Journal of Machine Learning Research 18 (2018) 1-40 Submitted 9/15; Revised 2/17; Published 4/18 Robust Topological Inference: Distance To a Measure and Kernel Distance Fr´ed´ericChazal
[email protected] Inria Saclay - Ile-de-France Alan Turing Bldg, Office 2043 1 rue Honor´ed'Estienne d'Orves 91120 Palaiseau, FRANCE Brittany Fasy
[email protected] Computer Science Department Montana State University 357 EPS Building Montana State University Bozeman, MT 59717 Fabrizio Lecci
[email protected] New York, NY Bertrand Michel
[email protected] Ecole Centrale de Nantes Laboratoire de math´ematiquesJean Leray 1 Rue de La Noe 44300 Nantes FRANCE Alessandro Rinaldo
[email protected] Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 Larry Wasserman
[email protected] Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 Editor: Mikhail Belkin 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 dis- tance 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 ker- nel 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.