Non-Negative Matrix Factorization Using Weighted Beta Divergence And Equality Constraints For Industrial Source Apportionment
Abdelhakim Limem, Gilles Delmaire, Matthieu Puigt, Gilles Roussel, Dominique Courcot

Abstract:
In this paper, we propose two weighted Non-negative Matrix Factorization (NMF) methods using a $\beta$-divergence cost function. This divergence is used as a dissimilarity measure which can be tuned by the parameter $\beta$. 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 $\beta$-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.