Orthogonal and unitary tensor decomposition from an algebraic perspective

Ada Boralevi, Jan Draisma, Emil Horobeţ, Elina Robeva

Research output: Contribution to journalArticleAcademicpeer-review

6 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)223-260
Number of pages38
JournalIsrael Journal of Mathematics
Volume222
Issue number1
DOIs
Publication statusPublished - 1 Oct 2017

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