Title:
Orthogonal and unitary tensor decomposition
Abstract:
Unlike matrices, which always have a singular value decomposition, higher-order tensors typically do not admit a decomposition in which the terms are pairwise orthogonal. The ones which do admit such a decomposition are called orthogonally decomposable (odeco) tensors.
In general, tensor decomposition is NP-hard, but the decomposition of odeco tensors can be found efficiently. Because of their efficient decomposition, odeco tensors have been used in machine learning, in particular for learning latent variables in statistical models, hence testing whether a tensor is odeco is rather useful.
Odeco tensors form a semi-algebraic set, so a priori a finite union of subsets described by polynomial equations and (weak or strict) polynomial inequalities. However, in this talk we will see that in fact, only *equations* are needed, and in fact only equations of degree at most 4.
A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras.