Uniqueness of Tensor Decompositions with Applications to Polynomial Identifiability
This work addresses the identifiability challenge in machine learning models like Hidden Markov and topic models, enabling more efficient learning algorithms by reducing reliance on non-degeneracy assumptions.
The paper tackles the problem of ensuring uniqueness in tensor decompositions by providing a robust version of Kruskal's theorem, which allows for approximate recovery of decompositions with small errors and implies polynomial sample identifiability for latent variable models under milder conditions.
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.