Robust Topological Inference: Distance To a Measure and Kernel Distance

Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topolog-ical 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 out-liers. 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 topologi-cal information but are robust to noise and outliers. Chazal et al. (2014) 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. 1. Introduction. Figure 1 shows three complex point clouds, based on a model used for simulating cosmology data. Visually, the three samples look very similar. Below the data plots are the persistence diagrams, which are summaries of topological features defined in Section 2. The persistence diagrams make it clearer that the third data set is from a different data generating process than the first two. This is an example of how topological features can summarize structure in point clouds. The field of topological data analysis (TDA) is concerned with defining such topological features; see Carlsson (2009). When performing TDA, it is important to use topological measures that are robust to noise. This paper explores some of these robust topological measures. Let P be a distribution with compact support S ⊂ R d. One way to describe the shape of S is by