Robust Topological Inference: Distance To a Measure and Kernel Distance

Frédéric Chazal (1) , Brittany Fasy (2) , Fabrizio Lecci (3) , Bertrand Michel (4) , Alessandro Rinaldo (3) , Larry Wasserman (3)

1 ABS - Algorithms, Biology, Structure
2 Department of Computer Science [Tulane]
3 Statistics Department, Carnegie Mellon University
4 DATASHAPE - Understanding the Shape of Data

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.

Domaines: Géométrie algorithmique [cs.CG]

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, 2018, 18 (159), pp.40. ⟨hal-01232217⟩