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

Preview

Text 10.1007_s11856-017-1588-6.pdf - Published Version Available under License Publisher holds Copyright. Download (382kB) | Preview

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.

Interest & Impact

Downloads

73 since deposited on 17 Apr 2018
42 in the past 12 months

Citations

5 Citations in Web of Science ®
6 Citations in Scopus

Search

in Google Scholar™

Services

Actions (login required)

Edit item

Actions (login required)

Edit item