Abstract
By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.
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Communicated by Gunnar Carlsson.
This research is partially supported by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.
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Edelsbrunner, H., Morozov, D. & Patel, A. Quantifying Transversality by Measuring the Robustness of Intersections. Found Comput Math 11, 345–361 (2011). https://doi.org/10.1007/s10208-011-9090-8
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DOI: https://doi.org/10.1007/s10208-011-9090-8
Keywords
- Smooth mappings
- Transversality
- Fixed points
- Contours
- Homology
- Filtrations
- Zigzag modules
- Persistence
- Perturbations
- Stability