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Despite the widespread application of latent factor analysis, existingmethods suffer from the following weaknesses: requiring the number of factorsto be known, lack of theoretical guarantees for learning the model structure,and nonidentifiability of the parameters due to rotation invariance propertiesof the likelihood. We address these concerns by proposing a fast correlationthresholding (CT) algorithm that simultaneously learns the number of latentfactors and a rotationally identifiable model structure. Our novel approachtranslates this structure learning problem into the search for so-calledindependent maximal cliques in a thresholded correlation graph that can beeasily constructed from the observed data. Our clique analysis technique scaleswell up to thousands of variables, while competing methods are not applicablein a reasonable amount of running time. We establish a finite-sample errorbound and high-dimensional consistency for the structure learning of ourmethod. Through a series of simulation studies and a real data example, we showthat the CT algorithm is an accurate method for learning the structure offactor analysis models and is robust to violations of its assumptions.

Structure Learning of Latent Factors via Clique Search on Correlation Thresholded Graphs