Joint Majorization-Minimization for Nonnegative Matrix Factorization with the $\beta$-divergence

TL;DR

This paper introduces a joint majorization-minimization approach for nonnegative matrix factorization with the $eta$-divergence, leading to faster multiplicative updates and significant computational savings across various datasets.

Contribution

It presents a novel joint MM scheme for NMF with $eta$-divergence, reducing computation time compared to classic methods.

Findings

Achieves up to 78% CPU time reduction.

Effective across diverse datasets like images, audio, and hyperspectral data.

Maintains solution quality while improving efficiency.

Abstract

This article proposes new multiplicative updates for nonnegative matrix factorization (NMF) with the $\beta$-divergence objective function. Our new updates are derived from a joint majorization-minimization (MM) scheme, in which an auxiliary function (a tight upper bound of the objective function) is built for the two factors jointly and minimized at each iteration. This is in contrast with the classic approach in which a majorizer is derived for each factor separately. Like that classic approach, our joint MM algorithm also results in multiplicative updates that are simple to implement. They however yield a significant drop of computation time (for equally good solutions), in particular for some $\beta$-divergences of important applicative interest, such as the squared Euclidean distance and the Kullback-Leibler or Itakura-Saito divergences. We report experimental results using diverse datasets: face images, an audio spectrogram, hyperspectral data and song play counts. Depending on the value of $\beta$ and on the dataset, our joint MM approach can yield CPU time reductions from about $13\%$ to $78\%$ in comparison to the classic alternating scheme.