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