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Geometric Inference for Probability Measures


Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset from the approximating point cloud data. It appears that the study of distance functions allows one to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers or with background noise. In this paper, we show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we introduce a notion of distance function to a probability distribution in ℝd. These functions share many properties with classical distance functions, which make them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, we show that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers. Moreover, in settings where empirical measures are considered, these functions can be easily evaluated, making them of particular practical interest.

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  1. N. Amenta, S. Choi, T.K. Dey, N. Leekha, A simple algorithm for homeomorphic surface reconstruction, Int. J. Comput. Geom. Appl. 12(1–2), 125–141 (2002).

    MathSciNet  MATH  Article  Google Scholar 

  2. F. Bolley, A. Guillin, C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Relat. 137(3), 541–593 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  3. F. Chazal, A. Lieutier, Stability and computation of topological invariants of solids in ℝn, Discrete Comput. Geom. 37(4), 601–617 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  4. F. Chazal, A. Lieutier, Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees, Comput. Geom. Theor. Appl. 40(2), 156–170 (2008).

    MathSciNet  MATH  Google Scholar 

  5. F. Chazal, S.Y. Oudot, Towards persistence-based reconstruction in Euclidean spaces, in Proc. 24th ACM Sympos. Comput. Geom. (2008), pp. 232–241.

    Google Scholar 

  6. F. Chazal, D. Cohen-Steiner, A. Lieutier, B. Thibert, Stability of curvature measures, Comput. Graph. Forum 28, 1485–1496 (2008) (proc. SGP 2009).

    Article  Google Scholar 

  7. F. Chazal, D. Cohen-Steiner, A. Lieutier, A sampling theory for compact sets in Euclidean space, Discrete Comput. Geom. 41(3), 461–479 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  8. F. Chazal, D. Cohen-Steiner, A. Lieutier, Normal cone approximation and offset shape isotopy, Comput. Geom. Theor. Appl. 42(6-7), 566–581 (2009).

    MathSciNet  MATH  Google Scholar 

  9. F. Chazal, D. Cohen-Steiner, Q. Mérigot, Boundary measures for geometric inference, Found. Comput. Math. 10, 221–240 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  10. F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983).

    MATH  Google Scholar 

  11. D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37(1), 103–120 (2007).

    MathSciNet  MATH  Article  Google Scholar 

  12. V. de Silva, G. Carlsson, Topological estimation using witness complexes, in Symposium on Point-Based Graphics, ETH, Zürich, Switzerland (2004).

    Google Scholar 

  13. H. Edelsbrunner, The union of balls and its dual shape, Discrete Comput. Geom. 13, 415–440 (1995).

    MathSciNet  MATH  Article  Google Scholar 

  14. H. Edelsbrunner, J. Harer, Computational Topology. An Introduction (American Mathematical Society, Providence, 2010).

    MATH  Google Scholar 

  15. H. Federer, Curvature measures, Trans. Am. Math. Soc. 93, 418–491 (1959).

    MathSciNet  MATH  Article  Google Scholar 

  16. S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry (Springer, Berlin, 1990).

    MATH  Google Scholar 

  17. K. Grove, Critical point theory for distance functions, in Proc. of Symposia in Pure Mathematics, vol. 54 (1993).

    Google Scholar 

  18. A. Lieutier, Any open bounded subset of ℝn has the same homotopy type as its medial axis, Comput. Aided Geom. Des. 36(11), 1029–1046 (2004).

    Google Scholar 

  19. Q. Mérigot, M. Ovsjanikov, L. Guibas, Robust Voronoi-based curvature and feature estimation, in Proc. SIAM/ACM Joint Conference on Geom. and Phys. Modeling (2009), pp. 1–12.

    Chapter  Google Scholar 

  20. P. Niyogi, S. Smale, S. Weinberger, A topological view of unsupervised learning from noisy data. Preprint (2008).

  21. P. Niyogi, S. Smale, S. Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. 39(1), 419–441 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  22. S. Peleg, M. Werman, H. Rom, A unified approach to the change of resolution: space and gray-level, IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 739–742 (1989).

    Article  Google Scholar 

  23. A. Petrunin, Semiconcave functions in Alexandrov’s geometry, in Surveys in differential geometry, vol. XI (International Press, Somerville, 2007), pp. 137–201.

    Google Scholar 

  24. V. Robins, Towards computing homology from finite approximations, Topol. Proc. 24, 503–532 (1999).

    MathSciNet  MATH  Google Scholar 

  25. Y. Rubner, C. Tomasi, L.J. Guibas, The earth mover’s distance as a metric for image retrieval, Int. J. Comput. Vis. 40(2), 99–121 (2000).

    MATH  Article  Google Scholar 

  26. C. Villani, Topics in Optimal Transportation (American Mathematical Society, Providence, 2003).

    MATH  Google Scholar 

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Correspondence to Frédéric Chazal.

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Communicated by Konstantin Mischaikow.

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Chazal, F., Cohen-Steiner, D. & Mérigot, Q. Geometric Inference for Probability Measures. Found Comput Math 11, 733–751 (2011).

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  • Geometric inference
  • Computational topology
  • Optimal transportation
  • Nearest neighbor
  • Surface reconstruction

Mathematics Subject Classification (2000)

  • 62G05
  • 62-07
  • 28A33