TY - JOUR

T1 - Orthogonal and unitary tensor decomposition from an algebraic perspective

AU - Boralevi, Ada

AU - Draisma, Jan

AU - Horobeţ, Emil

AU - Robeva, Elina

PY - 2017/10/1

Y1 - 2017/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85028694012&partnerID=8YFLogxK

U2 - 10.1007/s11856-017-1588-6

DO - 10.1007/s11856-017-1588-6

M3 - Article

AN - SCOPUS:85028694012

VL - 222

SP - 223

EP - 260

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -