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We develop necessary and sufficient conditions and a novel provablyconsistent and efficient algorithm for discovering topics (latent factors) fromobservations (documents) that are realized from a probabilistic mixture ofshared latent factors that have certain properties. Our focus is on the classof topic models in which each shared latent factor contains a novel word thatis unique to that factor, a property that has come to be known as separability.Our algorithm is based on the key insight that the novel words correspond tothe extreme points of the convex hull formed by the row-vectors of a suitablynormalized word co-occurrence matrix. We leverage this geometric insight toestablish polynomial computation and sample complexity bounds based on a fewisotropic random projections of the rows of the normalized word co-occurrencematrix. Our proposed random-projections-based algorithm is naturally amenableto an efficient distributed implementation and is attractive for modernweb-scale distributed data mining applications.

Necessary and Sufficient Conditions and a Provably Efficient Algorithm for Separable Topic Discovery