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.

翻译：本文证明了Minkowski型非局部周长到局部各向异性周长的Gamma收敛。该非局部模型描述了二元分类中对抗训练的正则化效应。该能量本质上依赖于表征相关类别似然度的两个分布之间的相互作用。我们仅假设分布具有有界$BV$密度，从而克服了通常对分布施加的严格正则性假设。在紧性导出的自然拓扑下，我们证明其Gamma收敛到加权周长，其中权重由两个密度的各向异性函数决定。尽管是局部的，这一尖锐界面极限反映了对对抗扰动的分类稳定性。我们进一步应用所得结果推导相关全变差的Gamma收敛，研究对抗训练的渐近行为，并证明非局部周长的图离散化形式的Gamma收敛。