Quantifying Transversality by Measuring the Robustness of Intersections

Authors:
Herbert Edelsbrunner;Dmitriy Morozov;Amit Patel
Affiliations:
Duke Univ., Departments of Computer Science and of Mathematics, Durham, NC, USA and Geomagic, Research Triangle Park, NC, USA and Inst. of Sci. and Technol. Austria, Klosterneuburg, Austria;Stanford University, Departments of Computer Science and of Mathematics, Stanford, CA, USA;Duke University, Departments of Computer Science and of Mathematics, Durham, NC, USA and Institute of Science and Technology Austria, Klosterneuburg, Austria
Venue:
Foundations of Computational Mathematics
Year:
2011

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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.