Joint Majorization-Minimization for Nonnegative Matrix Factorization with the $β$-divergence
This work addresses computational efficiency for practitioners using NMF in applications like image and audio processing, though it is incremental as it builds on existing MM methods.
The authors tackled the problem of slow computation in nonnegative matrix factorization with the β-divergence by proposing a joint majorization-minimization scheme, resulting in multiplicative updates that reduced CPU time by 13% to 78% across various datasets.
This article proposes new multiplicative updates for nonnegative matrix factorization (NMF) with the $β$-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 $β$-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 $β$ 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.