Orthogonal Decomposition of Symmetric Tensors

TL;DR

This paper investigates the properties of orthogonally decomposable symmetric tensors, providing formulas for eigenvectors and proposing polynomial equations that characterize the odeco tensor variety, with partial proofs and strong evidence.

Contribution

It introduces a formula for eigenvectors of odeco tensors and formulates polynomial equations conjectured to define the odeco variety, advancing understanding of tensor decomposition.

Findings

Derived a formula for all eigenvectors of odeco tensors

Formulated polynomial equations vanishing on the odeco variety

Provided partial proofs and evidence supporting the conjecture

Abstract

A real symmetric tensor is orthogonally decomposable (or odeco) if it can be written as a linear combination of symmetric powers of $n$ vectors which form an orthonormal basis of $\mathbb R^n$. Motivated by the spectral theorem for real symmetric matrices, we study the properties of odeco tensors. We give a formula for all of the eigenvectors of an odeco tensor. Moreover, we formulate a set of polynomial equations that vanish on the odeco variety and we conjecture that these polynomials generate its prime ideal. We prove this conjecture in some cases and give strong evidence for its overall correctness.