Convergence rates for persistence diagram estimation in topological data analysis

2014 Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. We show that the use of persistent homology can be naturally considered in general statistical frameworks. We establish convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled. Our approach relies on a recent and deep stability result for persistence that allows to relate our problem to support estimation problems (with respect to the Gromov-Hausdorff distance). Some numerical experiments are performed in various contexts to illustrate our results.