Exact multiplicative updates for convolutional $β$-NMF in 2D

In this paper we extend the convolutional NMF with the beta-divergence as cost function to two dimensions and derive exact multiplicative updates for its factors. Our updates correct and generalize the nonnegative matrix factor deconvolution, as proposed by Schmidt and Mφrup. We prove that the cost is non-increasing under the new updates for beta between 0 and 2. By numerical simulation we confirm that both the cost’s mean and standard deviation are monotonically decreasing in a consistent manner across the most common values for beta.