Multiplicative updates for convolutional NMF under β-divergence

We generalize the convolutional NMF by taking the β -divergence as the contrast function and present the exact multiplicative updates for its factors in closed form. The new updates unify the β -NMF and the convolutional NMF. We state why almost all existing updates are inexact and/or approximative w.r.t. the convolutional data model. In addition, we prove that the β -divergence is nonincreasing under our updates and confirm numerically that the updates are stable and that the convergence of the contrast function is consistent across the most common values of β .