Timeline for Optimal transport mapping between sets with a common subset
Current License: CC BY-SA 4.0
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| Dec 17, 2020 at 17:35 | comment | added | Gabe K | 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. | |
| Dec 17, 2020 at 14:55 | comment | added | Kashif | Why are the trajectories straight lines? | |
| Dec 17, 2020 at 10:25 | comment | added | Gabe K | 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. | |
| Dec 16, 2020 at 17:07 | history | edited | Kashif | CC BY-SA 4.0 |
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| Dec 16, 2020 at 16:32 | history | asked | Kashif | CC BY-SA 4.0 |