A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized Nonnegative Matrix Factorization

TL;DR

This paper provides a unified convergence analysis for the multiplicative update algorithm in regularized nonnegative matrix factorization, covering various divergences and regularizers, and proves convergence to stationary points.

Contribution

It offers a simple, unified proof of convergence for MU in NMF with diverse divergences and regularizers, extending theoretical guarantees.

Findings

Convergence of MU to stationary points in NMF.

Applicable to a wide range of divergences and regularizers.

Potential to influence analysis in machine learning and signal processing.

Abstract

The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the non-convex NMF optimization problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing.