It's All About Covers: Persistent Homology of Cover Refinements
The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth which can be exponential in the worst case, as for the Vietoris-Rips. We recast this problem at the level of covers, developing a framework in which filtrations and persistence modules can be constructed, analyzed, and compared through simple relations between covers rather than at the level of simplicial complexes. The guarantees propagate through any functor that preserves the contiguity of refinement maps, we give the example of two such functors: the Nerve and the Co-Nerve. Working at this level is drastically simpler, with stronger, more general consequences. We explore this perspective and show how it can be used to construct a robust approximation of the Vietoris-Rips filtration that is orders of magnitude smaller, while maintaining a log 3-interleaving unconditionally for any metric space. The resulting filtration restores near-linear scaling in the number of data points and enable us to efficiently capture homology at high degrees.