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Homotopy techniques for tensor decomposition and perfect identifiability

  • Jonathan D. Hauenstein EMAIL logo , Luke Oeding , Giorgio Ottaviani ORCID logo and Andrew J. Sommese
Published/Copyright: December 22, 2016

Abstract

Let T be a general complex tensor of format (n1,,nd). When the fraction ini/[1+i(ni-1)] is an integer, and a natural inequality (called balancedness) is satisfied, it is expected that T has finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions of T, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats (3,4,5) and (2,2,2,3) which have a unique decomposition as the sum of six and four decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the only identifiable cases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.

Award Identifier / Grant number: ACI-1460032

Award Identifier / Grant number: ACI-1440607

Award Identifier / Grant number: DMS-1262428

Award Identifier / Grant number: Young Faculty Award D14AP00052

Award Identifier / Grant number: Fellowship BR2014-110 TR14

Funding statement: The first three authors thank the Simons Institute for the Theory of Computing in Berkeley, CA for their generous support while in residence during the program on Algorithms and Complexity in Algebraic Geometry. Jonathan D. Hauenstein was also supported by DARPA YFA, NSF DMS-1262428, NSF ACI-1460032, and a Sloan Research Fellowship. Giorgio Ottaviani is member of GNSAGA. Andrew J. Sommese was partially supported by NSF ACI-1440607.

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Received: 2015-02-01
Revised: 2016-07-30
Published Online: 2016-12-22
Published in Print: 2019-08-01

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