Réduire la dimension des systèmes complexes : un regard sur l'émergence de la synchronisation

Les systèmes complexes se caractérisent par l’émergence de phénomènes macroscopiques qui ne s’expliquent pas uniquement par les propriétés de leurs composantes de base. La synchronisation est l’un de ces phénomènes par lequel les interactions entre des oscillateurs engendrent des mouvements collectifs coordonnés. Une représentation sous forme de graphe permet de modéliser précisément les interactions, alors que les oscillations se décrivent par des dynamiques non linéaires. Étant donné le lien subtil entre le graphe et la dynamique des oscillateurs, il est difficile de prédire l’émergence de la synchronisation. L’objectif de ce mémoire est de développer de nouvelles méthodes pour simplifier les systèmes complexes d’oscillateurs afin de révéler les mécanismes engendrant la synchronisation. À cette fin, nous introduisons des notions de la théorie des graphes et des systèmes dynamiques pour définir la synchronisation sur des bases mathématiques. Nous présentons ensuite des approches existantes sophistiquées pour réduire la dimension de dynamiques d’oscillateurs. Ces techniques sont toutefois limitées lorsque les dynamiques évoluent sur des graphes plus complexes. Nous développons alors une technique originale—spécialement adaptée pour les graphes aux propriétés plus hétérogènes—pour réduire la dimension de dynamiques non linéaires. En plus de généraliser des approches récentes, notre méthode dévoile plusieurs défis théoriques reliés à la simplification d’un système complexe. Entre autres, la réduction de la dimension du système se bute à une trichotomie : il faut favoriser la conservation des paramètres dynamiques, des propriétés locales du graphe ou des propriétés globales du graphe. Finalement, notre méthode permet de caractériser mathématiquement et numériquement l’émergence d’états exotiques de synchronisation.Complex systems are characterized by the emergence of macroscopic phenomena that cannot be explained by the properties of their basic constituents. Synchronization is one of these phenomena in which the interactions between oscillators generate coordinate collective behaviors. Graphs allow a precise representation of the interactions, while nonlinear dynamics describe the oscillations. Because of the subtle relationship between graphs and dynamics of oscillators, it is challenging to predict the emergence of synchronization. The goal of this master’s thesis is to develop new methods for simplifying complex systems of oscillators in order to reveal the mechanism causing synchronization. To this end, we introduce notions of graph theory and dynamical systems to define synchronization on sound mathematical basis. We then present existing sophisticated approaches for reducing the dimension of oscillator dynamics. Yet, the efficiency of these techniques is limited for dynamics on complex graphs. We thus develop an original method—specially adapted for graphs with heterogeneous properties—for reducing the dimensions of nonlinear dynamics. Our method generalizes previous approaches and uncovers multiple challenges related to the simplification of complex systems. In particular, the dimension reduction of a system comes up against a trichotomy: one must choose to conserve either the dynamical parameters, the local properties of the graph, or the global properties of the graph. We finally use our method to characterize mathematically and numerically the emergence of exotic synchronization states

Machine learning methodologies for high dimensional biomedical & bioinformatics applications

The impact of machine learning has been greatly expanded due to the increase in computational power in recent years, and has made a significant scientific contribution to many fields. This dissertation primarily investigates and expands the usage of certain machine learning methodologies on high-dimensional biomedical and bioinformatics applications. In particular, I aim to propose novel, data-driven clustering and feature extraction methods to uncover richer and more interpretable predictive features for classification problems. This dissertation considers three modern biomedical and bioinformatics problems in the context of text mining, computer vision and microbiome analysis. To address the different challenges in these applications, novels methods in matrix factorization, image registration, and deep learning are proposed. For the first project on text mining, we propose the semi-orthogonal non-negative matrix factorization as a topic model to investigate the potential of using triage notes to classify patient disposition in addressing the issue of emergency department crowding. For the second project on computer vision, we propose a novel implementation of the neural style transfer algorithm as an image preprocessing and registration method for skin lesion classification problems. For the third project on microbiome analysis, we discuss two works that have been done. First, we propose an analysis pipeline that implements the random forest model to identify food intake, along with a PCA-based approach to remove study batch effects and validate our classification results. Second, we proposed to incorporate the phylogenetic information of microbes as graphs via a graphical convolutional neural network to improve the classification performances for dietary and health outcomes.U of I OnlyAuthor requested U of Illinois access only (OA after 2yrs) in Vireo ETD syste