Robust Persistence Diagrams using Reproducing Kernels
This work addresses the problem of statistical robustness in topological data analysis for researchers and practitioners, representing an incremental improvement by applying existing robust methods to persistence diagrams.
The authors tackled the sensitivity of persistence diagrams to data perturbations by developing a framework using robust density estimators with reproducing kernels, resulting in less sensitivity to outliers and consistent estimators with convergence rates controlled by kernel smoothness, demonstrated on benchmark datasets.
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.