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The authors prove that variational inference in LDA converges to the ground truth model, in polynomial time, for two different case studies with different underlying assumptions about the structure of the data. In this analysis, the authors employ "thresholded" EM updates which estimate the per-topic word distribution based on the subset of documents where a given document dominates. The proofs, which are provided in a 35-page supplement, require assumptions about the number of words in a document that are uniquely associated with each topic, the number of topics per document, and the number documents in which a given word exclusively identifies a topic. I am not enough of a specialist to evaluate the provided proofs in detail, so I will restrict myself to relatively high level comments. Empirically speaking, variational inference can and does get stuck in local maxima.