Multiplicative Updates for Elastic Net Regularized Convolutional NMF Under $\beta$-Divergence

ONNEGATIVE matrix factorization finds its application in the fields of machine learning and in connection with inverse problems, mostly. It became immensely popular after Lee and Seung derived multiplicative update rules that made the up until then additive steps in the direction of the negative gradient obsolete [1]. In [2], they gave empirical evidence of their convergence to a stationary point, using (a) the squared Euclidean distance, and, (b) the generalized Kullback-Leibler divergence as the contrast function. The factorization's origins can be traced back to [3], [4]. To better deal with noisy data, the notion of a basis is (most commonly) abandoned in favor of an overcomplete frame or dictionary, and sparsity becomes a desired property of the coefficient matrix.