Algorithms for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence

TL;DR

This paper studies algorithms for nonnegative matrix factorization using the Kullback-Leibler divergence, proposing new methods with convergence guarantees and evaluating their performance on count data.

Contribution

It introduces three new algorithms for KL NMF with non-increasing objective functions and provides a global convergence guarantee for one of them.

Findings

Proposed algorithms guarantee non-increasing KL divergence.

Extensive experiments compare algorithm performances.

One algorithm has a proven global convergence guarantee.

Abstract

Nonnegative matrix factorization (NMF) is a standard linear dimensionality reduction technique for nonnegative data sets. In order to measure the discrepancy between the input data and the low-rank approximation, the Kullback-Leibler (KL) divergence is one of the most widely used objective function for NMF. It corresponds to the maximum likehood estimator when the underlying statistics of the observed data sample follows a Poisson distribution, and KL NMF is particularly meaningful for count data sets, such as documents or images. In this paper, we first collect important properties of the KL objective function that are essential to study the convergence of KL NMF algorithms. Second, together with reviewing existing algorithms for solving KL NMF, we propose three new algorithms that guarantee the non-increasingness of the objective function. We also provide a global convergence guarantee for one of our proposed algorithms. Finally, we conduct extensive numerical experiments to provide a comprehensive picture of the performances of the KL NMF algorithms.