  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.
