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On Metric and Statistical Properties of Topological Descriptors for geometric Data

Abstract : In the context of supervised Machine Learning, finding alternate representations, or descriptors, for data is of primary interest since it can greatly enhance the performance of algorithms. Among them, topological descriptors focus on and encode the topological information contained in geometric data. One advantage of using these descriptors is that they enjoy many good and desireable properties, due to their topological nature. For instance, they are invariant to continuous deformations of data. However, the main drawback of these descriptors is that they often lack the structure and operations required by most Machine Learning algorithms, such as a means or scalar products. In this thesis, we study the metric and statistical properties of the most common topological descriptors, the persistence diagrams and the Mappers. In particular, we show that the Mapper, which is empirically instable, can be stabilized with an appropriate metric, that we use later on to conpute confidence regions and automatic tuning of its parameters. Concerning persistence diagrams, we show that scalar products can be defined with kernel methods by defining two kernels, or embeddings, into finite and infinite dimensional Hilbert spaces.
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Submitted on : Tuesday, January 30, 2018 - 6:37:08 PM
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  • HAL Id : tel-01659347, version 2



Mathieu Carriere. On Metric and Statistical Properties of Topological Descriptors for geometric Data. Computational Geometry [cs.CG]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLS433⟩. ⟨tel-01659347v2⟩



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