# Robust Topological Inference: Distance To a Measure and Kernel Distance

Fr{\'e}d{\'e}ric Chazal, Brittany Fasy, Fabrizio Lecci, Bertr, Michel, Aless, ro Rinaldo, Larry Wasserman.

Year: 2018, Volume: 18, Issue: 159, Pages: 1−40

#### 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 \cite{chazal2011geometric}, and the kernel distance, introduced by \cite{phillips2014goemetric}, are smooth functions that provide useful topological information but are robust to noise and outliers. \cite{massart2014} 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.