Robust Topological Inference: Distance To a Measure and Kernel Distance

Frédéric Chazal 1, 2 Brittany Fasy 3 Fabrizio Lecci 4 Bertrand Michel 2 Alessandro Rinaldo 4 Larry Wasserman 4
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non-robust to noise and outliers. Even one outlier is deadly. The distance-to-a-measure (DTM), introduced by Chazal et al. (2011), and the kernel distance, introduced by Phillips et al. (2014), are smooth functions that provide useful topological information but are robust to noise and outliers. Chazal et al. (2015) derived concentration bounds for DTM. Building on these results, we derive limiting distributions and confidence sets, and we propose a method for choosing tuning parameters.
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Journal of Machine Learning Research, Journal of Machine Learning Research, 2017, To appear
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https://hal.inria.fr/hal-01232217
Contributeur : Frédéric Chazal <>
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Dernière modification le : jeudi 15 juin 2017 - 09:09:09
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Frédéric Chazal, Brittany Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, et al.. Robust Topological Inference: Distance To a Measure and Kernel Distance. Journal of Machine Learning Research, Journal of Machine Learning Research, 2017, To appear. <hal-01232217>

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