Abstract
Persistent homology captures the topology of a filtration—a one-parameter family of increasing spaces—in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.
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Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)
Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11(2), 149–187 (2005)
Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76(1), 1–12 (2008)
Chazal, F., Lieutier, A.: Weak feature size and persistent homology: computing homology of solids in ℝn from noisy data samples. In: Proceedings of ACM Symposium on Computational Geometry, pp. 255–262 (2005)
Cohen, D.C., Orlik, P.: Gauss–Manin connections for arrangements I. Eigenvalues. Compos. Math. 136(3), 299–316 (2003)
Collins, A., Zomorodian, A., Carlsson, G., Guibas, L.: A barcode shape descriptor for curve point cloud data. Comput. Graph. 28, 881–894 (2004)
de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Proceedings of IEEE/Eurographics Symposium on Point-Based Graphics, pp. 157–166 (2004)
de Silva, V., Ghrist, R., Muhammad, A.: Blind swarms for coverage in 2-D. In: Proceedings of Robotics: Science and Systems. http://www.roboticsproceedings.org/rss01/ (2005)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)
Frosini, P., Mulazzani, M.: Size homotopy groups for computation of natural size distances. Bull. Belg. Math. Soc. Simon Stevin 6(3), 455–464 (1999)
Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. New Ser. 45(1), 61–75 (2008)
Gromov, M.: Hyperbolic groups. In: Gersten, S. (ed.) Essays in Group Theory, pp. 75–263. Springer, New York (1987)
Gyulassy, A., Natarajan, V., Pascucci, V., Bremer, P.T., Hamann, B.: Topology-based simplification for feature extraction from 3D scalar fields. In: Proceedings of IEEE Visualization, pp. 275–280 (2005)
Knudson, K.P.: A refinement of multi-dimensional persistence. Homotopy Homol. Appl. 10, 259–281 (2008)
Lee, A., Mumford, D., Pedersen, K.: The nonlinear statistics of high-contrast patches in natural images. Int. J. Comput. Vis. 54(1–3), 83–103 (2003)
Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34. Springer, Berlin (1994)
Serre, J.-P.: Local Algebra. Springer, Berlin (2000)
Terao, H.: Moduli space of combinatorially equivalent arrangements of hyperplanes and logarithmic Gauss–Manin connections. Topol. Appl. 118(1–2), 255–274 (2002)
Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)
Zomorodian, A.: Computational topology. In: Atallah, M., Blanton, M. (eds.) Algorithms and Theory of Computation Handbook, 2nd edn. CRC, Boca Raton (2009) (in press)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)
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The first author was partially supported by NSF under grant DMS-0354543. The second author was partially supported by DARPA under grant HR 0011-06-1-0038 and by ONR under grant N 00014-08-1-0908. Both authors were partially supported by DARPA under grant HR 0011-05-1-0007.
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Carlsson, G., Zomorodian, A. The Theory of Multidimensional Persistence. Discrete Comput Geom 42, 71–93 (2009). https://doi.org/10.1007/s00454-009-9176-0
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DOI: https://doi.org/10.1007/s00454-009-9176-0