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From computation to comparison of tensor decompositions. (English) Zbl 1480.15031

The authors consider the following problem: given two tensors \(\mathcal A\) and \(\mathcal B\) of the same type, determine when the rank-1 tensors appearing in a decomposition of \(\mathcal A\) also determine (after rescaling) a decomposition of \(\mathcal B\). The authors move from the observation that, in the previous situation, for any subset \(S\) of the set of indices the matrix representation of the \(S\)-slices \(A_S\) and \(B_S\) of \(\mathcal A\) and \(\mathcal B\) respectively satisfy the condition on the spaces of columns \(\mathrm{Col}(B_S)\subset \mathrm{Col}(A_S)\), provided that the matrices \(A_S\) have full rank. They determine auxiliary technical conditions under which the converse holds, namely the inclusions \(\mathrm{Col}(B_S)\subset \mathrm{Col}(A_S)\) all together imply that \(\mathcal B\) has a decomposition with the same terms of a decomposition of \(\mathcal A\).
The authors extend their conditions to multilinear rank-(\(L_1,L_2,\dots\) ) decompositions of \(\mathcal A\) and \(\mathcal B\). They also show that their result holds even if the inclusions \(\mathrm{Col}(B_S)\subset \mathrm{Col}(A_S)\) are limited to some collections of subsets of the set of indices.
In the last part of the paper, the authors show applications of their result to problems arising from data analysis and pattern recognition.

MSC:

15A69 Multilinear algebra, tensor calculus
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References:

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