Robust Estimation under the Wasserstein Distance
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We study the problem of robust distribution estimation under the Wasserstein metric, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. We introduce a new outlier-robust Wasserstein distance 𝖶_p^ε which allows for ε outlier mass to be removed from its input distributions, and show that minimum distance estimation under 𝖶_p^ε achieves minimax optimal robust estimation risk. Our analysis is rooted in several new results for partial OT, including an approximate triangle inequality, which may be of independent interest. To address computational tractability, we derive a dual formulation for 𝖶_p^ε that adds a simple penalty term to the classic Kantorovich dual objective. As such, 𝖶_p^ε can be implemented via an elementary modification to standard, duality-based OT solvers. Our results are extended to sliced OT, where distributions are projected onto low-dimensional subspaces, and applications to homogeneity and independence testing are explored. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.
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