Robust Topological Inference: Distance To a Measure and Kernel Distance

TL;DR

This paper develops robust topological inference methods using distance-to-a-measure and kernel distance functions, providing theoretical guarantees and practical tools for noise-resistant persistent homology analysis.

Contribution

It introduces new limiting distributions, confidence sets, and parameter selection techniques for robust topological inference based on DTM and kernel distance.

Findings

Derived limiting distributions for DTM and kernel distance

Established confidence sets for topological features

Proposed a method for tuning parameter selection

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. (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.