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 language | English |
|---|---|
| Pages (from-to) | 223-260 |
| Number of pages | 38 |
| Journal | Israel Journal of Mathematics |
| Volume | 222 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 1 Oct 2017 |
Funding
Acknowledgments. AB and JD were supported by a Vidi grant from the Netherlands Organisation for Scientific Research (NWO). EH was supported by an NWO free competition grant. AB was partially supported by MIUR funds, PRIN 20102011 project Geometria delle varietà algebriche and Univer-sità degli Studi di TriesteFRA 2011 project Geometria e topologia delle varietà. AB is a member of INdAM-GNSAGA. ER was supported by the UC Berkeley Mathematics department and by an NSF Mathematical Sciences Postdoctoral Research Fellowship (DMS-1703821). We thank Nick Vannieuwenhoven for several remarks on a previous draft. Finally, we thank the organizers of the Fall 2014 workshop “Tensors in Computer Science and Geometry” at the Simons Institute for the Theory of Computing, where this project started in embryo.