论文标题
在对抗机器学习中出现的非本地周长的伽马连接
Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning
论文作者
论文摘要
在本文中，我们证明了Minkowski类型的非局部周长到局部各向异性周边的伽马连接。非局部模型描述了二进制分类中对抗训练的正则作用。能量本质上取决于两个分布建模相关类别的可能性之间的相互作用。我们仅假设它们已经限制了$ bv $密度来克服分布的典型严格规律性假设。在来自紧凑性的自然拓扑结构中，我们证明伽马连接到加权周长，并由两种密度的各向异性功能确定。尽管是本地的，但这种尖锐的界面限制反映了针对对抗扰动的分类稳定性。我们进一步应用结果来推断相关的总变化的伽马连接，研究对抗训练的渐近学，并证明非局部周长的图形离散化的伽玛相位。
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.
