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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.

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    $\begingroup$ To discuss optimal transport, you must specify the cost function. In context it seems like they are using the squared distance cost. To answer your question though, the optimal transport will not involve projection onto a common subspace. The trajectories of optimal transport w.r.t. squared distance will be straight lines and there's no reason to believe that they all pass through a lower dimensional subspace. Furthermore, if you project A to C, optimally transport on the subspace, then disintegrate to get a distribution on B, this is suboptimal except in degenerate cases. $\endgroup$
    – Gabe K
    Commented Dec 17, 2020 at 10:25
  • $\begingroup$ Why are the trajectories straight lines? $\endgroup$
    – Kashif
    Commented Dec 17, 2020 at 14:55
  • $\begingroup$ That's a central result in displacement interpolation. You can find more details in Chapter 7 of Villani's book. The basic idea can be described by the aphorism "A geodesic in the space of laws is the law of a geodesic,” which is on page 139. $\endgroup$
    – Gabe K
    Commented Dec 17, 2020 at 17:35

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