Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T01:44:41.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic analysis of the Ginzburg–Landau functional on point clouds

Part of: Existence theories Nonparametric inference

Published online by Cambridge University Press:  27 December 2018

Matthew Thorpe  and
Florian Theil
Matthew Thorpe
Affiliation:
University of Warwick, Coventry, CV4 7AL, UK ([email protected])
Florian Theil
Affiliation:
University of Warwick, Coventry, CV4 7AL, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Ginzburg-Landau FunctionalPhase TransitionsPDEs on GraphsTotal Variation on GraphsLarge Data AsymptoticsΓ-Convergence

MSC classification

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 62G20: Asymptotic properties
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 387 - 427
Copyright
Copyright © Royal Society of Edinburgh 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Present address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, email: [email protected]

References

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), 527–560.Google Scholar
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. 123–160 (Berlin, Heidelberg: Springer, 2003).Google Scholar
3Ando, R. K. and Zhang, T.. Learning on graph with Laplacian regularization. Advances in neural information processing systems 19 (2007), 25–32.Google Scholar
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).Google Scholar
5Baldi, A.. Weighted BV functions. Houst. J. Math. 27 (2001), 683–705.Google Scholar
6Bertozzi, A. L. and Flenner, A.. Diffuse interface models on graphs for classification of high dimensional data. Multiscale. Model. Simul. 10 (2012), 1090–1118.Google Scholar
7Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. and Hwang, D.-U.. Complex networks: structure and dynamics. Phys. Rep. 424 (2006), 175–308.Google Scholar
8Braides, A.. Γ-Convergence for Beginners (New York: Oxford University Press, 2002).Google Scholar
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. 1385–1393 (Lake Tahoe: Curran Associates, Inc., 2012).Google Scholar
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).Google Scholar
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).Google Scholar
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), 309–320.Google Scholar
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. 81–88, (2009).Google Scholar
14Caldarelli, G.. Scale Free Networks: Complex Webs in Nature and Technology (New York: Oxford University Press, 2007).Google Scholar
15Chambolle, A. and Pock, T.. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging. Vis. 40 (2011), 120–145.Google Scholar
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), 1–20.Google Scholar
17Dal Maso, G.. An Introduction to Γ-Convergence (New York: Springer, 1993).Google Scholar
18Danon, L., Diaz-Guilera, A., Duch, J. and Arenas, A.. Comparing community structure identification. J. Stat. Mech: Theory Exp. 2005 (2005).Google Scholar
19El Alaoui, A., Cheng, X., Ramdas, A., Wainwright, M. J. and Jordan, M. I.. Asymptotic behavior of ℓ p-based Laplacian regularization in semi-supervised learning. In 29th Annual Conference on Learning Theory, pp. 879–906 (2016).Google Scholar
20Evans, L. C. and Gariepy, R. F.. Measure Theory and Fine Properties of Functions (Boca Raton: CRC Press, 1992).Google Scholar
21Faloutsos, M., Faloutsos, P. and Faloutsos, C.. On power-law relationships of the internet topology. In ACM SIGCOMM Computer Communication Review, vol. 29, pp. 251–262 (1999).Google Scholar
22Fortunato, S.. Community detection in graphs. Phys. Rep. 3–5 (2010), 75–174.Google Scholar
23Gangbo, W. and J. McCann, R.. The geometry of optimal transportation. Acta Math. 177 (1996), 113–161.Google Scholar
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. 119–135 (Lisbon, Springer, 2015).Google Scholar
25García Trillos, N. and Slepčev, D.. Continuum limit of Total Variation on point clouds. Arch. Ration. Mech. Anal. 1–49, 2015).Google Scholar
26García Trillos, N. and Slepčev, D.. On the rate of convergence of empirical measures in ∞-transportation distance. Can. J. Math. 67 (2015), 1358–1383.Google Scholar
27García Trillos, N., Slepčev, D., von Brecht, J., Laurent, T. and Bresson, X.. Consistency of cheeger and ratio graph cuts. arXiv:1411.6590 (2014).Google Scholar
28Hennawi, J. F and Prochaska, J. X. Quasars probing quasars. II. The anisotropic clustering of optically thick absorbers around quasars. Astrophys. J. 655 (2007).Google Scholar
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. 934–936, (2012).Google Scholar
30Jylhä, H.. The L ∞ optimal transport: infinite cyclical monotonicity and the existence of optimal transport maps. Calc. Var. Partial. Differ. Equ. 52 (2015), 303–326.Google Scholar
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. 513–520. (Edinburgh, Omnipress, 2012).Google Scholar
32Leoni, G.. A First Course in Sobolev Spaces, vol. 105 (Providence: American Mathematical Society, 2009).Google Scholar
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), 177–187.Google Scholar
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), 063525Google Scholar
35MacKay, D. J. C.. Information Theory, Inference & Learning Algorithms (Cambridge: Cambridge University Press, 2002).Google Scholar
36Maz'ya, V.. Sobolev Spaces: With Applications to Elliptic Partial Differential Equations (Berlin: Springer, 2010).Google Scholar
37Modica, L.. The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98 (1987), 123–142.Google Scholar
38Penrose, M.. Random Geometric Graphs (New York: Oxford University Press, 2003).Google Scholar
39Porter, M. A., Onnela, J.-P. and Mucha, P. J.. Communities in networks. Notices of the AMS (2009).Google Scholar
40Strogatz, S. H.. Exploring complex networks. Nature 410(6825) (2001), 268–276.Google Scholar
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), 2444–2474.Google Scholar
42Van Gennip, Y. and Bertozzi, A. L.. Γ-convergence of graph Ginzburg–Landau functionals. Adv. Differ. Equ. 17(11–12) (2012), 1115–1180.Google Scholar
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), 67–83.Google Scholar
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), 3–65.Google Scholar
45Villani, C.. Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society, cop., (2003).Google Scholar
46Wasserman, S. and Faust, K.. Social Network Analysis: Methods and Applications (Cambridge: Cambridge University Press, 1994).Google Scholar
47Worsley, K. J., Andermann, M., Koulis, T., MacDonald, D. and Evans, A. C.. Detecting changes in nonisotropic images. Hum. Brain. Mapp. 8 (1999), 98–101.Google Scholar
48Zhou, X. and Belkin, M.. Semi-supervised learning by higher order regularization. In International Conference on Artificial Intelligence and Statistics, pp. 892–900 (2011).Google Scholar
49Zhou, D. and Schölkopf, B.. Pattern Recognition: 27th DAGM Symposium, Vienna, Austria, 31 August–2 September 2005. Proceedings, chapter Regularization on Discrete Spaces, pp. 361–368. (Berlin Heidelberg: Springer, 2005).Google Scholar
50Zhou, X. and Srebro, N.. Error analysis of Laplacian eigenmaps for semi-supervised learning. In AISTATS, pp. 901–908, (2011).Google Scholar