A Correlation Thresholding Algorithm for Learning Factor Analysis Models
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Factor analysis is a widely used method for modeling a set of observed variables by a set of (typically fewer) unobserved latent factors. The traditional method of Exploratory Factor Analysis (EFA), despite its widespread application, is often criticized for three well-known weaknesses: (1) requiring the number of factors to be known, (2) ad-hoc nature in learning the support of coefficients, and (3) nonidentifiability of the parameters due to the rotation invariance properties of the likelihood. While more recently developed penalized EFA methods partially address these issues, they remain computationally intense, and still require the number of latent factors to be given. To address these issues, we propose a fast algorithm that simultaneously learns the number of latent factors and a model structure that leads to identifiable parameters. Our novel approach is motivated from the perspective of graph theory, making use of thresholded correlation graphs from the observed data to learn the latent factor structure. We establish theoretical guarantees for our method with results for solution uniqueness and high-dimensional consistency of the structure learning as well as the parameter estimation. We also present a series of simulation studies and a real data example to test and demonstrate its performance. Our results show that the correlation thresholding algorithm is an accurate method for learning the structure of factor analysis models and is robust to violations of our assumptions. Moreover, we show that our algorithm scales well up to 1500 variables, while the competing methods are not applicable in a reasonable amount of running time.
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