Published Paper

Inserted: 8 dec 2022
Last Updated: 8 aug 2024

Journal: Calculus of Variations and Partial Differential Equations
Volume: 63
Number: 5
Pages: 114
Year: 2024
Doi: 10.1007/s00526-024-02721-9

Abstract:

In this paper we prove Gamma-convergence of a nonlocal perimeter of Minkowski type to a local anisotropic perimeter. The nonlocal model describes the regularizing effect of adversarial training in binary classifications. The energy essentially depends on the interaction between two distributions modelling likelihoods for the associated classes. We overcome typical strict regularity assumptions for the distributions by only assuming that they have bounded BV densities. In the natural topology coming from compactness, we prove Gamma-convergence to a weighted perimeter with weight determined by an anisotropic function of the two densities. Despite being local, this sharp interface limit reflects classification stability with respect to adversarial perturbations. We further apply our results to deduce Gamma-convergence of the associated total variations, to study the asymptotics of adversarial training, and to prove Gamma-convergence of graph discretizations for the nonlocal perimeter.

Keywords: Gamma-convergence, Nonlocal perimeter, Adversarial training, Random geometric graph