A roadmap for the computation of persistent homology

Persistent homology (PH) is a method used in topological data analysis (TDA)
to study qualitative features of data that persist across multiple scales. It
is robust to perturbations of input data, independent of dimensions and
coordinates, and provides a compact representation of the qualitative features
of the input. There has been recent progress, but the computation of PH remains
an open area with numerous important and fascinating challenges. The field of
PH computation is evolving rapidly, and new algorithms and software
implementations are being updated and released at a rapid pace. The purposes of
our article are to (1) introduce theory and computational methods for PH to a
broad range of computational scientists and (2) provide benchmarks of
state-of-the-art implementations for the computation of PH. We give a friendly
introduction to PH, navigate the pipeline for the computation of PH with an eye
towards applications, and use a range of synthetic and real-world data sets to
evaluate currently available open-source implementations for the computation of
PH. Based on our benchmarking, we indicate which algorithms and implementations
are best suited to different types of data sets. In an accompanying tutorial,
we provide guidelines for the computation of PH. We make publicly available all
scripts that we wrote for the tutorial, and we make available the processed
version of the data sets used in the benchmarking.