Orthogonal eigenvectors and singular vectors of tensors

TL;DR

This paper investigates tensors with orthogonal eigenvectors and singular vectors, proving generic uniqueness results and introducing a new tensor decomposition that generalizes existing methods.

Contribution

It establishes the generic uniqueness of orthogonal eigenvector and singular vector bases for symmetric and certain non-symmetric tensors, resolving a conjecture and proposing a new tensor decomposition.

Findings

Generic symmetric tensors have unique orthogonal eigenvector bases.

Most tensors with orthogonal singular vectors also have unique bases, except for order four binary tensors.

The new tensor decomposition generalizes orthogonally decomposable and Tucker decompositions.

Abstract

The spectral theorem says that a real symmetric matrix has an orthogonal basis of eigenvectors and that, for a matrix with distinct eigenvalues, the basis is unique (up to signs). In this paper, we study the symmetric tensors with an orthogonal basis of eigenvectors and show that, for a generic such tensor, the orthogonal basis is unique. This resolves a conjecture by Mesters and Zwiernik. We also study the non-symmetric setting. The singular value decomposition says that a real matrix has an orthogonal basis of singular vector pairs and that, for a matrix with distinct singular values, the basis is unique (up to signs). We describe the tensors with an orthogonal basis of singular vectors and show that a generic such tensor has a unique orthogonal basis, with one exceptional format: order four binary tensors. We use these results to propose a new tensor decomposition that generalizes an orthogonally decomposable decomposition and specializes the Tucker decomposition.