Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis
Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis
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发表时间：
2021-11
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通讯作者：
Sloan Nietert;Rachel Cummings;Ziv Goldfeld
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作者：
Sloan Nietert;Rachel Cummings;Ziv Goldfeld
Wasserstein距离源于最优传输（OT）理论，是概率分布之间的一种流行差异度量，在统计和机器学习中有各种应用。尽管其丰富的结构和证明效用，Wasserstein距离是敏感的离群值在考虑的分布，这阻碍了在实践中的适用性。我们提出了一个新的离群鲁棒Wasserstein距离$\mathsf{W}_p^\varepstein $，它允许$\varepstein $离群质量从每个污染分布中删除。在标准的矩假设下，$\mathsf{W}_p^\vareprogram $下的Huber $\vareprogram $-污染模型，以实现强大的鲁棒估计保证。我们制定的这个强大的距离相当于一个高度定期的优化问题，使自己更好地分析相比，以前考虑的框架。利用这一点，我们进行了深入的理论研究$\mathsf{W}_p^\vareps $，包括鲁棒性保证，最优扰动的特征，规律性，对偶性和统计估计。特别是，通过解耦优化变量，我们得到了一个简单的对偶形式的$\mathsf{W}_p^\vareps $，可以通过对标准的，基于对偶的OT求解器的基本修改来实现。我们说明了我们的框架的优点，通过应用程序生成建模与污染的数据集。
The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. We propose a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from each contaminated distribution. Under standard moment assumptions, $\mathsf{W}_p^\varepsilon$ is shown to achieve strong robust estimation guarantees under the Huber $\varepsilon$-contamination model. Our formulation of this robust distance amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of $\mathsf{W}_p^\varepsilon$, encompassing robustness guarantees, characterization of optimal perturbations, regularity, duality, and statistical estimation. In particular, by decoupling the optimization variables, we arrive at a simple dual form for $\mathsf{W}_p^\varepsilon$ that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.