A Statistical Approach to Topological Data Analysis

Abstract

Until very recently, topological data analysis and topological inference methods mostlyrelied on deterministic approaches. The major part of this habilitation thesis presents astatistical approach to such topological methods. We first develop model selection toolsfor selecting simplicial complexes in a given filtration. Next, we study the estimationof persistent homology on metric spaces. We also study a robust version of topologicaldata analysis. Related to this last topic, we also investigate the problem of Wassersteindeconvolution. The second part of the habilitation thesis gathers our contributions inother fields of statistics, including a model selection method for Gaussian mixtures, animplementation of the slope heuristic for calibrating penalties, and a study of Breiman’spermutation importance measure in the context of random forests