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Understanding proper distance measures between distributions is at the coreof several learning tasks such as generative models, domain adaptation,clustering, etc. In this work, we focus on mixture distributions that arisenaturally in several application domains where the data contains differentsub-populations. For mixture distributions, established distance measures suchas the Wasserstein distance do not take into account imbalanced mixtureproportions. Thus, even if two mixture distributions have identical mixturecomponents but different mixture proportions, the Wasserstein distance betweenthem will be large. This often leads to undesired results in distance-basedlearning methods for mixture distributions. In this paper, we resolve thisissue by introducing the Normalized Wasserstein measure. The key idea is tointroduce mixture proportions as optimization variables, effectivelynormalizing mixture proportions in the Wasserstein formulation. Using theproposed normalized Wasserstein measure leads to significant performance gainsfor mixture distributions with imbalanced mixture proportions compared to thevanilla Wasserstein distance. We demonstrate the effectiveness of the proposedmeasure in GANs, domain adaptation and adversarial clustering in severalbenchmark datasets.

Normalized Wasserstein Distance for Mixture Distributions with Applications in Adversarial Learning and Domain Adaptation