Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

Abstract

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined by equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity. We give partial results using in particular a connection with approximate simultaneous diagonalization (the so-called ASD property).