Skip to main content Accessibility help
×
Home

Asymptotic analysis of the Ginzburg–Landau functional on point clouds

  • Matthew Thorpe (a1) and Florian Theil (a1)

Abstract

The Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψn where n denotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.

Copyright

Footnotes

Hide All
*

Present address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, email: m.thorpe@maths.cam.ac.uk

Footnotes

References

Hide All
1Alberti, G. and Bellettini, G.. A nonlocal anisotropic model for phase transitions: asymptotic behaviour of rescaled energies. Eur. J. Appl. Math. Eur J Appl Math 3 (1998), 527560.
2Ambrosio, L. and Pratelli, A.. Existence and stability results in the L 1 theory of optimal transportation. In Optimal Transportation and Applications, volume 1813 of Lecture Notes in Mathematics, Caffarelli, L. A. and Salsa, S. (eds), pp. 123160 (Berlin, Heidelberg: Springer, 2003).
3Ando, R. K. and Zhang, T.. Learning on graph with Laplacian regularization. Advances in neural information processing systems 19 (2007), 2532.
4Attouch, H., Buttazzo, G. and Michaille, G.. Variational Analysis in Sobolev and BV Spaces: Applications to PDE's and Optimization. MPS-SIAM Series on Optimization (Philadelphia: Society for Industrial and Applied Mathematics, 2006).
5Baldi, A.. Weighted BV functions. Houst. J. Math. 27 (2001), 683705.
6Bertozzi, A. L. and Flenner, A.. Diffuse interface models on graphs for classification of high dimensional data. Multiscale. Model. Simul. 10 (2012), 10901118.
7Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. and Hwang, D.-U.. Complex networks: structure and dynamics. Phys. Rep. 424 (2006), 175308.
8Braides, A.. Γ-Convergence for Beginners (New York: Oxford University Press, 2002).
9Bresson, X., Laurent, T., Uminsky, D. and von Brecht, J.. Convergence and energy landscape for cheeger cut clustering. In Advances in Neural Information Processing Systems 25, Bartlett, P., Pereira, F. C. N., Burges, C. J. C., Bottou, L. and Weinberger, K. Q. (eds), pp. 13851393 (Lake Tahoe: Curran Associates, Inc., 2012).
10Bresson, X., Laurent, T., Uminsky, D. and von Brecht, J.. An adaptive total variation algorithm for computing the balanced cut of a graph. arXiv:1302.2717, (2013).
11Bridle, N. and Zhu, X.. p-voltages: Laplacian regularization for semi-supervised learning on high-dimensional data. In Eleventh Workshop on Mining and Learning with Graphs (MLG2013) (2013).
12Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A. and Wiener, J.. Graph structure in the web. Comput. Netw. 33 (2000), 309320.
13Bühler, T. and Hein, M.. Spectral clustering based on the graph p-Laplacian. In Proceedings of the 26th Annual International Conference on Machine Learning, pp. 8188, (2009).
14Caldarelli, G.. Scale Free Networks: Complex Webs in Nature and Technology (New York: Oxford University Press, 2007).
15Chambolle, A. and Pock, T.. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging. Vis. 40 (2011), 120145.
16Champion, T., De Pascale, L. and Juutinen, P.. The ∞-Wasserstein distance: Local solutions and existence of optimal transport maps. SIAM J. Math. Anal. 40 (2008), 120.
17Dal Maso, G.. An Introduction to Γ-Convergence (New York: Springer, 1993).
18Danon, L., Diaz-Guilera, A., Duch, J. and Arenas, A.. Comparing community structure identification. J. Stat. Mech: Theory Exp. 2005 (2005).
19El Alaoui, A., Cheng, X., Ramdas, A., Wainwright, M. J. and Jordan, M. I.. Asymptotic behavior of p-based Laplacian regularization in semi-supervised learning. In 29th Annual Conference on Learning Theory, pp. 879906 (2016).
20Evans, L. C. and Gariepy, R. F.. Measure Theory and Fine Properties of Functions (Boca Raton: CRC Press, 1992).
21Faloutsos, M., Faloutsos, P. and Faloutsos, C.. On power-law relationships of the internet topology. In ACM SIGCOMM Computer Communication Review, vol. 29, pp. 251262 (1999).
22Fortunato, S.. Community detection in graphs. Phys. Rep. 3–5 (2010), 75174.
23Gangbo, W. and J. McCann, R.. The geometry of optimal transportation. Acta Math. 177 (1996), 113161.
