Stochastic Matrix Factorization

TL;DR

This paper studies a restricted form of non-negative matrix factorization where one factor is stochastic, providing conditions for uniqueness, bounds on parameters, and a consistent estimator, with applications to topic modeling and image storage.

Contribution

It introduces necessary and sufficient conditions for the uniqueness of stochastic matrix factorizations and develops a consistent least squares estimator.

Findings

Conditions for unique factorization are established.

Bounds on model parameters are characterized.

A consistent estimator for the model is proposed.

Abstract

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.