Matthew Stephens

Peter Carbonetto, Abhishek Sarkar, Zihao Wang et al.

In an effort to develop topic modeling methods that can be quickly applied to large data sets, we revisit the problem of maximum-likelihood estimation in topic models. It is known, at least informally, that maximum-likelihood estimation in topic models is closely related to non-negative matrix factorization (NMF). Yet, to our knowledge, this relationship has not been exploited previously to fit topic models. We show that recent advances in NMF optimization methods can be leveraged to fit topic models very efficiently, often resulting in much better fits and in less time than existing algorithms for topic models. We also formally make the connection between the NMF optimization problem and maximum-likelihood estimation for the topic model, and using this result we show that the expectation maximization (EM) algorithm for the topic model is essentially the same as the classic multiplicative updates for NMF (the only difference being that the operations are performed in a different order). Our methods are implemented in the R package fastTopics.

Lei Sun, Matthew Stephens

The Normal Means problem plays a fundamental role in many areas of modern high-dimensional statistics, both in theory and practice. And the Empirical Bayes (EB) approach to solving this problem has been shown to be highly effective, again both in theory and practice. However, almost all EB treatments of the Normal Means problem assume that the observations are independent. In practice correlations are ubiquitous in real-world applications, and these correlations can grossly distort EB estimates. Here, exploiting theory from Schwartzman (2010), we develop new EB methods for solving the Normal Means problem that take account of unknown correlations among observations. We provide practical software implementations of these methods, and illustrate them in the context of large-scale multiple testing problems and False Discovery Rate (FDR) control. In realistic numerical experiments our methods compare favorably with other commonly-used multiple testing methods.