We give a robust version of the celebrated result of Kruskal on the uniqueness of tensor decompositions: we prove that given a tensor whose decomposition satisfies a robust form of Kruskal's rank condition, it is possible to approximately recover the decomposition if the tensor is known up to a sufficiently small (inverse polynomial) error. Kruskal's theorem has found many applications in proving the identifiability of parameters for various latent variable models and mixture models such as Hidden Markov models, topic models etc. Our robust version immediately implies identifiability using only polynomially many samples in many of these settings. This polynomial identifiability is an essential first step towards efficient learning algorithms for these models. Recently, algorithms based on tensor decompositions have been used to estimate the parameters of various hidden variable models efficiently in special cases as long as they satisfy certain "non-degeneracy" properties. Our methods give a way to go beyond this non-degeneracy barrier, and establish polynomial identifiability of the parameters under much milder conditions. Given the importance of Kruskal's theorem in the tensor literature, we expect that this robust version will have several applications beyond the settings we explore in this work.

我们给出了Kruskal定理的强鲁棒版本以及在张量分解中的应用，证明了通过多项式样本多项式鉴别性仅仅需要较小的误差即可对分解进行近似恢复，并展示了如何在混合变量模型中使用张量分解找到有效的学习算法。

张量分解的唯一性及其在多项式可辨识性中的应用