A Statistical Approach to Topological Data Analysis
Résumé
Until very recently, topological data analysis and topological inference methods mostly
relied on deterministic approaches. The major part of this habilitation thesis presents a
statistical approach to such topological methods. We first develop model selection tools
for selecting simplicial complexes in a given filtration. Next, we study the estimation
of persistent homology on metric spaces. We also study a robust version of topological
data analysis. Related to this last topic, we also investigate the problem of Wasserstein
deconvolution. The second part of the habilitation thesis gathers our contributions in
other fields of statistics, including a model selection method for Gaussian mixtures, an
implementation of the slope heuristic for calibrating penalties, and a study of Breiman’s
permutation importance measure in the context of random forests.
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