Reconstructing Embedded Graphs from Persistence Diagrams
| Autor: | Anna Schenfisch, Jordan Schupbach, Robin Belton, David L. Millman, Daniel Salinas, Samuel Micka, Lucia Williams, Rostik Mertz, Brittany Terese Fasy |
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| Rok vydání: | 2019 |
| Předmět: |
Discrete mathematics
Computational Geometry (cs.CG) FOS: Computer and information sciences Control and Optimization Correctness Plane (geometry) Inverse problem Space (mathematics) Oracle Computer Science Applications Set (abstract data type) Computational Mathematics Simplicial complex Computational Theory and Mathematics Computer Science - Computational Geometry Geometry and Topology Finite set Mathematics |
| DOI: | 10.48550/arxiv.1912.08913 |
| Popis: | The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes using a finite set of descriptors. A related inverse problem is the following: given a set of PDs (or an oracle we can query for persistence diagrams), what is underlying geometric simplicial complex? In this paper, we present an algorithm for reconstructing embedded graphs in $\mathbb{R}^d$ (plane graphs in $\mathbb{R}^2$) with $n$ vertices from $n^2 - n + d + 1$ directional (augmented) PDs. Additionally, we empirically validate the correctness and time-complexity of our algorithm in $\mathbb{R}^2$ on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm. Comment: 32 pages, 10 figures. This paper is an extended version of "Learning Simplicial Complexes from Persistence Diagrams" that appeared in the conference proceedings for the Canadian Conference on Computational Geometry (CCCG) 2018. This extended paper will appear in a special issue of the journal, Computational Geometry Theory and Applications (CGTA) |
| Databáze: | OpenAIRE |
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