论文标题
对抗性多类分类的多边缘最佳运输公式
The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classification
论文作者
论文摘要
我们研究了一个对抗性多类分类问题的家族，并根据以下方面提供了同等的重新纠正：1）论文中引入的广义重中心问题家族和2）多边形最佳运输问题家族，边际数量等于原始分类问题中的类别数量。这些新的理论结果揭示了多类分类中对抗性学习问题的丰富几何结构，并将最新结果扩展到二进制分类设置。我们结果的直接计算含义是，通过解决Barycenter问题及其双重问题，或MOT问题及其双重，我们可以为原始对抗性问题恢复最佳的稳健分类规则和最佳对抗性策略。合成和真实数据的示例说明了我们的结果。
We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results.
