Orthogonal and unitary tensor decomposition from an algebraic perspective

Boralevi, Ada; Draisma, Jan; Horobeţ, Emil; Robeva, Elina (2017). Orthogonal and unitary tensor decomposition from an algebraic perspective. Israel journal of mathematics, 222(1), pp. 223-260. Springer 10.1007/s11856-017-1588-6

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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.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Draisma, Jan

Subjects:

500 Science > 510 Mathematics

ISSN:

0021-2172

Publisher:

Springer

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

17 Apr 2018 10:16

Last Modified:

17 Apr 2018 10:16

Publisher DOI:

10.1007/s11856-017-1588-6

BORIS DOI:

10.7892/boris.109143

URI:

https://boris.unibe.ch/id/eprint/109143

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