
arXiv: 1604.04930
AbstractThe 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 Ψnwherendenotes 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.
Methods involving semicontinuity and convergence; relaxation, Ginzburg-Landau functional, Existence of optimal solutions to problems involving randomness, phase transitions, large data asymptotics, Mathematics - Analysis of PDEs, Asymptotic properties of nonparametric inference, PDEs on graphs, FOS: Mathematics, QA, total variation on graphs, \(\Gamma\)-convergence, Analysis of PDEs (math.AP)
Methods involving semicontinuity and convergence; relaxation, Ginzburg-Landau functional, Existence of optimal solutions to problems involving randomness, phase transitions, large data asymptotics, Mathematics - Analysis of PDEs, Asymptotic properties of nonparametric inference, PDEs on graphs, FOS: Mathematics, QA, total variation on graphs, \(\Gamma\)-convergence, Analysis of PDEs (math.AP)
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