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Continuum Limit of Total Variation on Point Clouds

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Abstract

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when the cut capacity, and more generally total variation, on these graphs is a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. We obtain almost optimal conditions on the scaling, as the number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γ-convergence to hold. Taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds.

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References

  1. Agueh M.: Finsler structure in the p-Wasserstein space and gradient flows. C. R. Math. Acad. Sci. Paris. 350, 35–40 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ajtai M., Komlós J., Tusnády G.: On optimal matchings. Combinatorica. 4, 259–264 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alberti G., Bellettini G.: A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies. European J. Appl. Math. 9, 261–284 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambrosio L., Fusco N., Pallara D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000)

    Google Scholar 

  5. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: In: Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics, Birkhäuser Basel (2008)

  6. Andreev K., Racke H.: Balanced graph partitioning. Theory Comput. Syst. 39, 929–939 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arias-Castro E., Pelletier B., Pudlo P.: The normalized graph cut and Cheeger constant: from discrete to continuous. Adv. Appl. Probab. 44, 907–937 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arora S., Rao S., Vazirani U.: Expander flows, geometric embeddings and graph partitioning. J. ACM (JACM). 56, 5 (2009)

    Article  MathSciNet  Google Scholar 

  9. Baldi A.: Weighted BV functions. Houston J. Math. 27, 683–705 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Ball, J.M., Zarnescu, A.: Partial regularity and smooth topology-preserving approximations of rough domains. arXiv:1312.5156 (2013) (arXiv preprint)

  11. Belkin M., Niyogi P.: Convergence of Laplacian eigenmaps. Adv. Neural Inf. Process. Syst. (NIPS) 19, 129 (2007)

    Google Scholar 

  12. Belkin M., Niyogi P.: Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. Syst. Sci. 74, 1289–1308 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bertozzi A.L., Flenner A.: Diffuse interface models on graphs for classification of high dimensional data. Multiscale Model. Simul. 10, 1090–1118 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bourgain J., Brezis H., Mironescu P. et al.: Another look at sobolev spaces. In: Optimal control and partial differential equations, A volume in honour of A. Benssoussans 60th birthday, pp. 439455. IOS Press, Amsterdam (2001)

    Google Scholar 

  15. Boykov Y., Veksler O., Zabih R.: Fast approximate energy minimization via graph cuts. Pattern Anal. Mach. Intell. IEEE Trans. 23, 1222–1239 (2001)

    Article  Google Scholar 

  16. Braides A.: Gamma-Convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications Series. Oxford University Press. Incorporated, Oxford (2002)

    Google Scholar 

  17. Braides A., Yip N.K.: A quantitative description of mesh dependence for the discretization of singularly perturbed nonconvex problems. SIAM J. Numer. Anal. 50, 1883–1898 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bresson, X., Laurent, T.: Asymmetric cheeger cut and application to multi-class unsupervised clustering. CAM Report, pp. 1–8 (2012)

  19. Bresson, X., Laurent, T., Uminsky, D., von Brecht, J.: Multiclass total variation clustering. In: Burges, C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K. (eds.) Advances in Neural Information Processing Systems, vol. 26, pp. 1421–1429 (2013)

  20. Bresson, X., Laurent, T., Uminsky, D., von Brecht, J. H.: Convergence and energy landscape for cheeger cut clustering. In: P. L. Bartlett, F. C. N. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger, (eds.) Advances in Neural Information Processing Systems (NIPS), pp. 1394–1402 (2012)

  21. Bresson, X., Laurent, T., Uminsky, D., von Brecht, J. H.: An adaptive total variation algorithm for computing the balanced cut of a graph. arXiv preprint arXiv:1302.2717. (2013)

  22. Bresson, X., Tai, X.-C., Chan, T. F., Szlam, A.: Multi-class transductive learning based on l1 relaxations of cheeger cut and mumford-shah-potts model. UCLA CAM Rep. 12–03 (2012)

  23. Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young measures on topological spaces of Mathematics and its Applications, vol. 571, Kluwer Academic Publishers, Dordrecht, 2004

  24. Chambolle A., Giacomini A., Lussardi L.: Continuous limits of discrete perimeters. M2AN Math. Model. Numer. Anal. 44, 207–230 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dal Maso G.: An Introduction to Γ-convergence. Springer, New York (1993)

    Book  Google Scholar 

  26. Delling, D., Fleischman, D., Goldberg, A., Razenshteyn, I., Werneck, R.: An exact combinatorial algorithm for minimum graph bisection. Math. Program. 152(2), 417–458 (2014)

  27. Esedōglu S., Otto F.: Threshold dynamics for networks with arbitrary surface tensions. Commun. Pure Appl. Math. 68, 808–864 (2015)

    Article  MathSciNet  Google Scholar 

  28. Feige U., Krauthgamer R.: A polylogarithmic approximation of the minimum bisection. SIAM Rev. 48, 99–130 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. García Trillos, N., Slepčev, D.: On the rate of convergence of empirical measures in ∞-transportation distance. Can. J. Math.(2015) (online first)

  30. Giné, E., Koltchinskii, V.: Empirical graph Laplacian approximation of Laplace-Beltrami operators: large sample results. In: High dimensional probability, of IMS Lecture Notes Monogr. Ser., Inst. Math. Statist., Beachwood, OH, vol. 51, pp. 238–259 (2006)

