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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 component in 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 whereas 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 complex. While 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 smaller for any given data dimension. However, its construction requires the computation of a Delaunay Triangulation that has high computational complexity. As a result, it is difficult to construct a Delaunay Complex 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 of k-Simplices (where k ≪ d) to quickly generate a reduced sparsified complex sufficient to compute homologies up to k-subspace, irrespective of the data dimensionality d. 

Topological Data Analysis (TDA) is a data mining technique to characterize the topological features of data. Persistent Homology (PH) is an important tool of TDA that has been applied to a wide range of applications. However its time and space complexities motivates a need for new methods to compute the PH of high-dimensional data. An important, and memory intensive, element in the computation of PH is the complex constructed from the input data. In general, PH tools use and focus on optimizing simplicial complexes; less frequently cubical complexes are also studied. This paper develops a method to construct polytopal complexes (or complexes constructed of any mix of convex polytopes) in any dimension Rn . In general, polytopal complexes are significantly smaller than simplicial or cubical complexes. This paper includes an experimental assessment of the impact that polytopal complexes have on memory complexity and output results of a PH computation. 

Topological Data Analysis is a machine learning method that summarizes the topological features of a space. Persistent Homology (PH) can identify these topological features as they persist within a point cloud; persisting in respect to the connectedness of the point cloud at increasing distances. The utility of PH is apparent in several fields including bioinformatics, network security, and object classification. However, the memory complexity of PH limits the application to relatively small point clouds for low-dimensional topological feature identification. For this reason, numerous approaches to optimize and approximate the PH have been introduced for providing results over large point clouds. One solution, Partitioned Persistent Homology (PPH), has shown favorable approximation on a single node with significant performance improvement. However, the single-node approach is limited by the available system memory, leading to the need for a distributed approach for additional (especially memory) resources. This paper studies a distributed version of PPH for use with large point clouds over a high-performance compute cluster. Experimental results of the distributed algorithm against previous studies is presented along with scalability of the distributed library.