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:在本文中,我们证明Minkowski型非局部周边与局部厌食性周界的伽马混杂。非本地模型描述了二进制分类方面的对抗性培训的常规效果。能量主要取决于两个分布类模拟可能性之间的相互作用。我们克服了典型的关于分配的严格常规假设,仅假设它们与$BV美元密度相联。在来自紧凑度的自然地形学中,我们证明伽马混杂到一个加权周边,其重量由两种密度的厌食功能决定。尽管这种尖锐的界面限制是局部的,它反映了对抗性扰动的分类稳定性。我们进一步应用我们的结果来推断相关的总变异的伽马-趋同性,研究对抗性训练的零点,并证明非本地周边的图形离散化的伽马-趋同性。