Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning
Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning
复制标题
对抗性机器学习中出现的非局部周界的伽玛收敛
DOI：
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发表时间：
2022
影响因子：
2.1
通讯作者：
Kerrek Stinson
中科院分区：
文献类型：
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作者：
Leon Bungert;Kerrek Stinson
本文证明了Minkowski型非局部周长到局部各向异性周长的Gamma收敛性。非局部模型描述了二元分类中对抗训练的正则化效果。能量基本上取决于两个分布之间的相互作用，为相关的类建模的可能性。我们克服了典型的严格的规则性假设的分布，只假设他们有界BV密度。在紧性的自然拓扑中，我们证明了Gamma-收敛到一个权由两个密度的各向异性函数决定的加权周长。尽管是本地的，这个尖锐的接口限制反映了分类的稳定性相对于对抗扰动。我们进一步应用我们的结果来推导相关总变分的Gamma收敛，研究对抗训练的渐近性，并证明非局部周长的图离散化的Gamma收敛。
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.
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