Normalized Wasserstein Distance for Mixture Distributions with Applications in Adversarial Learning and Domain Adaptation
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Understanding proper distance measures between distributions is at the core of several learning tasks such as generative models, domain adaptation, clustering, etc. In this work, we focus on mixture distributions that arise naturally in several application domains where the data contains different sub-populations. For mixture distributions, established distance measures such as the Wasserstein distance do not take into account imbalanced mixture proportions. Thus, even if two mixture distributions have identical mixture components but different mixture proportions, the Wasserstein distance between them will be large. This often leads to undesired results in distance-based learning methods for mixture distributions. In this paper, we resolve this issue by introducing Normalized Wasserstein distance. The key idea is to introduce mixture proportions as optimization variables, effectively normalizing mixture proportions in the Wasserstein formulation. Using the proposed normalized Wasserstein distance, instead of the vanilla one, leads to significant gains working with mixture distributions with imbalanced mixture proportions. We demonstrate effectiveness of the proposed distance in GANs, domain adaptation, adversarial clustering and hypothesis testing over mixture of Gaussians, MNIST, CIFAR-10, CelebA and VISDA datasets.
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