| Popis: |
The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when Gaussian distributions extend more generally to elliptical contoured distributions, similar results also hold true. In this case, Gaussian distributions are regarded as elliptical contoured distribution with generator function $e^{-x/2}$. However, there are few results about optimal transport for elliptical contoured distributions with different generator functions. In this paper, we degenerate elliptical contoured distributions to radial contoured distributions and study their optimal transport and prove their Wasserstein barycenter is still radial contoured. For general elliptical contoured distributions, we give two numerical counterexamples to show that the Wasserstein barycenter of elliptical contoured distributions does not have to be elliptical contoured. |
