  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.
