Stochastic Matrix Factorization

This paper considers a restriction to nonnegative matrix factorization in which at least one matrix factor is stochastic. That is, the elements of the matrix factors are nonnegative and the columns of one matrix factor sum to 1. This restriction includes topic models, a popular method for analyzing unstructured data. It also includes a method for storing and finding pictures. The paper presents necessary and sufficient conditions on the observed data such that the factorization is unique. In addition, the paper characterizes natural bounds on the parameters for any observed data and presents a consistent least squares estimator. The results are illustrated using a topic model analysis of PhD abstracts in economics and the problem of storing and retrieving a set of pictures of faces. The views expressed in this article are those of the author and do not necessarily reflect those of the Federal Trade Commission. I'm grateful for continued discussions about this problem with Devesh Raval and Nathan Wilson.