关于张量分解的Kruskal定理的一个推广
A generalization of Kruskal's theorem on tensor decomposition
AI总结：
该研究提出乘积张量集合的分裂定理，将Kruskal张量分解定理的k秩条件弱化为标准秩、唯一性结论放宽为集合分裂，采用全新证明技术推广了原定理，还得到张量秩下界等相关结果。
摘要：
Kruskal定理指出，若乘积张量的所谓k秩足够大，则乘积张量的和构成唯一的张量秩分解。我们证明了针对乘积张量集合的“分裂定理”：该定理将Kruskal定理中的k秩条件弱化为标准的秩概念，同时将唯一性结论放宽为乘积张量集合发生分裂（即作为拟阵是不连通的）。我们的分裂定理可推导出Kruskal定理的一个推广形式。尽管现有文献中已有若干Kruskal定理的扩展结果，但这些结果均使用Kruskal最初的置换引理，因此当k秩低于某一阈值时，仍无法验证分解的唯一性。我们的推广采用了全新的证明技术，涵盖了其中许多扩展结果，且能在该阈值以下验证唯一性。作为分裂定理的推论，我们还得到了若干其他关于张量分解的有用结果：我们证明了张量秩和Waring秩的精确下界，将Sylvester矩阵秩不等式推广到张量情形；我们还证明了非秩张量分解的全新唯一性结果。
英文摘要：
Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a "splitting theorem" for sets of product tensors, in which the k-rank condition of Kruskal's theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e. is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal's theorem. While several extensions of Kruskal's theorem are already present in the literature, all of these use Kruskal's original permutation lemma, and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions, and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester's matrix rank inequality to tensors. We also prove novel uniqueness results for non-rank tensor decompositions.
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