  Optimal Transport (OT) distances such as Wasserstein have been used in
several areas such as GANs and domain adaptation. OT, however, is very
sensitive to outliers (samples with large noise) in the data since in its
objective function, every sample, including outliers, is weighed similarly due
to the marginal constraints. To remedy this issue, robust formulations of OT
with unbalanced marginal constraints have previously been proposed. However,
employing these methods in deep learning problems such as GANs and domain
adaptation is challenging due to the instability of their dual optimization
solvers. In this paper, we resolve these issues by deriving a
computationally-efficient dual form of the robust OT optimization that is
amenable to modern deep learning applications. We demonstrate the effectiveness
of our formulation in two applications of GANs and domain adaptation. Our
approach can train state-of-the-art GAN models on noisy datasets corrupted with
outlier distributions. In particular, our optimization computes weights for
training samples reflecting how difficult it is for those samples to be
generated in the model. In domain adaptation, our robust OT formulation leads
to improved accuracy compared to the standard adversarial adaptation methods.
Our code is available at https://github.com/yogeshbalaji/robustOT.
