Abstract
Topological data analysis tools enjoy increasing popularity in a wide range of applications, such as Computer graphics, Image analysis, Machine learning, and Astronomy for extracting information. However, due to computational complexity, processing large numbers of samples of higher dimensionality quickly becomes infeasible. This contribution is twofold: We present an efficient novel sub-sampling strategy inspired by Coulomb’s law to decrease the number of data points in d-dimensional point clouds while preserving its homology. The method is not only capable of reducing the memory and computation time needed for the construction of different types of simplicial complexes but also preserves the size of the voids in d-dimensions, which is crucial e.g. for astronomical applications. Furthermore, we propose a technique to construct a probabilistic description of the border of significant cycles and cavities inside the point cloud. We demonstrate and empirically compare the strategy in several synthetic scenarios and an astronomical particle simulation of a dwarf galaxy for the detection of superbubbles (supernova signatures).
Highlights
Topological data analysis (TDA) provides exploration tools for increasingly diverse applications in various domains, ranging from Biology and medical images [1], mapping disease spaces [2], and Astronomy [3]
Superbubbles are of great astronomical interest but typically measured by eye in available catalogues and automatic tools are highly desirable. This contribution extends A Sub-sampling Approach for Preserving topological structures ASAP2 [23], that reduces the computational cost suitable for different types of Persistent homology (PH) filtration, general ddimensional point clouds, and large number of samples
In order to fully capture the properties of such cavities, taking advantage of their low dimensional nature, we propose a modified version of the Generative Topographic Mapping (GTM): geodesic GTM
Summary
Topological data analysis (TDA) provides exploration tools for increasingly diverse applications in various domains, ranging from Biology and medical images [1], mapping disease spaces [2], and Astronomy [3]. Superbubbles are of great astronomical interest but typically measured by eye in available catalogues and automatic tools are highly desirable This contribution extends A Sub-sampling Approach for Preserving topological structures ASAP2 [23], that reduces the computational cost suitable for different types of PH filtration, general ddimensional point clouds, and large number of samples. In order to fully capture the properties of such cavities, taking advantage of their low dimensional nature, we propose a modified version of the GTM: geodesic GTM (gGTM) Through this formulation, the topological features of the modelled structures are accounted for by embedding a closed low dimensional latent space onto the ambient space of the point cloud. We compare to state-of-the-art methods in several controlled experiments and investigate a snapshot of an astronomical particle simulation by computing the number and size of superbubbles within a jelly-fish like dwarf galaxy
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