Orthogonal and unitary tensor decomposition from an algebraic perspective

TL;DR

This paper provides an algebraic geometric analysis of orthogonally decomposable tensors, characterizing their structure as real-algebraic varieties and revealing connections to semisimple algebras across different tensor types.

Contribution

It introduces a novel algebro-geometric framework for understanding orthogonal tensor decompositions, including explicit polynomial characterizations and algebraic connections.

Findings

Orthogonally decomposable tensors form real-algebraic varieties defined by degree at most four polynomials.

Different tensor scenarios (ordinary, symmetric, alternating) have distinct polynomial characterizations.

A surprising link exists between orthogonally decomposable tensors and semisimple algebras.

Abstract

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors. More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras---associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting.