Multiplicative Updates for Convolutional NMF Under $\beta$-Divergence

TL;DR

This paper introduces a generalized convolutional NMF framework using the $eta$-divergence, providing exact multiplicative updates that unify existing models and improve stability and convergence.

Contribution

It derives the correct closed-form multiplicative updates for convolutional NMF under $eta$-divergence, correcting previous inexact methods and unifying different NMF variants.

Findings

Updates are stable across various $eta$ values.

Convergence performance is consistent.

Existing updates are often inexact and approximate.

Abstract

In this letter, we generalize the convolutional NMF by taking the $\beta$-divergence as the contrast function and present the correct multiplicative updates for its factors in closed form. The new updates unify the $\beta$-NMF and the convolutional NMF. We state why almost all of the existing updates are inexact and approximative w.r.t. the convolutional data model. We show that our updates are stable and that their convergence performance is consistent across the most common values of $\beta$.