Find correspondences between a set of 3D gaussian distributions (quadrics or ellipsoids) and a set of 2d gaussian distributions (conics or ellipses) about pot HOT 1 CLOSED

Hi,

With BGWB (which is not implemented in POT), you need discrete distributions. A possible avenue would be to sample from your 3D and 2D gaussians, compute a projection P: R^3 -> R^2 which maps your samples, as well as a transport plan $\pi$ which will tell you which 3d sample matches which 2d sample.

The minimisation problem in question would be $\min_{P}W_2^2(P$# $\mu, \nu)$, where $\mu$ is your 3D samples and $\nu$ is your 2D samples. You could solve this using BCD or SGD (similarly to Generalised Wasserstein Barycentres ), but this could yield poor results numerically.

To use Gaussians and not samples, you could look see each Gaussian as a point on the Bures-Wasserstein manifold. To compute an assignment between your 3D gaussians and 2D gaussians, you could then consider the Gromov-Wasserstein distance, which can correspond elements of different metric spaces.

In general, your problem also makes me think of Domain Adaptation, which has been tackled with OT successfully.

Thanks for the interesting question, I don't think we can give an immediate answer with accessible tools :D
Let us know how your thoughts go along!

from pot.