Compact Boundary Matrix from a Polytopal Complex for Computing Persistent Homology

Persistent Homology (PH) is a tool of Topological Data Analysis (TDA) that records the persistence of homologies in data. The persistence of the homologies is computed from a filtration of the data created at a set of increasing connectivity distances $\left(0=\epsilon_{0}, \epsilon_{1}, \cdots, \epsilon_{\max } \leq \infty\right)$. Unfortunately the time and space complexity of computing PH from simplicial complexes (Vietoris-Rips, Cech, or Alpha) or cubical complexes limits its scope of use to small and low dimensional data. This paper examines the use of a Polytopal Complex to represent the data using maximal polytopes as elements of the filtered complexes. Furthermore, the approach further reduces size of the polytopal complex with an externally supplied approximation factor $\delta$. The approximation preserves the homology of the space that extends beyond $\delta$. Experimental results shows a significant reduction in space requirements. The reduction in space complexity and the compromise in PH computation is proportional to $\delta$. In addition to efficient construction of polytopal complexes, this work computes the homology of space using a compact boundary matrix representation made possible by the polytopal complex. The polytopal complex also provides a significant reduction in topological representations of high dimensional data.