  We present an optimal mass transport framework on the space of Gaussian
mixture models, which are widely used in statistical inference. Our method
leads to a natural way to compare, interpolate and average Gaussian mixture
models. Basically, we study such models on a certain submanifold of probability
densities with certain structure. Different aspects of this framework are
discussed and several examples are presented to illustrate the results. This
method represents our first attempt to study optimal transport problems for
more general probability densities with structures.
