[pt] Novos métodos de fatoração de matrizes introduzem restrições às matrizes decompostas, permitindo tipos únicos de análise. Uma modificação significativa é a fatoração de matrizes binárias para matrizes binárias. Esta técnica pode revelar subconjuntos comuns e mistura de subconjuntos, tornando-a útil em uma variedade de aplicações, como análise de cesta de mercado, modelagem de tópicos e sistemas de recomendação. Apesar das vantagens, as abordagens atuais enfrentam um trade-off entre precisão, escalabilidade e explicabilidade. Enquanto os métodos baseados em gradiente descendente são escaláveis, eles geram altos erros de reconstrução quando limitados para matrizes binárias. Por outro lado, os métodos heurísticos não são escaláveis. Para superar isso, essa tese propõe um procedimento de pós-processamento para discretizar matrizes obtidas por gradiente descendente. Esta nova abordagem recupera o erro de reconstrução após a limitação e processa com sucesso matrizes maiores dentro de um prazo razoável. Testamos esta técnica a muitas aplicações, incluindo um novo pipeline para descobrir e visualizar padrões em processos petroquímicos em batelada. / [en] Novel methods for matrix factorization introduce constraints to the decomposed matrices, allowing for unique kinds of analysis. One significant modification is the binary matrix factorization for binary matrices. This technique can reveal common subsets and mixing of subsets, making it useful in a variety of applications, such as market basket analysis, topic modeling, and recommendation systems. Despite the advantages, current approaches face a trade-off between accuracy, scalability, and explainability. While gradient descent-based methods are scalable, they yield high reconstruction errors when thresholded for binary matrices. Conversely, heuristic methods are not scalable. To overcome this, this thesis propose a post-processing procedure for discretizing matrices obtained by gradient descent. This novel approach recovers the reconstruction error post-thresholding and successfully processes larger matrices within a reasonable timeframe. We apply this technique to many applications including a novel pipeline for discovering and visualizing patterns in petrochemical batch processes.

Cette thèse aborde le problème de la décomposition booléenne des tableaux multidimensionnels de données binaires par modèle de mélange post non-linéaire. Dans la première partie, nous introduisons une nouvelle approche pour la factorisation booléenne en matrices binaires (FBMB) fondée sur un modèle de mélange post non-linéaire. Contrairement aux autres méthodes de factorisation de matrices binaires existantes, fondées sur le produit matriciel classique, le modèle proposé est équivalent au modèle booléen de factorisation matricielle lorsque les entrées des facteurs sont exactement binaires et donne des résultats plus interprétables dans le cas de sources binaires corrélées, et des rangs d'approximation matricielle plus faibles. Une condition nécessaire et suffisante d'unicité pour la FBMB est également fournie. Deux algorithmes s'appuyant sur une mise à jour multiplicative sont proposés et illustrés dans des simulations numériques ainsi que sur un jeu de données réelles. La généralisation de cette approche au cas de tableaux multidimensionnels (tenseurs) binaires conduit à la factorisation booléenne de tenseurs binaires (FBTB). La démonstration de la condition nécessaire et suffisante d’unicité de la décomposition booléenne de tenseurs binaires repose sur la notion d'indépendance booléenne d'une famille de vecteurs. L'algorithme multiplicatif fondé sur le modèle de mélange post non-linéaire est étendu au cas multidimensionnel. Nous proposons également un nouvel algorithme, plus efficace, s'appuyant sur une stratégie de type AO-ADMM (Alternating Optimization -ADMM). Ces algorithmes sont comparés à ceux de l'état de l'art sur des données simulées et sur un jeu de données réelles / 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