Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning

2022/11/28 by Leon Bungert, Kerrek Stinson, Bungert, Leon +1 · 4 citations
Computer Science · Mathematics · #28A75 #49J45 #60D05 #68R10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2211.15223

openalex publication_date 2022/11/28 · openalex created_date 2022/12/10 · openalex updated_date 2026/07/28

Abstract

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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