Stochastic Matrix Factorization
This work addresses the uniqueness issue in stochastic matrix factorization, which is incremental for applications like topic modeling and image analysis.
The paper tackles the problem of ensuring uniqueness in non-negative matrix factorization with stochastic constraints, establishing necessary and sufficient conditions for unique factorization and providing a consistent estimator, illustrated with topic modeling on economics PhD abstracts and face image retrieval.
This paper considers a restriction to non-negative matrix factorization in which at least one matrix factor is stochastic. That is, the elements of the matrix factors are non-negative 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.