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Stability and Statistical properties of topological information inferred from metric data

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. It is usually computed from filtrations built on top of data sets sampled from some unknown (metric) space, providing "topological signatures" revealing the structure of the underlying space. To ensure the relevance of such signatures, it is necessary to prove that they come with stability properties with respect to the way data are sampled. In this talk, after a short introduction to persistent homology, we will present a few results on the stability of persistence diagrams (the mathematical objects encoding the persistent homology information) built on top of general metric spaces. We will show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. The results presented in this talk are joint works with B. Fasy, M. Glisse, C. Labruère, F. Lecci, B. Michel, A. Rinaldo and L. Wasserman.

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46