Optimal Transport for Gaussian Mixture Models

Abstract

We introduce an optimal mass transport framework on the space of Gaussian mixture models. These models are widely used in statistical inference. Specifically, we treat the Gaussian mixture models as a submanifold of probability densities equipped with the Wasserstein metric. The topology induced by optimal transport is highly desirable and natural because, in contrast to total variation and other metrics, the Wasserstein metric is weakly continuous (i.e., convergence is equivalent to the convergence of moments). Thus, our approach provides natural ways to compare, interpolate, and average Gaussian mixture models. Moreover, the approach has low computational complexity. Different aspects of the framework are discussed, and examples are presented for illustration purposes.