Rohit P. Singh

Persistent Homology (PH) is a method of Topological Data Analysis that characterizes the topological structure of a space. Unfortunately, the computation of PH for high-dimensional and big data is not possible due to the exponential growth of the constructed complex. Fortunately, sparsification techniques can substantially reduce the size of the complex. This paper examines a sparsification technique ($\beta$-Sparsification) that produces a complex reduction capability that is scalable to a user-specified value $\beta$. At $\beta =0$ this scaling generates complexes that can have the same 1-Skeleton as the Vietoris–Rips complex; $\beta =1$ produces a Delaunay complex, and other values of $\beta$ produce a range of (unnamed) complexes. Experiments with $\beta$-Sparsification reveal that the topology of the sparsified simplicial complex is preserved for $0\le \beta \le 1$; for $\beta >1$, the complex begins to lose (potentially insignificant) topological features.