Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence

TL;DR

This paper introduces a primal-dual algorithm based on Chambolle-Pock for non-negative matrix factorization with Kullback-Leibler divergence, improving convergence speed and solution quality over traditional methods.

Contribution

It presents the first primal-dual approach for NMF with KL divergence, enabling closed-form computations and better convergence properties.

Findings

Faster convergence than existing algorithms on synthetic and real datasets

Achieves better local optima in face recognition and music source separation

Extends naturally to NMF via alternating optimization

Abstract

Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrices. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses such as the least-squares loss. In this article, we propose a first-order primal-dual algorithm for non-negative decomposition problems (where one factor is fixed) with the KL divergence, based on the Chambolle-Pock algorithm. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic examples, face recognition or music source separation datasets, it is either faster than existing algorithms, or leads to improved local optima, or both.