems using whitening, and provides a new eigenproblem-based method for orthogonal tensor decomposition with demonstrated effectiveness.

Contribution

The paper introduces a novel eigenproblem-based method for orthogonal symmetric tensor decomposition, simplifying the process when such a decomposition exists.

Findings

Method reduces tensor decomposition to a matrix eigenproblem

Numerical results demonstrate the method's effectiveness

Orthogonal tensor decomposition can be achieved efficiently

Abstract

We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued, pairwise orthogonal vectors. Such decompositions do not generally exist, but we show that some symmetric tensor decomposition problems can be converted to orthogonal problems following the whitening procedure proposed by Anandkumar et al. (2012). If an orthogonal decomposition of an $m$-way $n$-dimensional symmetric tensor exists, we propose a novel method to compute it that reduces to an $n \times n$ symmetric matrix eigenproblem. We provide numerical results demonstrating the effectiveness of the method.