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.
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Distributed Computation of Persistent Homology from Partitioned Big Data
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.
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- Award ID(s):
- 1909096
- NSF-PAR ID:
- 10297300
- Date Published:
- Journal Name:
- IEEE International Conference on Cluster Computing
- Page Range / eLocation ID:
- 344 to 354
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely, persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the sugar, fish, flower, and gravel dataset produced for the study of mesoscale organization of clouds by Rasp et al. We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and we explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that readers of this paper will leave with a better understanding of TDA and persistent homology, will be able to identify problems and datasets of their own for which persistent homology could be helpful, and will gain an understanding of the results they obtain from applying the included GitHub example code.
Significance Statement Information such as the geometric structure and texture of image data can greatly support the inference of the physical state of an observed Earth system, for example, in remote sensing to determine whether wildfires are active or to identify local climate zones. Persistent homology is a branch of topological data analysis that allows one to extract such information in an interpretable way—unlike black-box methods like deep neural networks. The purpose of this paper is to explain in an intuitive manner what persistent homology is and how researchers in environmental science can use it to create interpretable models. We demonstrate the approach to identify certain cloud patterns from satellite imagery and find that the resulting model is indeed interpretable.
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Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several ℓ 1 minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: 1) optimization is effective in reducing the size of cycle representatives, though the extent of the reduction varies according to the dimension and distribution of the underlying data, 2) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis, in most data sets we consider, 3) the choice of linear solvers matters a lot to the computation time of optimizing cycles, 4) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, 5) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in { ‐ 1,0,1 } and therefore, are also solutions to a restricted ℓ 0 optimization problem, and 6) we obtain qualitatively different results for generators in Erdős-Rényi random clique complexes than in real-world and synthetic point cloud data.more » « less
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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 performedmore » « less
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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
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/Cluster48925.2021.00050
