Gamma-convergence of graph Ginzburg-Landau functionals

TL;DR

This paper investigates the Gamma-convergence of graph-based Ginzburg-Landau functionals, analyzing limits as the interface parameter approaches zero and as the number of graph nodes increases, connecting discrete and continuum models.

Contribution

It establishes the Gamma-convergence results for graph Ginzburg-Landau functionals, linking them to graph cut and total variation in the continuum limit, including nonlocal models.

Findings

Graph cut objective recovered as epsilon approaches zero.

Continuum limit related to total variation seminorm on 4-regular graphs.

Analysis of simultaneous limits for epsilon and graph size.

Abstract

We study Gamma-convergence of graph based Ginzburg-Landau functionals, both the limit for zero diffusive interface parameter epsilon->0 and the limit for infinite nodes in the graph m -> infinity. For general graphs we prove that in the limit epsilon -> 0 the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4-regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg-Landau functional. For both functionals we also study the simultaneous limit epsilon -> 0 and m -> infinity, by expressing epsilon as a power of m and taking m -> infinity. Finally we investigate the continuum limit for a nonlocal means type functional on a completely connected graph.