通过对偶方法揭示对抗训练中总变差的数学结构。
Duality for the Adversarial Total Variation
- 用对偶理论分析非局部总变差的子微分性质。
- 推导出非局部梯度与散度的积分公式。
- 适用于连续函数与有界函数空间，适合数学优化研究者。
二值分类器的对抗训练可重述为包含非局部总变差的正则化风险最小化问题。基于这一视角，我们利用对偶技术刻画了该总变差的子微分。为此，推导出非局部总变差的对偶表示形式及相关的积分换序公式，涉及非局部梯度与散度算子。我们在紧致度量空间上的连续无穷远处消失函数空间，以及欧氏域上的本质有界函数空间中建立了此类对偶关系。此外，在若干附加条件下，给出了这些情形下子微分的显式刻画。
原文摘要 · Abstract (English)
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.
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