What would the optimal transport mapping between two sets with a common subset be? The problem I'm thinking about is the following:

I'm in $\mathbb{R}^n$ and I have two distributions $\mu$ and $\nu$ lying in subspaces $A$ and $B$ that have some overlap. Furthermore, $A$ and $B$ have a common subspace $C$ at their intersection and $A=C+U$ and $B=C+V$, where $U$ and $V$ are orthogonal to $C$.

How could I compute the optimal transport mapping here?

In this paper, https://arxiv.org/pdf/1712.01504.pdf only the case where the spaces were of full rank were considered. But I thought the optimal transport map in the above case could be a combination of projecting onto $C$ and then the optimal transport mapping now of $A|_C$ to $B|_C$.

Am I correct? I don't see how to show this.

What would the optimal transport mapping between two sets with a common subset be? The problem I'm thinking about is the following:

I'm in $\mathbb{C}^n$ and I have two distributions $\mu$ and $\nu$ lying in subspaces $A$ and $B$ that have some overlap. Furthermore, $A$ and $B$ have a common subspace $C$ at their intersection and $A=C+U$ and $B=C+V$, where $U$ and $V$ are orthogonal to $C$.

How could I compute the optimal transport mapping here?

In this paper, https://arxiv.org/pdf/1712.01504.pdf only the case where the spaces were of full rank were considered. But I thought the optimal transport map in the above case could be a combination of projecting onto $C$ and then the optimal transport mapping now of $A|_C$ to $B|_C$.

Am I correct? I don't see how to show this.