Non-negative matrix factorization using weighted beta divergence and equality constraints for industrial source apportionment

Abdelhakim Limem (1) , Gilles Delmaire (1) , Matthieu Puigt (1) , Gilles Roussel (1) , Dominique Courcot (2)

1 LISIC - Laboratoire d'Informatique Signal et Image de la Côte d'Opale
2 UCEIV - Unité de Chimie Environnementale et Interactions sur le Vivant

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.

Mots clés: β-divergence; NMF; weigthed factorization; equality constraints; multiplicative updates

Domaines: Traitement du signal et de l'image [eess.SP]

Abdelhakim Limem, Gilles Delmaire, Matthieu Puigt, Gilles Roussel, Dominique Courcot. Non-negative matrix factorization using weighted beta divergence and equality constraints for industrial source apportionment. 23rd IEEE International Workshop on Machine Learning for Signal Processing (MLSP 2013), Sep 2013, Southampton, United Kingdom. ⟨10.1109/MLSP.2013.6661903⟩. ⟨hal-01367327⟩