Extract couplings or transport plan between two sets of gaussian distributions of different dimensions

[ttsesm]

Hello,

I have set of 3D gaussian distributions, i.e quadrics or Ellipsoids, in the 3d space and a set of 2D gaussian distributions, i.e. conics or ellipses, projected on an plane. For each ellipsoid/ellipse of my two sets of gaussian distributions I have the mean, μ, and the covariance matrices Σ.

My question is, based on this information is there an approach that I could possibly obtain some good coupling or a transport plan so that I can match an ellipsoid with an ellipse. Each distribution in each of the sets can be considered as individual distributions or combined. Moreover, it is not necessary that all distributions in one set match with all distributions on the other set. It can be that only few ellipsoids match with some of the ellipses in the other set. I am trying to figure out whether geometrically or statistically is possible to get something based on OT.

The ideas that we are examining at the moment are either through a minimisation problem as $min_P W_{2}^{2}(P\#\mu,\nu)$, where $\mu$ is my 3D Gaussians, and $\nu$ is my 2D Gaussians, and $P$ being a linear map $\mathbb{R}^{3} \to \mathbb{R}^{2}$ where I could use the wasserstein distance to build my cost matrix.

The other idea is to see each Gaussian as a point on the Bures-Wasserstein manifold. To compute an assignment between the 3D gaussians and 2D gaussians, we could then consider the Gromov-Wasserstein distance, which can correspond elements of different metric spaces apparently. Meaning that for our 3D gaussians, we compute the distance matrix between the Gaussians $g_i=(m_i,\Sigma_i)$ using the Bures-Wasserstein distance, this yields a distance matrix $C_1$. For the 2D gaussians, we do the same, yielding a distance matrix $C_2$. Then, once I have cost distance matrices of my two sets we could compute the Gromov-Wasserstein plan $T$, whose entry $i,j$ matches the 3D Gaussian $i$ with the 2D Gaussian $j$.

Would the above make sense, or is there any other suggestion for working with different sizes distributions that could yield better results. I am also having a look on the tutorials to get some idea how to implement the above and if there is any suggestion towards which tutorial to have a better look I would appreciate it.

Thanks.