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, ϵ0to ϵ∞, 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.

Persistent Homology (PH) computes the persistence of homologies in data. Unfortunately, PH suffers from exponential complexity. Two important mechanisms used to attack this complexity are: (i) using Alpha complexes; and (ii) setting an upper bound on the dimensions of homology group dimensions reported. Unfortunately Alpha complexes cannot be constructed in a bounded subspace$\mathbb{R}^{s}$of the ambient dimension$\mathbb{R}^{n}(s < n)$, so they cannot be used together. This paper explores a mechanism that uses sparsified$\beta^{s}$-complexes to build Alpha complexes in bounded subspaces. The approach leverages Space-Partitioning$(SP)$trees to manage the generation of subspace$\beta^{s}$-meshes that are sufficient to preserve$H_{s}$homology features for data in$\mathbb{R}^{n}(s < n)$. This approach constructs sparsified$\beta^{s}$-complexes significantly faster and for larger data than previously possible. The sparsified$\beta^{s}$-complexes produce a family of complexes scalable to$\beta$; members of this family approximate a Vietoris-Rips complex when$\beta=0$and an Alpha complex in when$\beta=1$. Experimental results with$\beta^{s}$-complexes at$\beta=0$and$\beta=1$produce results from PH that mirror those using conventional constructions of Vietoris-Rips and Alpha complexes.

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.

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 (H0-H10) 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.