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.

Topological Data Analysis (TDA) has demonstrated notable success in data mining by measuring the presence of topological structure in data. Persistent Homology (PH), one popular tool of TDA, examines the nested sequence of graphs formed over a data filtration to characterize unique homology classes. PH suffers from exponential complexity, limiting the approach to relatively small data sets and lower homology class applications. The Euler Characteristic Curve (ECC), a metric closely related to PH, can be computed more efficiently and, in some cases, can be used as a direct replacement to the applications of PH. A recent technique has been introduced to separate the ECC into dimensional components representing the homology classes identified by PH. This study examines the dimensional ECC, provides an improved algorithm, and introduces interpretation of the results as proximity-series representations of topological features; exploration of how to compare and classify ECC curves is detailed and shown in the context of MRA Brain Artery scan classification. Experimental results with ECC demonstrate the effectiveness and scalability to big data sets beyond that of current persistent homology applications.

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 is a computational method of data mining in the field of Topological Data Analysis. Large-scale data analysis with persistent homology is computationally expensive and memory intensive. The performance of persistent homology has been rigorously studied to optimize data encoding and intermediate data structures for high-performance computation. This paper provides an application-centric survey of the High-Performance Computation of Persistent Homology. Computational topology concepts are reviewed and detailed for a broad data science and engineering audience.

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.

Topological Data Analysis (TDA) utilizes concepts from topology to analyze data. In general, TDA considers objects similar based on a topological invariant. Topological invariants are properties of the topological space that are homeomorphic; resilient to deformation in the space. The Euler-Poincaré Characteristic is a classic topological invariant that represents the alternating sum of the vertices, edges, faces, and higherorder cells of a closed surface. Tracking the Euler characteristic over a topological filtration produces an Euler Characteristic Curve (ECC). This study introduces a computational technique to determine the ECC of $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ data; the technique generalizes to higher dimensions. This technique separates landscapes of lowerorder homologies utilizing triangulations of the space.

Persistent Homology (PH) is computationally expensive and is thus generally employed with strict limits on the (i) maximum connectivity distance and (ii) dimensions of homology groups to compute (unless working with trivially small data sets). As a result, most studies with PH only work with H0and H1homology groups. This paper examines the identification and isolation of regions of data sets where high dimensional topological features are suspected to be located. These regions are analyzed with PH to characterize the high dimensional homology groups contained in that region. Since only the region around a suspected topological feature is analyzed, it is possible to identify high dimension homologies piecewise and then assemble the results into a scalable characterization of the original data set.

An emerging method for data analysis is called Topological Data Analysis (TDA). TDA is based in the mathematical field of topology and examines the properties of spaces under continuous deformation. One of the key tools used for TDA is called persistent homology which considers the connectivity of points in a d-dimensional point cloud at different spatial resolutions to identify topological properties (holes, loops, and voids) in the space. Persistent homology then classifies the topological features by their persistence through the range of spatial connectivity. Unfortunately the memory and run-time complexity of computing persistent homology is exponential and current tools can only process a few thousand points in $\mathbb{R}^{3}$. Fortunately, the use of data reduction techniques enables persistent homology to be applied to much larger point clouds. Techniques to reduce the data range from random sampling of points to clustering the data and using the cluster centroids as the reduced data. While several data reduction approaches appear to preserve the large topological features present in the original point cloud, no systematic study comparing the efficacy of different data clustering techniques in preserving the persistent homology results has been performed. This paper explores the question of topology preserving data reductions and describes formally when and how topological features can be mischaracterized or lost by data reduction techniques. The paper also performs an experimental assessment of data reduction techniques and resilient effects on the persistent homology. In particular, data reduction by random selection is compared to cluster centroids extracted from different data clustering algorithms.

Persistent homology is a method of data analysis that is based in the mathematical field of topology. Unfortunately, the run-time and memory complexities associated with computing persistent homology inhibit general use for the analysis of big data. For example, the best tools currently available to compute persistent homology can process only a few thousand data points in ℝ3. Several studies have proposed using sampling or data reduction methods to attack this limit. While these approaches enable the computation of persistent homology on much larger data sets, the methods are approximate. Furthermore, while they largely preserve the results of large topological features, they generally miss reporting information about the small topological features that are present in the data set. While this abstraction is useful in many cases, there are data analysis needs where the smaller features are also significant (e.g., brain artery analysis). This paper explores a combination of data reduction and data partitioning to compute persistent homology on big data that enables the identification of both large and small topological features from the input data set. To reduce the approximation errors that typically accompany data reduction for persistent homology, the described method also includes a mechanism of “upscaling” the data circumscribing the large topological features that are computed from the sampled data. The designed experimental method provides significant results for improving the scale at which persistent homology can be performed.

Persistent homology is used for computing topological features of a space at different spatial resolutions. It is one of the main tools from computational topology that is applied to the problems of data analysis. Despite several attempts to reduce its complexity, persistent homology remains expensive in both time and space. These limits are such that the largest data sets to which the method can be applied have the number of points of the order of thousands in ℝ3. This paper explores a technique intended to reduce the number of data points while preserving the salient topological features of the data. The proposed technique enables the computation of persistent homology on a reduced version of the original input data without affecting significant components of the output. Since the run time of persistent homology is exponential in the number of data points, the proposed data reduction method facilitates the computation in a fraction of the time required for the original data. Moreover, the data reduction method can be combined with any existing technique that simplifies the computation of persistent homology. The data reduction is performed by creating small groups of similar data points, called nano-clusters, and then replacing the points within each nano-cluster with its cluster center. The persistence homology of the reduced data differs from that of the original data by an amount bounded by the radius of the nano-clusters. The theoretical analysis is backed by experimental results showing that persistent homology is preserved by the proposed data reduction technique.

This paper presents a solution to the approximate k-means clustering problem for very large distributed datasets. Distributed data models have gained popularity in recent years following the efforts of commercial, academic and government organizations, to make data more widely accessible. Due to the sheer volume of available data, in-memory single-core computation quickly becomes infeasible, requiring distributed multiprocessing. Our solution achieves comparable clustering performance to other popular clustering algorithms, with improved overall complexity growth while being amenable to distributed processing frameworks such as Map-Reduce. Our solution also maintains certain guarantees regarding data privacy deanonimization.