Affiliation:
1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA
2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA
3. Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Firenze, Italy
Abstract
AbstractLetTbe a general complex tensor of format{(n_{1},\dots,n_{d})}. When the fraction{\prod_{i}n_{i}/[1+\sum_{i}(n_{i}-1)]}is an integer, and a natural inequality (called balancedness) is satisfied, it is expected thatThas finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions ofT, 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 onlyidentifiablecases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.
Funder
National Science Foundation
Defense Advanced Research Projects Agency
Alfred P. Sloan Foundation
Subject
Applied Mathematics,General Mathematics
Reference140 articles.
1. Identifiability of homogeneous polynomials and Cremona transformations;Preprint,2016
2. A proof of the set-theoretic version of the salmon conjecture;J. Algebra,2012
3. Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning;Discrete Contin. Dyn. Syst. Ser. S,2011
4. Equations for secant varieties of Veronese and other varieties;Ann. Mat. Pura Appl. (4),2013
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