Persistent Cohomology Of Cover Refinements

Abstract

Topological data analysis (TDA) is a new approach to analyzing complex data which often helps reveal otherwise hidden patterns by highlighting various geometrical and topological features of the data. Persistent homology is a key in the TDA toolbox. It measures topological features of data that persist across multiple scales and thus are robust with respect to noise. Persistent homology has had many successful applications, but there is room for improvement. For large datasets, computation of persistent homology often takes a significant amount of time. Several approaches have been proposed to try to remedy this issue, such as witness complexes, but those approaches present their own difficulties. In this work, we show that one can leverage a well-known data structure in computer science called a cover tree. It allows us to create a new construction that avoids difficulties of witness complex and can potentially provide a significant computational speed up. Moreover, we prove that the persistence diagrams obtained using our novel construction are actually close in a certain rigorously defined way to persistence diagrams which we obtain using the standard approach based on Cech complexes. This quantifiable coarse computation of persistence diagrams has the potential to be used in many applications where features with a low persistence are known to be less important.