The Annals of Applied Probability

Limit theorems for persistence diagrams

Yasuaki Hiraoka, Tomoyuki Shirai, and Khanh Duy Trinh

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Abstract

The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 2740-2780.

Dates
Received: December 2016
Revised: August 2017
First available in Project Euclid: 28 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1535443233

Digital Object Identifier
doi:10.1214/17-AAP1371

Mathematical Reviews number (MathSciNet)
MR3847972

Zentralblatt MATH identifier
06974764

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B10: Convergence of probability measures
Secondary: 55N20: Generalized (extraordinary) homology and cohomology theories

Keywords
Point process persistence diagram persistent Betti number random topology

Citation

Hiraoka, Yasuaki; Shirai, Tomoyuki; Trinh, Khanh Duy. Limit theorems for persistence diagrams. Ann. Appl. Probab. 28 (2018), no. 5, 2740--2780. doi:10.1214/17-AAP1371. https://projecteuclid.org/euclid.aoap/1535443233


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References

  • [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York.
  • [2] Błaszczyszyn, B. and Yogeshwaran, D. (2015). Clustering comparison of point processes, with applications to random geometric models. In Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Math. 2120 31–71. Springer, Cham.
  • [3] Bobrowski, O. and Kahle, M. (2014). Topology of random simplicial complexes: A survey. Topology in Statistical Inference, the Proceedings of Symposia in Applied Mathematics. To appear. Available at arXiv:1409.4734.
  • [4] Carlsson, G. (2009). Topology and data. Bull. Amer. Math. Soc. (N.S.) 46 255–308.
  • [5] Chazal, F., de Silva, V., Glisse, M. and Oudot, S. (2016). The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, Cham.
  • [6] Chazal, F., de Silva, V. and Oudot, S. (2014). Persistence stability for geometric complexes. Geom. Dedicata 173 193–214.
  • [7] Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd ed. Wiley Series in Probability and Statistics. Wiley, Chichester.
  • [8] Cohen-Steiner, D., Edelsbrunner, H. and Harer, J. (2007). Stability of persistence diagrams. Discrete Comput. Geom. 37 103–120.
  • [9] Edelsbrunner, H. and Harer, J. L. (2010). Computational Topology. Amer. Math. Soc., Providence, RI.
  • [10] Edelsbrunner, H., Letscher, D. and Zomorodian, A. (2002). Topological persistence and simplification. Discrete Comput. Geom. 28 511–533.
  • [11] Gameiro, M., Hiraoka, Y., Izumi, S., Kramar, M., Mischaikow, K. and Nanda, V. (2015). A topological measurement of protein compressibility. Jpn. J. Ind. Appl. Math. 32 1–17.
  • [12] Ghosh, S. (2012). Rigidity and Tolerance in Gaussian zeroes and Ginibre eigenvalues: Quantitative estimates. Preprint. Available at arXiv:1211.3506.
  • [13] Ghosh, S. and Peres, Y. (2017). Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues. Duke Math. J. 166 1789–1858.
  • [14] Greschonig, G. and Schmidt, K. (2000). Ergodic decomposition of quasi-invariant probability measures. Colloq. Math. 84/85 495–514.
  • [15] Hatcher, A. (2002). Algebraic Topology. Cambridge Univ. Press, Cambridge.
  • [16] Hiraoka, Y., Nakamura, T., Hirata, A., Escolar, E. G., Matsue, K. and Nishiura, Y. (2016). Hierarchical structures of amorphous solids characterized by persistent homology. Proc. Natl. Acad. Sci. USA 113 7035–7040.
  • [17] Hiraoka, Y. and Shirai, T. (2016). Tutte polynomials and random-cluster models in Bernoulli cell complexes. In Stochastic Analysis on Large Scale Interacting Systems. RIMS Kôkyûroku Bessatsu, B59 289–304. Res. Inst. Math. Sci. (RIMS), Kyoto.
  • [18] Hiraoka, Y. and Shirai, T. (2017). Minimum spanning acycle and lifetime of persistent homology in the Linial–Meshulam process. Random Structures Algorithms 51 315–340.
  • [19] Hiraoka, Y. and Tsunoda, K. (2016). Limit theorems for random cubical homology. Preprint. Available at arXiv:1612.08485.
  • [20] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51. Amer. Math. Soc., Providence, RI.
  • [21] Kaczynski, T., Mischaikow, K. and Mrozek, M. (2004). Computational Homology. Applied Mathematical Sciences 157. Springer, New York.
  • [22] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45 553–573.
  • [23] Kahle, M. (2014). Topology of random simplicial complexes: A survey. In Algebraic Topology: Applications and New Directions. Contemp. Math. 620 201–221. Amer. Math. Soc., Providence, RI.
  • [24] Kallenberg, O. (1986). Random Measures, 4th ed. Akademie-Verlag, Berlin.
  • [25] Karamchandani, N., Manjunath, D., Yogeshwaran, D. and Iyer, S. K. (2006). Evolving random geometric graph models for mobile wireless networks. In 2006 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks 1–7. IEEE.
  • [26] Kusano, G., Fukumizu, K. and Hiraoka, Y. (2016). Persistence weighted Gaussian kernel for topological data analysis. In Proceedings of the 33rd International Conference on Machine Learning 2004–2013.
  • [27] Miyoshi, N. and Shirai, T. (2014). A cellular network model with Ginibre configured base stations. Adv. in Appl. Probab. 46 832–845.
  • [28] Nakamura, T., Hiraoka, Y., Hirata, A., Escolar, E. G. and Nishiura, Y. (2015). Persistent homology and many-body atomic structure for medium-range order in the glass. Nanotechnol. 26 304001.
  • [29] Owada, T. and Adler, R. J. (2017). Limit theorems for point processes under geometric constraints (and topological crackle). Ann. Probab. 45 2004–2055. DOI:10.1214/16-AOP1106.
  • [30] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
  • [31] Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005–1041.
  • [32] Pugh, C. and Shub, M. (1971). Ergodic elements of ergodic actions. Compos. Math. 23 115–122.
  • [33] Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York.
  • [34] Riley, S., Eames, K., Isham, V., Mollison, D. and Trapman, P. (2015). Five challenges for spatial epidemic models. Epidemics 10 68–71.
  • [35] Saadatfar, M., Takeuchi, H., Robins, V., Francois, N. and Hiraoka, Y. (2017). Pore configuration landscape of granular crystallization. Nat. Commun. 8 15082.
  • [36] Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Funct. Anal. 205 414–463. DOI:10.1016/S0022-1236(03)00171-X.
  • [37] Trinh, K. D. (2017). A remark on the convergence of Betti numbers in the thermodynamic regime. Pac. J. Math. Ind. 9 Art. 4, 7. DOI:10.1186/s40736-017-0029-0.
  • [38] Yogeshwaran, D. and Adler, R. J. (2015). On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 3338–3380.
  • [39] Yogeshwaran, D., Subag, E. and Adler, R. J. (2017). Random geometric complexes in the thermodynamic regime. Probab. Theory Related Fields 167 107–142.
  • [40] Zomorodian, A. and Carlsson, G. (2005). Computing persistent homology. Discrete Comput. Geom. 33 249–274.