  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 the Normalized Wasserstein measure. The key idea is to
introduce mixture proportions as optimization variables, effectively
normalizing mixture proportions in the Wasserstein formulation. Using the
proposed normalized Wasserstein measure leads to significant performance gains
for mixture distributions with imbalanced mixture proportions compared to the
vanilla Wasserstein distance. We demonstrate the effectiveness of the proposed
measure in GANs, domain adaptation and adversarial clustering in several
benchmark datasets.

Machine Learning;Machine Learning