Let T be a general complex tensor of format (n_1,...,n_d). 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 6 and 4 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.
Homotopy techniques for tensor decomposition and perfect identifiability / Hauenstein J.D.; Oeding L.; Ottaviani G.; Sommese A.J.. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - STAMPA. - 2019:(2019), pp. 1-22. [10.1515/crelle-2016-0067]
Homotopy techniques for tensor decomposition and perfect identifiability
Ottaviani G.;
2019
Abstract
Let T be a general complex tensor of format (n_1,...,n_d). 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 6 and 4 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.File | Dimensione | Formato | |
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