One of the primary tools of topological data analysis is persistent homology, where, in the most general setting, one associates to a dataset a filtration of chain complexes. Homology of this filtration then allows one to identify persistent topological features of the dataset.
In this talk, based on joint work with Chris Kapulkin, we propose a new algorithm for computing persistent homology of a filtration of chain complexes. We suggest that persistence can be viewed as replacing a filtration with a quasi-isomorphic sequence of chain complexes whose differentials are all zero. This viewpoint leads to a new implementation for persistent homology, which we call RedZeD (for Reduction to Zero Differentials).
The most common form of persistent homology is of Vietoris-Rips filtrations, and is the main motivation of this work. In this setting, our viewpoint allows for a further improvement to RedZeD. When implemented, in most cases, we see an improvement in homology in degree one over Ripser, the current fastest general algorithm for Vietoris-Rips persistence, developed by Ulrich Bauer.