Structure and Stability of the 1-Dimensional Mapper

Mathieu Carrière 1 Steve Oudot 1
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : Given a continuous function $f:X\to\mathbb{R}$ and a cover $\mathcal{I}$ of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover $f^{-1}(\mathcal{I})$. Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of $f$, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework that relates the structure of the Mapper to the one of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover $\mathcal{I}$ goes to zero.
Type de document :
Communication dans un congrès
Proceedings of the International Symposium on Computational Geometry, Jun 2016, Boston, United States. Proceedings of the International Symposium on Computational Geometry, 2016, <http://socg2016.cs.tufts.edu/>
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https://hal.inria.fr/hal-01247511
Contributeur : Steve Oudot <>
Soumis le : mardi 22 décembre 2015 - 11:17:15
Dernière modification le : samedi 18 février 2017 - 01:14:35

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  • HAL Id : hal-01247511, version 1
  • ARXIV : 1511.05823

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Mathieu Carrière, Steve Oudot. Structure and Stability of the 1-Dimensional Mapper. Proceedings of the International Symposium on Computational Geometry, Jun 2016, Boston, United States. Proceedings of the International Symposium on Computational Geometry, 2016, <http://socg2016.cs.tufts.edu/>. <hal-01247511>

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