Asymptotic analysis of the Ginzburg–Landau functional on point clouds

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.

[1]  Nathan Srebro,et al.  Error Analysis of Laplacian Eigenmaps for Semi-supervised Learning , 2011, AISTATS.

[2]  Nicolás García Trillos,et al.  Continuum Limit of Total Variation on Point Clouds , 2014, Archive for Rational Mechanics and Analysis.

[3]  Annalisa Baldi,et al.  WEIGHTED BV FUNCTIONS , 2001 .

[4]  W. Marsden.  I and J , 2012 .

[5]  C. Villani.  Topics in Optimal Transportation , 2003 .

[6]  E. Linder,et al.  Cosmological constraints from the anisotropic clustering analysis using BOSS DR9 , 2013, 1311.5226.

[7]  Martin J. Wainwright,et al.  Asymptotic behavior of ℓp-based Laplacian regularization in semi-supervised learning , 2016, ArXiv.

[8]  Arjuna Flenner,et al.  Diffuse Interface Models on Graphs for Classification of High Dimensional Data , 2012, SIAM Rev..

[9]  Bernhard Schölkopf,et al.  Regularization on Discrete Spaces , 2005, DAGM-Symposium.

[10]  Michael I. Jordan,et al.  Revisiting k-means: New Algorithms via Bayesian Nonparametrics , 2011, ICML.

[11]  Florian Theil,et al.  Convergence of the k-Means Minimization Problem using Γ-Convergence , 2015, SIAM J. Appl. Math..

[12]  Guido Caldarelli,et al.  Scale-Free Networks , 2007 .

[13]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[14]  A. Bertozzi,et al.  $\Gamma$-convergence of graph Ginzburg-Landau functionals , 2012, Advances in Differential Equations.

[15]  G. Leoni.  A First Course in Sobolev Spaces , 2009 .

[16]  Mikhail Belkin,et al.  Semi-supervised Learning by Higher Order Regularization , 2011, AISTATS.

[17]  Xavier Bresson,et al.  Consistency of Cheeger and Ratio Graph Cuts , 2014, J. Mach. Learn. Res..

[18]  Yuhong Yang,et al.  Information Theory, Inference, and Learning Algorithms , 2005 .

[19]  Leon Danon,et al.  Comparing community structure identification , 2005, cond-mat/0505245.

[20]  Heikki Jylhä.  The $$L^\infty $$L∞ optimal transport: infinite cyclical monotonicity and the existence of optimal transport maps , 2015 .

[21]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[22]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[23]  Quasars Probing Quasars. II. The Anisotropic Clustering of Optically Thick Absorbers around Quasars , 2006, astro-ph/0606084.

[24]  J. Prochaska.  Quasars Probing Quasars , 2006 .

[25]  Arjuna Flenner,et al.  Diffuse Interface Models on Graphs for Classification of High Dimensional Data , 2012, Multiscale Model. Simul..

[26]  Jin-Feng Wang,et al.  A spatial scan statistic for nonisotropic two-level risk cluster. , 2012, Statistics in medicine.

[27]  Andrea Braides.  Gamma-Convergence for Beginners , 2002 .

[28]  Vladimir Maz’ya,et al.  Sobolev Spaces: with Applications to Elliptic Partial Differential Equations , 2011 .

[29]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[30]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[31]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[32]  A. Bertozzi,et al.  Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs , 2013, 1307.0045.

[33]  Kyle Luh,et al.  Community Detection Using Spectral Clustering on Sparse Geosocial Data , 2012, SIAM J. Appl. Math..

[34]  Giovanni Alberti,et al.  A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies , 1998, European Journal of Applied Mathematics.

[35]  L. Goddard.  Information Theory , 1962, Nature.

[36]  Xavier Bresson,et al.  Convergence and Energy Landscape for Cheeger Cut Clustering , 2012, NIPS.

[37]  FaloutsosMichalis,et al.  On power-law relationships of the Internet topology , 1999 .

[38]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[39]  Tong Zhang,et al.  Learning on Graph with Laplacian Regularization , 2006, NIPS.

[40]  Andrea Braides.  Local Minimization, Variational Evolution and Γ-Convergence , 2013 .

[41]  Andrea Braides.  Γ-convergence for beginners , 2002 .

[42]  Thierry Champion,et al.  The ∞-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps , 2008, SIAM J. Math. Anal..

[43]  Stanley Wasserman,et al.  Social Network Analysis: Methods and Applications , 1994 .

[44]  D. Slepčev,et al.  On the Rate of Convergence of Empirical Measures in ∞-transportation Distance , 2015, Canadian Journal of Mathematics.

[45]  Xavier Bresson,et al.  An Adaptive Total Variation Algorithm for Computing the Balanced Cut of a Graph , 2013, 1302.2717.

[46]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[47]  David J. C. MacKay,et al.  Information Theory, Inference, and Learning Algorithms , 2004, IEEE Transactions on Information Theory.

[48]  Nicolás García Trillos,et al.  On the rate of convergence of empirical measures in $\infty$-transportation distance , 2014, 1407.1157.

[49]  L. Ambrosio,et al.  Existence and stability results in the L 1 theory of optimal transportation , 2003 .

[50]  Arjuna Flenner,et al.  Multiclass Semi-Supervised Learning on Graphs using Ginzburg-Landau Functional Minimization , 2013, ICPRAM.

[51]  Guido Caldarelli,et al.  Scale-Free Networks , 2007 .

[52]  A. Bertozzi,et al.  Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs , 2013, 1307.0045.

[53]  Matthias Hein,et al.  Spectral clustering based on the graph p-Laplacian , 2009, ICML '09.

[54]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[55]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[56]  Alan C. Evans,et al.  Detecting changes in nonisotropic images , 1999, Human brain mapping.

[57]  Mason A. Porter,et al.  Communities in Networks , 2009, ArXiv.

[58]  Ahmed El Alaoui,et al.  Asymptotic behavior of \(\ell_p\)-based Laplacian regularization in semi-supervised learning , 2016, COLT.

[59]  L. Modica.  The gradient theory of phase transitions and the minimal interface criterion , 1987 .

[60]  Mason A. Porter,et al.  Multislice Modularity Optimization in Community Detection and Image Segmentation , 2012, 2012 IEEE 12th International Conference on Data Mining Workshops.

[61]  Xiaojin Zhu,et al.  p-voltages: Laplacian Regularization for Semi-Supervised Learning on High-Dimensional Data , 2013 .

[62]  A. Bertozzi,et al.  Γ-CONVERGENCE OF GRAPH GINZBURG–LANDAU FUNCTIONALS , 2012 .