A generalization of Kruskal’s theorem on tensor decomposition

Kruskal's theorem states that a sum of product tensors constitutes unique tensor rank decomposition if the so-called k-ranks are large. We prove "splitting theorem" for sets tensors, in which k-rank condition is weakened to standard notion rank, and conclusion uniqueness relaxed statement set splits (i.e. disconnected as matroid). Our splitting implies generalization theorem. While several extensions already present literature, all these use original permutation lemma, hence still cannot certify when below certain threshold. uses completely new proof technique, contains many extensions, can this obtain other useful results on decompositions consequences our sharp lower bounds Waring extend Sylvester's matrix inequality tensors. also novel non-rank decompositions.