SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 449-474, January 2021.
Decompositions of higher-order tensors into sums of simple terms are ubiquitous. We show that in order to verify that two tensors are generated by the same (possibly scaled) terms it is not necessary to compute the individual decompositions. In general the explicit computation of such a decomposition may have high complexity and can be ill-conditioned. We now show that under some assumptions the verification can be reduced to a comparison of both the column and row spaces of the corresponding matrix representations of the tensors. We consider rank-1 terms as well as low multilinear rank terms (also known as block terms) and show that the number of the terms and their multilinear rank can be inferred as well. The comparison relies only on numerical linear algebra and can be done in a numerically reliable way. We also illustrate how our results can be applied to solve a multilabel classification problem that appears in the context of blind source separation.

中文翻译：

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从计算到张量分解的比较

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 449-474 页，2021 年 1 月。
将高阶张量分解为简单项的和是普遍存在的。我们表明，为了验证两个张量是由相同的（可能缩放的）项生成的，没有必要计算单独的分解。通常，这种分解的显式计算可能具有很高的复杂性并且可能是病态的。我们现在表明，在某些假设下，验证可以简化为张量的相应矩阵表示的列和行空间的比较。我们考虑秩 1 项以及低多重线性秩项（也称为块项），并表明也可以推断出项的数量及其多重线性秩。比较仅依赖于数值线性代数，并且可以以数值上可靠的方式进行。