We study the problem of robust distribution estimation under the Wasserstein distance, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. Given $n$ samples from an unknown distribution $\mu$, of which $\varepsilon n$ are adversarially corrupted, we seek an estimate for $\mu$ with minimal Wasserstein error. To address this task, we draw upon two frameworks from OT and robust statistics: partial OT (POT) and minimum distance estimation (MDE). We prove new structural properties for POT and use them to show that MDE under a partial Wasserstein distance achieves the minimax-optimal robust estimation risk in many settings. Along the way, we derive a novel dual form for POT that adds a sup-norm penalty to the classic Kantorovich dual for standard OT. Since the popular Wasserstein generative adversarial network (WGAN) framework implements Wasserstein MDE via Kantorovich duality, our penalized dual enables large-scale generative modeling with contaminated datasets via an elementary modification to WGAN. Numerical experiments demonstrating the efficacy of our approach in mitigating the impact of adversarial corruptions are provided.

翻译：我们研究了在Wasserstein距离下的鲁棒分布估计问题，该距离是一种基于最优传输理论的常用概率分布差异度量。给定从未知分布$\mu$中抽取的$n$个样本，其中$\varepsilon n$个样本受到对抗性破坏，我们寻求对$\mu$的估计，使其Wasserstein误差最小。为解决此问题，我们结合了最优传输和鲁棒统计中的两个框架：部分最优传输与最小距离估计。我们证明了部分最优传输的新结构性质，并利用这些性质证明：在许多设定下，基于部分Wasserstein距离的最小距离估计能够达到极小极大最优的鲁棒估计风险。在此过程中，我们推导出部分最优传输的一种新颖对偶形式，该形式在经典Kantorovich对偶基础上增加了上确界范数惩罚项。由于流行的Wasserstein生成对抗网络框架通过Kantorovich对偶实现Wasserstein最小距离估计，我们提出的惩罚对偶形式通过对WGAN进行基础修改，即可实现基于污染数据集的大规模生成建模。数值实验证明了我们的方法在减轻对抗性破坏影响方面的有效性。