A Comparative Study of Gamma Markov Chains for Temporal Non-Negative Matrix Factorization

TL;DR

This paper reviews four Gamma Markov chains used in temporal non-negative matrix factorization, highlights their limitations, introduces a new model BGAR(1) with a stationary distribution, and compares these models on a prediction task.

Contribution

It introduces the BGAR(1) process as a novel Gamma Markov chain with a well-defined stationary distribution for temporal NMF.

Findings

All four reviewed models lack a stationary distribution.

BGAR(1) overcomes the stationary distribution limitation.

Model comparison shows differences in prediction performance.

Abstract

Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.