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：本文证明了一种闵可夫斯基型非局部周长在Gamma收敛意义下趋于局部各向异性周长。该非局部模型描述了二分类任务中对抗训练的规则化效应。该能量本质上依赖于刻画两类似然分布之间的相互作用。我们无需对分布施加严格的规则性假设，仅需假设分布具有有界的BV密度。在由紧性导出的自然拓扑下，我们证明了Gamma收敛性，收敛极限为加权周长，其权重由两个密度的各向异性函数确定。尽管该尖锐界面极限是局部的，但它反映了对抗扰动下分类的稳定性。我们进一步将结果应用于推导相应全变分的Gamma收敛性、研究对抗训练的渐近行为，并证明该非局部周长的图离散化近似也具有Gamma收敛性。