Piecewise Computation of Persistent Homology

Persistent Homology (PH) is a widely used tool of Topological Data Analysis (TDA) that measures the persistence of homological features present in data. PH is computed on a sequence of nested sub-complexes that form a filtration ${{\mathcal{K}}_{\mathcal{F}}}$ of data. In general, each nested sub-complex is defined by a scale parameter ϵ that defines the connectivity distances used to for its construction. These sub-complexes are arranged by increasing distances, ϵ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> to ϵ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf>, in the filtration; PH is then computed from ${{\mathcal{K}}_{\mathcal{F}}}$. However, due to its exponential space and time complexity, computing PH on big data is beyond the capabilities of contemporary machines. This paper explores the Piecewise computation of PH (PwPH) using subset constructions of the filtrations of data. PwPH is a framework for solutions that compute PH of data. In some embodiments, PwPH can be support computations of PH can be assembled into a complete picture of the homologies; in others, PwPH can only compute an approximation of the PH. The general solution of PwPH can be organized to use significantly less memory and provides a partition of computational PH elements that is, in some embodiments, embarrassingly parallel. This paper explores two complementary foundations to organize filtrations for PwPH.