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In this paper we prove Gamma-convergence of a nonlocal perimeter of Minkowskitype to a local anisotropic perimeter. The nonlocal model describes theregularizing effect of adversarial training in binary classifications. Theenergy essentially depends on the interaction between two distributionsmodelling likelihoods for the associated classes. We overcome typical strictregularity assumptions for the distributions by only assuming that they havebounded $BV$ densities. In the natural topology coming from compactness, weprove Gamma-convergence to a weighted perimeter with weight determined by ananisotropic function of the two densities. Despite being local, this sharpinterface limit reflects classification stability with respect to adversarialperturbations. We further apply our results to deduce Gamma-convergence of theassociated total variations, to study the asymptotics of adversarial training,and to prove Gamma-convergence of graph discretizations for the nonlocalperimeter.

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