  Despite the widespread application of latent factor analysis, existing
methods suffer from the following weaknesses: requiring the number of factors
to be known, lack of theoretical guarantees for learning the model structure,
and nonidentifiability of the parameters due to rotation invariance properties
of the likelihood. We address these concerns by proposing a fast correlation
thresholding (CT) algorithm that simultaneously learns the number of latent
factors and a rotationally identifiable model structure. Our novel approach
translates this structure learning problem into the search for so-called
independent maximal cliques in a thresholded correlation graph that can be
easily constructed from the observed data. Our clique analysis technique scales
well up to thousands of variables, while competing methods are not applicable
in a reasonable amount of running time. We establish a finite-sample error
bound and high-dimensional consistency for the structure learning of our
method. Through a series of simulation studies and a real data example, we show
that the CT algorithm is an accurate method for learning the structure of
factor analysis models and is robust to violations of its assumptions.
