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Homotopy techniques for tensor decomposition and perfect identifiability. (English) Zbl 1440.15019

An element of the complex vector space \(\mathbb{C}^{n_1} \otimes \dots \otimes \mathbb{C}^{n_d}\) is a tensor of format \((n_1,\dots,n_d)\). The rank of such a tensor \[ T \in \mathbb{C}^{n_1}\otimes \dots \otimes \mathbb{C}^{n_d} \] is defined to be the smallest integer \(r\) such that \[ T = \sum_{i = 1}^r v_1^i \otimes \dots \otimes v_d^i, \] where \(v_j^i \in \mathbb{C}^{n_j}\). Such an expression is called a decomposition of \(T\) into a sum of decomposable summands.
A tensor of format \((n_1,\dots,n_d)\) is called perfect if the quantity \[ R(n_1,\dots,n_d) := \frac{ \prod_{i=1}^d n_i }{ 1 - d + \sum_{i=1}^d n_i } \] is an integer. The condition that a general tensor of format \((n_1,\dots,n_d)\) is perfect is necessary for it to admit finitely many decompositions into a sum of decomposable summands.
Within this context, the authors study the problem of uniqueness for decompositions of perfect tensor formats. Their main results pertain to general tensors of formats \((3,4,5)\) and \((2,2,2,3)\).
Specifically, for the case of general tensors of format \((3,4,5)\), the authors prove that each such general tensor admits a unique decomposition into a sum of six decomposable summands. For the case of general tensors of format \((2,2,2,3)\), they prove that such tensors admit a unique decomposition into a sum of four decomposable summands.
An interesting aspect of the proof of these results is the technique of Koszul flattenings developed in [J. M. Landsberg and G. Ottaviani, Ann. Mat. Pura Appl. (4) 192, No. 4, 569–606 (2013; Zbl 1274.14058)]. Another aspect is the use of monodromy loops (see [A. J. Sommese et al., NATO Sci. Ser. II, Math. Phys. Chem. 36, 297–315 (2001; Zbl 0990.65051)]).
Finally, the authors formulate a conjecture about the perfect format of general tensors admitting a unique decomposition into decomposable summands.

MSC:

15A69 Multilinear algebra, tensor calculus
14Q15 Computational aspects of higher-dimensional varieties
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References:

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