  31. Gobbino M.: Finite difference approximation of the Mumford-Shah functional. Comm. Pure Appl. Math. 51, 197–228 (1998)

    Article  MathSciNet  Google Scholar 

  32. Gobbino M., Mora M.G.: Finite-difference approximation of free-discontinuity problems. Proc. Roy. Soc. Edinburgh Sect. A. 131, 567–595 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Goel A., Rai S., Krishnamachari B.: Sharp thresholds for monotone properties in random geometric graphs. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 580–586. ACM, New York (2004)

    Google Scholar 

  34. Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity in wireless networks. In: Stochastic analysis, control, optimization and applications, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, pp. 547–566 (1999)

  35. Hein, M., Audibert, J.-Y., Von Luxburg, U.: From graphs to manifolds–weak and strong pointwise consistency of graph Laplacians. In: Learning theory, pp. 470–485. Springer, New York, 2005

  36. Hein, M., Bühler, T.: An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. In: Advances in Neural Information Processing Systems (NIPS), pp. 847–855, 2010

  37. Hein, M., Setzer, S.: Beyond spectral clustering - tight relaxations of balanced graph cuts. In: Advances in Neural Information Processing Systems (NIPS) (Eds. J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Weinberger) pp. 2366–2374, 2011

  38. Leighton T., Shor P.: Tight bounds for minimax grid matching with applications to the average case analysis of algorithms. Combinatorica 9, 161–187 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  39. Leoni, G.: A first course in Sobolev spaces of Graduate Studies in Mathematics. vol. 105, American Mathematical Society, Providence, 2009

  40. Maier M., von Luxburg U., Hein M.: How the result of graph clustering methods depends on the construction of the graph. ESAIM Probab. Stat., 17, 370–418 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Merkurjev E., Kostić T., Bertozzi A.L.: An mbo scheme on graphs for classification and image processing. SIAM J. Imaging Sci. 6, 1903–1930 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Modica L., Mortola S.: Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. B (5) 14, 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

  43. Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26, 101–174 (2001)

    Article  MATH  Google Scholar 

  44. Pedregal, P.: Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and their Applications, 30, Birkhäuser Verlag, Basel (1997)

  45. Penrose M.: A strong law for the longest edge of the minimal spanning tree. Ann. Probab. 27, 246–260 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ponce A.C.: A new approach to Sobolev spaces and connections to Γ-convergence. Calc. Var. Partial Differ. Equ. 19, 229–255 (2004)

    Article  MathSciNet  Google Scholar 

  47. Rangapuram, S.S., Hein, M.: Constrained 1-spectral clustering. In: International conference on Artificial Intelligence and Statistics (AISTATS), pp. 1143–1151, 2012

  48. Savin O., Valdinoci E.: Γ-convergence for nonlocal phase transitions. Ann. Inst. H. Poincaré Anal. Non Linéaire. 29, 479–500 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Shi J., Malik J.: Normalized cuts and image segmentation. Pattern Anal. Mach. Intell. IEEE Trans. 22, 888–905 (2000)

    Article  Google Scholar 

  50. Shor P.W., Yukich J.E.: Minimax grid matching and empirical measures. Ann. Probab. 19, 1338–1348 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  51. Singer A.: From graph to manifold Laplacian: the convergence rate. Appl. Comput. Harmon. Anal. 21, 128–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  52. Szlam, A., Bresson, X.: A total variation-based graph clustering algorithm for cheeger ratio cuts. UCLA CAM Report, pp. 1–12 (2009)

  53. Szlam, A., Bresson, X.: Total variation and cheeger cuts., In: ICML (Eds. J. Frnkranz and T. Joachims) Omnipress pp. 1039–1046 (2010)

  54. Talagrand M.: The transportation cost from the uniform measure to the empirical measure in dimension ≥  3. Ann. Probab. 22, 919–959 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  55. Talagrand, M.: The generic chaining. Springer Monographs in Mathematics Springer-Verlag, Berlin, 2005

  56. Talagrand, M.: Upper and lower bounds of stochastic processes of Modern Surveys in Mathematics. vol. 60. Springer-Verlag, Berlin Heidelberg, 2014

  57. Talagrand M., Yukich J.E.: The integrability of the square exponential transportation cost. Ann. Appl. Probab. 3, 1100–1111 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  58. Ting, D., Huang, L., Jordan, M.I.: An analysis of the convergence of graph Laplacians. In: Proceedings of the 27th International Conference on Machine Learning, 2010

  59. van Gennip Y., Bertozzi A.L.: Γ-convergence of graph Ginzburg-Landau functionals. Adv. Differential Equations 17, 1115–1180 (2012)

    MathSciNet  MATH  Google Scholar 

  60. Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics. American Mathematical Society, Providence, US (2003)

  61. von Luxburg U., Belkin M., Bousquet O.: Consistency of spectral clustering. Ann. Statist. 36, 555–586 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  62. Weyl H.: On the Volume of Tubes. Amer. J. Math. 61, 461–472 (1939)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Nicolás García Trillos & Dejan Slepčev

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  1. Nicolás García Trillos
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  2. Dejan Slepčev
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Correspondence to Dejan Slepčev.

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Communicated by F. Otto

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García Trillos, N., Slepčev, D. Continuum Limit of Total Variation on Point Clouds. Arch Rational Mech Anal 220, 193–241 (2016). https://doi.org/10.1007/s00205-015-0929-z

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