×

zbMATH — the first resource for mathematics

Continuation of point clouds via persistence diagrams. (English) Zbl 1415.55006
Summary: In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton-Raphson continuation method in this setting. Given an original point cloud \(P\), its persistence diagram \(D\), and a target persistence diagram \(D'\), we gradually move from \(D\) to \(D'\), by successively computing intermediate point clouds until we finally find a point cloud \(P'\) having \(D'\) as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.

MSC:
55U10 Simplicial sets and complexes in algebraic topology
55N35 Other homology theories in algebraic topology
65H10 Numerical computation of solutions to systems of equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Carlsson, G., Topology and Data, Bull. Amer. Math. Soc., 46, 255-308 (2009) · Zbl 1172.62002
[2] Edelsbrunner, H.; Harer, J., Computational Topology: An Introduction (2010), AMS · Zbl 1193.55001
[3] Edelsbrunner, H.; Letscher, D.; Zomorodian, A., Topological Persistence and simplification, Discrete Comput. Geom., 28, 4, 511-533 (2002) · Zbl 1011.68152
[4] Zomorodian, A.; Carlsson, G., Computing Persistent homology, Discrete Comput. Geom., 33, 2, 249-274 (2005) · Zbl 1069.55003
[6] Nakamura, T.; Hiraoka, Y.; Hirata, A.; Escolar, E. G.; Nishiura, Y., Persistent homology and many-body atomic structure for medium-range order in the glass, Nanotechnology, 26, 304001 (2015)
[7] Gameiro, M.; Hiraoka, Y.; Izumi, S.; Kramar, M.; Mischaikow, K.; Nanda, V., Topological measurement of protein compressibility via persistent diagrams, Japan J. Indust. Appl. Math., 32, 1-17 (2015) · Zbl 1320.55004
[8] de Silva, V.; Ghrist, R., Coverage in sensor networks via persistent homology, Algebr. Geom. Topol., 7, 339-358 (2007) · Zbl 1134.55003
[9] Allgower, E. L.; Georg, K., Introduction to Numerical Continuation Methods (2003), SIAM · Zbl 1036.65047
[10] (Krauskopf, B.; Osinga, H.; Galán-Vioque, J., Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems (2007), Springer) · Zbl 1117.65005
[11] Edelsbrunner, H.; Mücke, E., Three-dimensional alpha shapes, ACM Trans. Graph., 13, 1, 43-72 (1994) · Zbl 0806.68107
[13] Chazal, F.; de Silva, V.; Oudot, S., Persistence stability for geometric complexes, Geom. Dedicata, 173, 1, 193-214 (2014) · Zbl 1320.55003
[15] Bauer, U.; Kerber, M.; Reininghaus, J., Clear and compress: Computing Persistent homology in chunks, (In Topological Methods in Data Analysis and Visualization III: Theory, Algorithms, and Applications (2014), Springer), 103-117 · Zbl 1326.68299
[16] Federer, H., Geometric Measure Theory (1996), Springer-Verlag · Zbl 0874.49001
[17] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (2000), SIAM · Zbl 0949.65053
[18] Ben-Israel, A.; Greville, T., Generalized Inverses: Theory and Applications (2003), Springer · Zbl 1026.15004
[19] Horn, R. A.; Johnson, C. R., Matrix Analysis (1990), Cambridge University Press · Zbl 0704.15002
[20] Ben-Israel, A., A Newton-Raphson method for the solution of systems of equations, J. Math. Anal. Appl., 15, 243-252 (1966) · Zbl 0139.10301
[21] Ben-Israel, A., A modified Newton-Raphson method for the solution of systems of equations, Israel J. Math., 3, 94-98 (1965) · Zbl 0134.32603
[22] de Berg, M.; Cheong, O.; van Kreveld, M.; Overmars, M., Computational Geometry: Algorithms and Applications (2008), Springer · Zbl 1140.68069
[23] Bubenik, P., Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16, 77-102 (2015) · Zbl 1337.68221
[25] Mischaikow, K.; Nanda, V., Morse theory for filtrations and efficient computation of persistent homology, Discrete Comput. Geom., 50, 330-353 (2013) · Zbl 1278.57030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.