Persistence Homology of Proximity Hyper-Graphs for Higher Dimensional Big Data

Persistent Homology (PH) is a method of Topological Data Analysis that analyzes the topological structure of data to help data scientists infer relationships in the data to assist in informed decision- making. A significant c omponent i n the computation of PH is the construction and use of a complex that represents the topological structure of the data. Some complex types are fast to construct but space inefficient w hereas others are costly to construct and space efficient. Unfortunately, existing complex types are not both fast to construct and compact.This paper works to increase the scope of PH to support the computation of low dimensional homologies (H 0 -H 10 ) in high-dimension, big data. In particular, this paper exploits the desirable properties of the Vietoris-Rips Complex (VR-Complex) and the Delaunay Complex in order to construct a sparsified complex. The VR-Complex uses a distance matrix to quickly generate a complex up to the desired homology dimension. In contrast, the Delaunay Complex works at the dimensionality of the data to generate a sparsified c omplex. W hile construction of the VR-Complex is fast, its size grows exponentially by the size and dimension of the data set; in contrast, the Delaunay complex is significantly s maller f or a ny g iven d ata dimension. However, its construction requires the computation of a Delaunay Triangulation that has high computational complexity. As a result, it is difficult t o c onstruct a D elaunay C omplex for data in dimensions d > 6 that contains more than a few hundred points. The techniques in this paper enable the computation of topological preserving sparsification o f k -Simplices (where k ≪ d) to quickly generate a reduced sparsified complex sufficient t o c ompute h omologies u p t o k -subspace, irrespective of the data dimensionality d.