Robust Persistence Diagrams using Reproducing Kernels

TL;DR

This paper introduces a robust framework for persistence diagrams using reproducing kernels, making them less sensitive to outliers and providing statistical guarantees, with demonstrated improvements on benchmark datasets.

Contribution

The paper develops a kernel-based method for robust persistence diagrams, establishing their statistical consistency and confidence bands, improving robustness against outliers.

Findings

Robust persistence diagrams are less sensitive to outliers.

The framework achieves consistency in bottleneck distance.

Demonstrated superior performance on benchmark datasets.

Abstract

Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams are very sensitive to perturbations in the input space. In this work, we develop a framework for constructing robust persistence diagrams from superlevel filtrations of robust density estimators constructed using reproducing kernels. Using an analogue of the influence function on the space of persistence diagrams, we establish the proposed framework to be less sensitive to outliers. The robust persistence diagrams are shown to be consistent estimators in bottleneck distance, with the convergence rate controlled by the smoothness of the kernel. This, in turn, allows us to construct uniform confidence bands in the space of persistence diagrams. Finally, we demonstrate the superiority of the proposed approach on benchmark datasets.