NMF methods aim to factorize a non-negative observation matrix X as the product X = G F between two non-negative matrices G and F. Although these approaches have been studied with great interest in the scientific community, they often suffer from a lack of robustness to data and to initial conditions, and provide multiple solutions. To this end and in order to reduce the space of admissible solutions, we propose to inform NMF, thus placing our work in between regression and classic blind factorization. In addition, some cost functions called αβ-divergences are used, so that the resulting NMF methods are robust to outliers in the data.
Three types of constraints are introduced on the matrix F, i.e., (i) the exact or (ii) the bounded knowledge of some components, and (iii) the sum to 1 of each line of F. Update rules are proposed so that all these constraints are taken into account by mixing multiplicative methods with projection.
Moreover, we propose to constrain the structure of the matrix G by the use of a physical model, in order to discern the sources which do contribute to the observed data.
The considered application---consisting of source identification of particulate matter in the air around an industrial area on the French northern coast---showed the interest of the proposed methods. Through a series of experiments on both synthetic and real data, we show the contribution of different informations to make the factorization results more consistent in terms of physical interpretation and less dependent of the initialization.
Matthieu Puigt received both the Bachelor and first year of M.S. degrees, in Pure and Applied Mathematics, in 2001 and 2002 respectively, from the Université de Perpignan, France. He then received the M.S. degree in Signal, Image Processing, and Acoustics, from the Université Paul Sabatier Toulouse 3, Toulouse, France, in 2003, and his Ph.D. in Signal Processing from the Université de Toulouse in 2007. Currently, he is an Associate Professor with the Université du Littoral Côte d'Opale, in Calais and Saint-Omer, France. Prior to this position, he was a Postdoctoral lecturer at the Université Paul Sabatier Toulouse 3 and the Laboratoire d'Astrophysique de Toulouse-Tarbes from 2007 to 2009. From September 2009 to June 2010, he held an Assistant Professor position in the University for Information Science and Technology, in Ohrid, FYROM. From August 2010 to July 2012, he was a Marie Curie postdoctoral fellow in the Signal Processing Lab of the Institute of Computer Science of the Foundation for Research and Technology - Hellas, Heraklion, Crete, Greece.
Matthieu Puigt's current research interests include linear and nonlinear signal processing, with an emphasis on sparse and non-negative signal processing and especially blind source separation methods and their applications to acoustics, astrophysics, and pollution monitoring.
