Adaptive Approximation of Persistent Homology

Abstract

We study an important preprocessing step for the efficient calculation of persistent homology: coarsening a set of points while controlling the quality of the induced persistence diagram. This coarsening step is instrumental in reducing the overall runtime of state-of-the-art algorithms such as Ripser, GUDHI, or PHAT. For this, we adaptively sparsify the set of points and carefully define a dissimilarity function between the remaining points. This function takes into account local properties given by a separate function defined on the point set as well as the relation of the removed points to the subsample. It is then used to build simplicial filtrations and calculate their persistent homology. We assess the quality of our approach both theoretically by proving topological approximation guarantees and empirically by using the bottleneck distance. Since our subsample is calculated adaptively, we also prove adaptive properties of our result. Our results show that we can significantly reduce the size of the point sample with only very moderate bottleneck distance to the ground truth; in particular, we still are able to capture the salient topological features of the input through our approximated persistence diagrams.