When are Overcomplete Topic Models Identifiable? Uniqueness of Tensor Tucker Decompositions with Structured Sparsity

摘要

Overcomplete latent representations have been very popular for unsupervisedfeature learning in recent years. In this paper, we specify which overcompletemodels can be identified given observable moments of a certain order. Weconsider probabilistic admixture or topic models in the overcomplete regime,where the number of latent topics can greatly exceed the size of the observedword vocabulary. While general overcomplete topic models are not identifiable,we establish generic identifiability under a constraint, referred to as topicpersistence. Our sufficient conditions for identifiability involve a novel setof "higher order" expansion conditions on the topic-word matrix or thepopulation structure of the model. This set of higher-order expansionconditions allow for overcomplete models, and require the existence of aperfect matching from latent topics to higher order observed words. Weestablish that random structured topic models are identifiable w.h.p. in theovercomplete regime. Our identifiability results allows for general(non-degenerate) distributions for modeling the topic proportions, and thus, wecan handle arbitrarily correlated topics in our framework. Our identifiabilityresults imply uniqueness of a class of tensor decompositions with structuredsparsity which is contained in the class of Tucker decompositions, but is moregeneral than the Candecomp/Parafac (CP) decomposition.