  We develop necessary and sufficient conditions and a novel provably
consistent and efficient algorithm for discovering topics (latent factors) from
observations (documents) that are realized from a probabilistic mixture of
shared latent factors that have certain properties. Our focus is on the class
of topic models in which each shared latent factor contains a novel word that
is 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 to
the extreme points of the convex hull formed by the row-vectors of a suitably
normalized word co-occurrence matrix. We leverage this geometric insight to
establish polynomial computation and sample complexity bounds based on a few
isotropic random projections of the rows of the normalized word co-occurrence
matrix. Our proposed random-projections-based algorithm is naturally amenable
to an efficient distributed implementation and is attractive for modern
web-scale distributed data mining applications.
