On provable exact low-rank recovery in topic models

In the past few years, probabilistic topic models have been developed and applied to problems in text document classification and computer vision. Such models provide a probabilistic framework for characterizing a corpus of documents (or images) in the bag-of-words representation. Key feature of such models is that a low dimensional representation is facilitated through latent topic variables. Most inference algorithms in topic models assume a fixed number of topics and determine the number of topics empirically. In this paper, we consider the problem of identifying the number of topics in topic models. We present a rank minimization framework and provide sufficient conditions, which guarantee exact recovery of the number of topics. Moreover, we propose a heuristic convex relaxation to the rank minimization. Using simulations, we show that the proposed convex relaxation provides exact rank recovery under the sufficient conditions proposed for the rank minimization problem.