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Optimal transport (OT) is known to be sensitive against outliers because ofits marginal constraints. Outlier robust OT variants have been proposed basedon the definition that outliers are samples which are expensive to move. Inthis paper, we show that this definition is restricted by considering the casewhere outliers are closer to the target measure than clean samples. We showthat outlier robust OT fully transports these outliers leading to poorperformances in practice. To tackle these outliers, we propose to detect themby relying on a classifier trained with adversarial training to classify sourceand target samples. A sample is then considered as an outlier if the predictionfrom the classifier is different from its assigned label. To decrease theinfluence of these outliers in the transport problem, we propose to eitherremove them from the problem or to increase the cost of moving them by usingthe classifier prediction. We show that we successfully detect these outliersand that they do not influence the transport problem on several experimentssuch as gradient flows, generative models and label propagation.

On making optimal transport robust to all outliers