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Could you advise me please on what to read on the "inverse" problem: suppose I have a source measure, a target measure and I observe the solution to optimal transport problem -- can I "back out" the cost function e.g. find a cost function such that the observed transport is optimal conditional on this cost function?

Thanks.

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    $\begingroup$ Well, one probably needs further regularity requirements on the cost function, otherwise $c(x,y)=0$ for all $(x,y)\in spt(\pi)$ and $c(x,y)=+\infty$ otherwise trivially does the job ($\pi$ being the transference plan) $\endgroup$
    – leo monsaingeon
    Commented Apr 30, 2020 at 23:33
  • $\begingroup$ Thanks! Yes, I agree that the space of functions should be restricted (and may be with very stringent requirements to get uniqueness). I am looking for a systematic study of this. $\endgroup$
    – Sergey Egiev
    Commented Apr 30, 2020 at 23:44
  • $\begingroup$ Even with regularity enforced you should not expect uniqueness, at least not for a single "observation". For, you can always add a (smooth) non-negative function vanishing on $spt(\pi)$, at least when this support is smooth. So I guess it's more about requiring "structure", really (e.g. strict convexity, twist conditions, etc...) I'm sorry I can't help you more, I've never heard of such a problem. $\endgroup$
    – leo monsaingeon
    Commented Apr 30, 2020 at 23:52
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    $\begingroup$ To add another complication, in one dimension for any cost of the form $c(x,y)=h(x-y)$ for $h$ convex, the solution to optimal transport will be the identical, and is given by the so-called monotone map. $\endgroup$
    – Gabe K
    Commented May 1, 2020 at 0:00
  • $\begingroup$ By Lagrange multipliers, such cost functions have the form $C(x,y) =a(x) +b(y)+ c(x,y)$ where $c$ is nonnegative everywhere and $0$ on the support of the transport map. If you're asking for functions that satisfy linear conditions and inequalities such as being a metric, or strict convexity as suggested by Leo, then finding a suitable cost function is a linear programming problem. $\endgroup$
    – Will Sawin
    Commented May 1, 2020 at 0:02

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See "Inverse Optimal Transport" By Stuart and Wolfram, SIAM J. App. Math, 80(1), 2020, and "Learning to Match via Inverse Optimal Transport" by Li et al., JMLR 2019.

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