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.

翻译：我们研究的是对抗性多级分类问题,并提供相当的重新表述,其内容包括:(1) 本文中介绍的普遍巴氏中心问题;(2) 多种边际最佳运输问题,其边际人数与最初分类问题中的班级数相等;这些新的理论结果显示,在多级分类中,对抗性学习问题具有丰富的几何结构,并将最近的结果扩大到限于二元分类设置;我们结果的一个直接计算意义是,通过解决巴氏中心问题及其双重问题,或者解决MOT问题及其双重问题,我们可以恢复最佳的稳健分类规则,以及最初对抗性问题的最佳对抗战略;合成和真实数据的例子说明了我们的结果。