Stochastic Convergence of Persistence Landscapes and Silhouettes

ABSTRACT

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landscape, introduced by Bubenik, which converts a diagram into a well-behaved real-valued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory.

References

1. S. Balakrishnan, B. Fasy, F. Lecci, A. Rinaldo, A. Singh, and L. Wasserman. Statistical inference for persistent homology, 2013. arXiv 1303.7117.
2. P. Bubenik. Statistical topology using persistence landscapes, 2012. arXiv 1207.6437.
3. F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules, July 2012. arXiv 1207.3674.
4. F. Chazal, V. de Silva, and S. Oudot. Persistence stability for geometric complexes. Geom. Dedicata, Dec. 2013.
5. F. Chazal, B. T. Fasy, F. Lecci, A. Rinaldo, A. Singh, and L. Wasserman. On the bootstrap for persistence diagrams and landscapes, 2013. arXiv 1311.0376.
6. F. Chazal, C. Labruère, M. Glisse, and B. Michel. Optimal rates of convergence for persistence diagrams in topological data analysis. arXiv 1305.6239, 2013.
7. V. Chernozhukov, D. Chetverikov, and K. Kato. Gaussian approximation of suprema of empirical processes, 2012. arXiv 1212.6885.
8. V. Chernozhukov, D. Chetverikov, and K. Kato. Anti-concentration and honest adaptive confidence bands, 2013. arXiv 1303.7152.
9. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete Comput. Geom., 37(1):103--120, 2007.
10. P. Diaconis, S. Holmes, and M. Shahshahani. Sampling from a manifold, 2012. arXiv 1206.6913.
11. H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Disc. Comput. Geom., 28(4):511--533, July 2002.
12. B. Efron. Bootstrap methods: another look at the jackknife. The Annals of Statistics, pages 1--26, 1979.
13. B. Efron and R. Tibshirani. An introduction to the bootstrap, volume 57. CRC press, 1993.
14. E. Munch, P. Bendich, K. Turner, S. Mukherjee, J. Mattingly, and J. Harer. Probabilistic Fréchet means and statistics on vineyards, 2013. arXiv 1307.6530.
15. M. Talagrand. Sharper bounds for Gaussian and empirical processes. The Annals of Probability, 22(1):28--76, 1994.
16. K. Turner, Y. Mileyko, S. Mukherjee, and J. Harer. Fréchet means for distributions of persistence diagrams, 2012. arXiv 1206.2790.
17. A. van der Vaart. Asymptotic Statistics, volume 3. Cambridge UP, 2000.
18. A. van der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Verlag, 1996.
19. A. Zomorodian and G. Carlsson. Computing persistent homology. Disc. Comp. Geom., 33(2):249--274, 2005.

Index Terms

• Mathematics of computing

  • Discrete mathematics