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Can neural networks learn persistent homology features?
Topological data analysis uses tools from topology – the mathematical ar...
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A flat persistence diagram for improved visualization of persistent homology
Visualization in the emerging field of topological data analysis has pro...
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Persistent Homology of Geospatial Data: A Case Study with Voting
A crucial step in the analysis of persistent homology is transformation ...
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The Generalized Persistent Nerve Theorem
In this paper a parameterized generalization of a good cover filtration ...
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Persistence of the Conley Index in Combinatorial Dynamical Systems
A combinatorial framework for dynamical systems provides an avenue for c...
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Optimisation of Spectral Wavelets for Persistence-based Graph Classification
A graph's spectral wavelet signature determines a filtration, and conseq...
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A Framework for Differential Calculus on Persistence Barcodes
We define notions of differentiability for maps from and to the space of...
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Filtration Simplification for Persistent Homology via Edge Contraction
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex K. These topological changes are summarized in persistence diagrams. We propose two contraction operators which when applied to K and its associated filtration, bound the perturbation in the persistence diagrams. The first assumes that the underlying space of K is a 2-manifold and ensures that simplices are paired with the same simplices in the contracted complex as they are in the original. The second is for arbitrary d-complexes, and bounds the bottleneck distance between the initial and contracted p-dimensional persistence diagrams. This is accomplished by defining interleaving maps between persistence modules which arise from chain maps defined over the filtrations. In addition, we show how the second operator can efficiently compose across multiple contractions. We conclude with experiments demonstrating the second operator's utility on manifolds.
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