Learning Topic Models: Identifiability and Finite-Sample Analysis

Topic models provide a useful text-mining tool for learning, extracting, and discovering latent structures in large text corpora. Although a plethora of methods have been proposed for topic modeling, lacking in the literature is a formal theoretical investigation of the statistical identifiability and accuracy of latent topic estimation. In this article, we propose a maximum likelihood estimator (MLE) of latent topics based on a specific integrated likelihood that is naturally connected to the concept, in computational geometry, of <i>volume minimization</i>. Our theory introduces a new set of geometric conditions for topic model identifiability, conditions that are weaker than conventional separability conditions, which typically rely on the existence of pure topic documents or of anchor words. Weaker conditions allow a wider and thus potentially more fruitful investigation. We conduct finite-sample error analysis for the proposed estimator and discuss connections between our results and those of previous investigations. We conclude with empirical studies employing both simulated and real datasets. Supplementary materials for this article are available online.