More Like this

1. Topology Preserving Data Reduction for Computing Persistent Homology

   https://doi.org/10.1109/BigData50022.2020.9378216  

   Malott, Nicholas O.; Sens, Aaron M.; Wilsey, Philip A. (December 2020, International Workshop on Big Data Reduction)

   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 R3. 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. 

   more » « less
2. Fast Computation of Persistent Homology with Data Reduction and Data Partitioning

   https://doi.org/10.1109/BigData47090.2019.9006572  

   Malott, Nicholas O.; Wilsey, Philip A. (December 2019, 2019 IEEE International Conference on Big Data)

   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 R^3. 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 

   more » « less
3. Polytopal Complex Construction and Use in Persistent Homology

   https://doi.org/10.1109/ICDMW58026.2022.00087  

   Singh, Rohit P.; Wilsey, Philip A. (November 2022, ICDM Workshop on High Dimensional Data Mining)

   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. 

   more » « less
4. A Survey on the High Performance Computation of Persistent Homology

   https://doi.org/10.1109/TKDE.2022.3147070  

   Malott, Nicholas; Chen, Shangye; Wilsey, Philip (January 2022, IEEE Transactions on Knowledge and Data Engineering)

   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. 

   more » « less
5. Data Reduction and Feature Isolation for Computing Persistent Homology on High Dimensional Data

   https://doi.org/10.1109/BigData52589.2021.9671839  

   Verma, Rishi R.; Malott, Nicholas O.; Wilsey, Philip A. (December 2021, Workshop on Applications of Topological Data Analysis to "Big Data")

   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 H0 and H1 homology 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. 

   more » « less