Non-negative Matrix Factorization using weighted beta divergence and equality constraints for industrial source apportionment

In this paper, we propose two weighted Non-negative Matrix Factorization (NMF) methods using a β-divergence cost function. This divergence is used as a dissimilarity measure which can be tuned by the parameter β. The weights allow to deal with the uncertainty associated to each data sample. Our first approach consists of generalizing weighted NMF methods proposed with specific divergences or norms to the β-divergence. In our second approach, we assume that some components of the factorization are known and we use them to inform our NMF algorithm. We thus consider a specific parameterization which involves these constraints. In particular, we propose specific multiplicative update rules for the minimization of this parameterization with a weighted divergence. Lastly, some experiments on simulated mixtures of particulate matter sources show the relevance of these approaches.