  Recently, a relaxed formulation of optimal transport for Gaussian mixtures
has been proposed, which is based on the explicit formulation between
Gaussians. The Gaussian distributions can be viewed as special elliptical
contoured distributions generated from exponential function. In literature,
there are few research about optimal transport between elliptical contoured
distributions generated from different functions. In this paper, we first study
optimal transport between radial contoured distributions generated from
different functions and show theirWasserstein barycenter is still radial. Then
we introduce a relaxed Wasserstein-type distance for mixtures with radial
contoured components. We also consider the corresponding barycenter problem and
connect it with a multimarginal problem.
