## The Annals of Applied Probability

### Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

#### Abstract

We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 2814-2854.

Dates
First available in Project Euclid: 28 August 2018

https://projecteuclid.org/euclid.aoap/1535443235

Digital Object Identifier
doi:10.1214/17-AAP1375

Mathematical Reviews number (MathSciNet)
MR3847974

Zentralblatt MATH identifier
06974766

#### Citation

Owada, Takashi. Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes. Ann. Appl. Probab. 28 (2018), no. 5, 2814--2854. doi:10.1214/17-AAP1375. https://projecteuclid.org/euclid.aoap/1535443235

#### References

• [1] Adler, R. J., Bobrowski, O., Borman, M. S., Subag, E. and Weinberger, S. (2010). Persistent homology for random fields and complexes. In Borrowing Strength: Theory Powering Applications—a Festschrift for Lawrence D. Brown. Inst. Math. Stat. (IMS) Collect. 6 124–143. IMS, Beachwood, OH.
• [2] Adler, R. J., Bobrowski, O. and Weinberger, S. (2014). Crackle: The homology of noise. Discrete Comput. Geom. 52 680–704.
• [3] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
• [4] Balkema, G. and Embrechts, P. (2007). High Risk Scenarios and Extremes: A Geometric Approach. European Mathematical Society (EMS), Zürich.
• [5] Balkema, G., Embrechts, P. and Nolde, N. (2010). Meta densities and the shape of their sample clouds. J. Multivariate Anal. 101 1738–1754.
• [6] Balkema, G., Embrechts, P. and Nolde, N. (2013). The shape of asymptotic dependence. In Prokhorov and Contemporary Probability Theory. Springer Proc. Math. Stat. 33 43–67. Springer, Heidelberg.
• [7] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• [8] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• [9] Björner, A. (1995). Topological methods. In Handbook of Combinatorics, Vols 1, 2 1819–1872. Elsevier, Amsterdam.
• [10] Bobrowski, O. and 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.
• [11] Bobrowski, O., Kahle, M. and Skraba, P. (2017). Maximally persistent cycles in random geometric complexes. Ann. Appl. Probab. 27 2032–2060.
• [12] Bobrowski, O. and Mukherjee, S. (2015). The topology of probability distributions on manifolds. Probab. Theory Related Fields 161 651–686.
• [13] Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16 77–102.
• [14] Carlsson, G. (2014). Topological pattern recognition for point cloud data. Acta Numer. 23 289–368.
• [15] Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.
• [16] Dabaghian, Y., Memoli, F., Frank, L. and Carlsson, G. (2012). A topological paradigm for hippocampal spatial map formation using persistent homology. PLoS Comput. Biol. 8 e1002581.
• [17] Dabrowski, A. R., Dehling, H. G., Mikosch, T. and Sharipov, O. (2002). Poisson limits for $U$-statistics. Stochastic Process. Appl. 99 137–157.
• [18] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
• [19] de Silva, V. and Ghrist, R. (2007). Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7 339–358.
• [20] Decreusefond, L., Schulte, M. and Thäle, C. (2016). Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry. Ann. Probab. 44 2147–2197.
• [21] Duy, T. K., Hiraoka, Y. and Shirai, T. (2016). Limit theorems for persistence diagrams. Available at arXiv:1612.08371.
• [22] Edelsbrunner, H. and Harer, J. L. (2010). Computational Topology: An Introduction. Amer. Math. Soc., Providence, RI.
• [23] Edelsbrunner, H., Letscher, D. and Zomorodian, A. (2002). Topological persistence and simplification. Discrete Comput. Geom. 28 511–533.
• [24] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
• [25] Fasy, B. T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S. and Singh, A. (2014). Confidence sets for persistence diagrams. Ann. Statist. 42 2301–2339.
• [26] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed. Krieger, Melbourne, FL.
• [27] Ghrist, R. (2008). Barcodes: The persistent topology of data. Bull. Amer. Math. Soc. (N.S.) 45 61–75.
• [28] Ghrist, R. (2014). Elementary Applied Topology, Createspace.
• [29] Hatcher, A. (2002). Algebraic Topology. Cambridge Univ. Press, Cambridge.
• [30] 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.
• [31] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45 553–573.
• [32] Kahle, M. and Meckes, E. (2013). Limit theorems for Betti numbers of random simplicial complexes. Homology, Homotopy Appl. 15 343–374.
• [33] Kahle, M. and Meckes, E. (2016). Erratum to “Limit theorems for Betti numbers of random simplicial complexes.” Homology, Homotopy Appl. 18 129–142.
• [34] Kusano, G., Fukumizu, K. and Hiraoka, Y. (2016). Persistence weighted Gaussian kernel for topological data analysis. In ICML’16 Proceedings of the 33rd International Conference on International Conference on Machine Learning, Vol. 48, pp. 2004–2013.
• [35] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York–Berlin.
• [36] Martin, S., Thompson, A., Coutsias, E. A. and Watson, J. (2010). Topology of cyclo-octane energy landscape. J. Chem. Phys. 132 234115.
• [37] Munkres, J. R. (1984). Elements of Algebraic Topology, 1st ed. Benjamin-Cummings, San Francisco, CA.
• [38] Niyogi, P., Smale, S. and Weinberger, S. (2008). Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39 419–441.
• [39] Owada, T. (2017). Functional central limit theorem for subgraph counting processes. Electron. J. Probab. 22 Paper No. 17, 38.
• [40] Owada, T. and Adler, R. J. (2017). Limit theorems for point processes under geometric constraints (and topological crackle). Ann. Probab. 45 2004–2055.
• [41] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
• [42] Port, A., Gheorghita, I., Guth, D., Clark, J. M., Liang, C., Dasu, S. and Marcolli, M. (2015). Persistent topology of syntax. Available at arXiv:1507.05134.
• [43] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
• [44] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
• [45] Schulte, M. and Thäle, C. (2012). The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stochastic Process. Appl. 122 4096–4120.
• [46] Vick, J. W. (1994). Homology Theory: An Introduction to Algebraic Topology, 2nd ed. Graduate Texts in Mathematics 145. Springer, New York.
• [47] Yogeshwaran, D. and Adler, R. J. (2015). On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 3338–3380.
• [48] Yogeshwaran, D., Subag, E. and Adler, R. J. (2017). Random geometric complexes in the thermodynamic regime. Probab. Theory Related Fields 167 107–142.
• [49] Zomorodian, A. and Carlsson, G. (2005). Computing persistent homology. Discrete Comput. Geom. 33 249–274.