4
$\begingroup$

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.

$\endgroup$
9
  • 3
    $\begingroup$ 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) $\endgroup$ 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. $\endgroup$ 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. $\endgroup$ Apr 30, 2020 at 23:52
  • 1
    $\begingroup$ 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
    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
    May 1, 2020 at 0:02

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.