Décomposition booléenne des tableaux multi-dimensionnels de données binaires : une approche par modèle de mélange post non-linéaire

Abstract : This work is dedicated to the study of boolean decompositions of binary multidimensional arrays using a post nonlinear mixture model. In the first part, we introduce a new approach for the boolean factorization of binary matrices (BFBM) based on a post nonlinear mixture model. Unlike the existing binary matrix factorization methods, the proposed method is equivalent to the boolean factorization model when the matrices are strictly binary and give thus more interpretable results in the case of correlated sources and lower rank matrix approximations compared to other state-of-the-art algorithms. A necessary and suffi-cient condition for the uniqueness of the BFBM is also provided. Two algorithms based on multiplicative update rules are proposed and tested in numerical simulations, as well as on a real dataset. The gener-alization of this approach to the case of binary multidimensional arrays (tensors) leads to the boolean factorisation of binary tensors (BFBT). The proof of the necessary and sufficient condition for the boolean decomposition of binary tensors is based on a notion of boolean independence of binary vectors. The multiplicative algorithm based on the post nonlinear mixture model is extended to the multidimensional case. We also propose a new algorithm based on an AO-ADMM (Alternating Optimization-ADMM) strategy. These algorithms are compared to state-of-the-art algorithms on simulated and on real data
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https://hal.univ-lorraine.fr/tel-02079639
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Mamadou Diop. Décomposition booléenne des tableaux multi-dimensionnels de données binaires : une approche par modèle de mélange post non-linéaire. Traitement du signal et de l'image [eess.SP]. Université de Lorraine, 2018. Français. ⟨NNT : 2018LORR0222⟩. ⟨tel-02079639⟩

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