Adversarially Robust Topological Inference

TL;DR

This paper introduces a robust statistical framework for topological data analysis that effectively handles outliers by using a median-of-means distance function, ensuring consistent and near-optimal inference of persistent homology.

Contribution

It proposes a median-of-means variant of the distance function for robust persistent homology inference, with proven statistical properties and improved outlier resilience.

Findings

The MoM Dist provides robustness against outliers in topological inference.

Sublevel and weighted filtrations using MoM Dist are consistent estimators.

The method achieves near minimax-optimal performance in adversarial scenarios.

Abstract

The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance function are used in the computation of persistent homology -- a backbone of the topological data analysis pipeline. Despite its stability to perturbations in the Hausdorff distance, persistent homology is highly sensitive to outliers. In this work, we develop a framework of statistical inference for persistent homology in the presence of outliers. Drawing inspiration from recent developments in robust statistics, we propose a \textit{median-of-means} variant of the distance function (\textsf{MoM Dist}) and establish its statistical properties. In particular, we show that, even in the presence of outliers, the sublevel filtrations and weighted filtrations induced by \textsf{MoM Dist} are both consistent estimators of the true underlying population counterpart and exhibit near minimax-optimal performance in adversarial settings. Finally, we demonstrate the advantages of the proposed methodology through simulations and applications.