Informed Weighted Non-Negative Matrix Factorization Using αβ-Divergence Applied to Source Apportionment

Abstract

In this paper, we propose informed weighted non-negative matrix factorization (NMF) methods using an α β -divergence cost function. The available information comes from the exact knowledge/boundedness of some components of the factorization—which are used to structure the NMF parameterization—together with the row sum-to-one property of one matrix factor. In this contribution, we extend our previous work which partly involved some of these aspects to α β -divergence cost functions. We derive new update rules which are extendthe previous ones and take into account the available information. Experiments conducted for several operating conditions on realistic simulated mixtures of particulate matter sources show the relevance of these approaches. Results from a real dataset campaign are also presented and validated with expert knowledge.

Information Theory Applications in Signal Processing

“…Five of the articles [ 16 , 17 , 18 , 19 , 20 ] are devoted to the analysis of the latent components of the observations. In [ 16 ], the authors suggest a nonlinear estimation of the partial correlation coefficients, due to its potential applications in graph signal processing.…”

“…The works in [ 17 , 18 ] present some unsupervised learning criteria based on the family of Alpha–Beta divergences and validate them through both synthetic and real experiments. These generalized dissimilarity measures are governed by the two real parameters, and , that smoothly connect several classical divergences (see [ 21 ]).…”

“…In [ 17 ], Sarmiento et al obtain the closed-form expressions for the -centroids, and propose an original - k -means algorithm for centroid-based clustering. Delmaire et al propose in [ 18 ] several informed and weighted -non-negative matrix factorization methods. The contribution reveals that the combination of the factorization criterion with a set of structural constraints (which are derived from some available information in the chemical source apportionment problem) enhances the robustness of the decomposition with respect to possible contaminations by noise and outliers.…”