Asymptotic Analysis of the Ginzburg-Landau Functional on Point Clouds

TL;DR

This paper analyzes the asymptotic behavior of the Ginzburg-Landau functional on point clouds with anisotropic interactions, showing its convergence to a total variation-based surface energy and extending previous isotropic results.

Contribution

It generalizes the asymptotic analysis of Ginzburg-Landau functionals from isotropic to anisotropic interaction potentials on point clouds.

Findings

Limit of Ginzburg-Landau functionals relates to total variation norm.

Extended convergence results to anisotropic interaction potentials.

Provided rate of convergence for the asymptotic behavior.

Abstract

The Ginzburg-Landau functional is a phase transition model which is suitable for clustering or classification type problems. We study the asymptotics of a sequence of Ginzburg-Landau functionals with anisotropic interaction potentials on point clouds $\Psi_n$ where $n$ denotes the number data points. In particular we show the limiting problem, in the sense of $\Gamma$-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.