24Garcia-Cardona, C., Flenner, A. and Percus, A. G.. Multiclass semi-supervised learning on graphs using Ginzburg–Landau functional minimization. In Pattern Recognition Applications and Methods, pp. 119135 (Lisbon, Springer, 2015).
25García Trillos, N. and Slepčev, D.. Continuum limit of Total Variation on point clouds. Arch. Ration. Mech. Anal. 149, 2015).
26García Trillos, N. and Slepčev, D.. On the rate of convergence of empirical measures in ∞-transportation distance. Can. J. Math. 67 (2015), 13581383.
27García Trillos, N., Slepčev, D., von Brecht, J., Laurent, T. and Bresson, X.. Consistency of cheeger and ratio graph cuts. arXiv:1411.6590 (2014).
28Hennawi, J. F and Prochaska, J. X. Quasars probing quasars. II. The anisotropic clustering of optically thick absorbers around quasars. Astrophys. J. 655 (2007).
29Hu, H., van Gennip, Y., Hunter, B., Bertozzi, A. L. and Porter, M. A.. Multislice modularity optimization in community detection and image segmentation. In Data Mining Workshops (ICDMW), 2012 IEEE 12th International Conference on, pp. 934936, (2012).
30Jylhä, H.. The L optimal transport: infinite cyclical monotonicity and the existence of optimal transport maps. Calc. Var. Partial. Differ. Equ. 52 (2015), 303326.
31Kulis, B. and Jordan, M. I.. Revisiting k-means: New algorithms via Bayesian nonparametrics. In Proceedings of the 29th International Conference on Machine Learning (ICML-12) Langford, J. and Pineau, J.. (eds), pp. 513520. (Edinburgh, Omnipress, 2012).
32Leoni, G.. A First Course in Sobolev Spaces, vol. 105 (Providence: American Mathematical Society, 2009).
33Li, X.-Z., Wang, J.-F., Yang, W.-Z., Li, Z.-J. and Lai, S.-J.. A spatial scan statistic for nonisotropic two-level risk cluster. Stat. Med. 31 (2012), 177187.
34Linder, E. V., Oh, M., Okumura, T., G. Sabiu, C. and Song, Y.-S.. Cosmological constraints from the anisotropic clustering analysis using BOSS DR9. Phys. Rev. D 89 (2014), 063525
35MacKay, D. J. C.. Information Theory, Inference & Learning Algorithms (Cambridge: Cambridge University Press, 2002).
36Maz'ya, V.. Sobolev Spaces: With Applications to Elliptic Partial Differential Equations (Berlin: Springer, 2010).
37Modica, L.. The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98 (1987), 123142.
38Penrose, M.. Random Geometric Graphs (New York: Oxford University Press, 2003).
39Porter, M. A., Onnela, J.-P. and Mucha, P. J.. Communities in networks. Notices of the AMS (2009).
40Strogatz, S. H.. Exploring complex networks. Nature 410(6825) (2001), 268276.
41Thorpe, M., Theil, F., Johansen, A. M. and Cade, N.. Convergence of the k-means minimization problem using Γ-convergence. SIAM J. Appl. Math. 75 (2015), 24442474.
42Van Gennip, Y. and Bertozzi, A. L.. Γ-convergence of graph Ginzburg–Landau functionals. Adv. Differ. Equ. 17(11–12) (2012), 11151180.
43van Gennip, Y., Hunter, B., Ahn, R., Elliott, P., Luh, K., Halvorson, M., Reid, S., Valasik, M., Wo, J., Tita, G., Bertozzi, A. and Brantingham, P.. Community detection using spectral clustering on sparse geosocial data. SIAM J. Appl. Math. 73 (2013), 6783.
44van Gennip, Y., Guillen, N., Osting, B. and Bertozzi, A.. Mean curvature, threshold dynamics, and phase field theory on finite graphs. Milan J. Math. 82 (2014), 365.
45Villani, C.. Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society, cop., (2003).
46Wasserman, S. and Faust, K.. Social Network Analysis: Methods and Applications (Cambridge: Cambridge University Press, 1994).
47Worsley, K. J., Andermann, M., Koulis, T., MacDonald, D. and Evans, A. C.. Detecting changes in nonisotropic images. Hum. Brain. Mapp. 8 (1999), 98101.
48Zhou, X. and Belkin, M.. Semi-supervised learning by higher order regularization. In International Conference on Artificial Intelligence and Statistics, pp. 892900 (2011).
49Zhou, D. and Schölkopf, B.. Pattern Recognition: 27th DAGM Symposium, Vienna, Austria, 31 August–2 September 2005. Proceedings, chapter Regularization on Discrete Spaces, pp. 361368. (Berlin Heidelberg: Springer, 2005).
50Zhou, X. and Srebro, N.. Error analysis of Laplacian eigenmaps for semi-supervised learning. In AISTATS, pp. 901908, (2011).

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed