Duality for the Adversarial Total Variation

TL;DR

This paper explores the duality properties of nonlocal total variation in the context of adversarial training, providing new mathematical characterizations and dual representations.

Contribution

It establishes a duality framework for nonlocal total variation, including subdifferential characterizations and dual representations in various function spaces.

Findings

Derived a dual representation of nonlocal total variation.

Provided a characterization of the subdifferential using duality techniques.

Extended duality results to different function spaces and conditions.

Abstract

Adversarial training of binary classifiers can be reformulated as regularized risk minimization involving a nonlocal total variation. Building on this perspective, we establish a characterization of the subdifferential of this total variation using duality techniques. To achieve this, we derive a dual representation of the nonlocal total variation and a related integration of parts formula, involving a nonlocal gradient and divergence. We provide such duality statements both in the space of continuous functions vanishing at infinity on proper metric spaces and for the space of essentially bounded functions on Euclidean domains. Furthermore, under some additional conditions we provide characterizations of the subdifferential in these settings.