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Modèles de mélange pour la caractérisation topologique de données multidimensionnelles

Maxime Maillot 1
1 LIMA - Laboratoire Information, Modèles, Apprentissage [Gif-sur-Yvette]
DM2I - Département Métrologie Instrumentation & Information : DRT/LIST/DM2I
Abstract : This phD report describes a model enabling to learn the topology of a variety underlying in a sample. The problem was conceptualized first, then a model was proposed to solve it: the Generative Simplicial Complex. Partly geometrical and partly statistical, this model provides a statistics formalism to geometrical problem. The main interest lies in the possibility of choosing the right model through an objective statistical criterion. The main idea is to build an initial Delaunay complex from well positioned points in relation to the data. Then, we optimize the parameters (weight and variance) by maximizing likelihood through the EM algorithm and we simplify this complex, keeping only the relevant components (those whose weight is not zero). The number of peaks used is selected using the BIC criterion, which empirically gives a model that matches the Betti numbers of the varieties underlying the data.
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Maxime Maillot. Modèles de mélange pour la caractérisation topologique de données multidimensionnelles. Topologie générale [math.GN]. UTC Compiègne, 2015. Français. ⟨tel-02271267⟩

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