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