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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References
V.I. Arnold, Catastrophe Theory, 3rd edn. (Springer, Berlin, 1992).
G. Carlsson, V. de Silva, Zigzag persistence, Found. Comput. Math. 10, 367–405 (2010).
D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37, 103–120 (2007).
H. Edelsbrunner, J. Harer, Computational Topology. An Introduction (Am. Math. Soc., Providence, 2010).
H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28, 511–533 (2002).
H. Edelsbrunner, D. Morozov, A. Patel, The stability of the apparent contour of an orientable 2-manifold, in Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, ed. by V. Pascucci, X. Tricoche, H. Hagen, J. Tierny (Springer, Heidelberg, 2011).
H. Gu, T.R. Chase, D.C. Cheney, T. Bailey, D. Johnson, Identifying correcting, and avoiding errors in computer-aided design models which affect interoperability, J. Comput. Inf. Sci. Eng. 1, 156–166 (2001).
V. Guillemin, A. Pollack, Differential Topology (Prentice Hall, Englewood Cliffs, 1974).
A. Hatcher, Algebraic Topology (Cambridge Univ. Press, Cambridge, 2002).
K. Popper, The Logic of Scientific Discovery (Basic Books, New York, 1959).
S. Smale, Book review on Catastrophe Theory: Selected Papers by E.C. Zeeman, Bull. Am. Math. Soc. 84, 1360–1468 (1978).
R. Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models (Addison–Wesley, Reading, 1989). Translated from the French by D.H. Fowler.
A. von Schemde, B. von Stengel, Strategic characterization of the index of an equilibrium, in Sympos. Algor. Game Theory. Lecture Notes Comput. Sci., vol. 4997 (Springer, Berlin, 2008), pp. 242–254.
H. Whitney, The self-intersections of a smooth n-manifold in 2n-space, Ann. Math. 45, 220–246 (1944).
H. Whitney, On singularities of mappings of Euclidean space. I. Mappings of the plane to the plane, Ann. Math. 62, 374–410 (1955).
E.C. Zeeman (ed.), Catastrophe Theory: Selected Papers, 1972–1977 (Addison–Wesley, Reading, 1977). London, England, 1978.
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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