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Contributions to geometric inference for manifolds and to the statistical study of persistence diagrams

Authors
  • Divol, Vincent
Publication Date
Aug 30, 2021
Source
HAL
Keywords
Language
English
License
Unknown
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Abstract

Topological data analysis (or TDA for short) consists in a set of methods aiming to extract topological and geometric information from complex nonlinear datasets. This field is here tackled from two different perspectives.First, we consider techniques from geometric inference, whose goal is to reconstruct geometric invariants of a manifold thanks to a random sample. We study from one hand the question of building an adaptive manifold estimator, and on the other hand the question of reconstructing the probability generating the observations.Second, we study persistent homology theory in TDA. A central tool in this theory, the persistence diagram, allows one to summarize in a multiscale fashion a dataset. We participate to the statistical study of persistence diagrams in several ways: first, by studying the metric structure of the space of persistence diagrams, and second, by defining a notion of linear expectation in this space. Diverse properties of this average object are then exhibited (asymptotic behavior, regularity, etc.)

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