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Let us consider $\mu_1, \mu_2$ and $\mu_3$ three probability measures living on $[0,1]^{k_1}, [0,1]^{k_2}$ and $[0,1]^k$respectively, with $k_1 +k_2=k$. Let us denote by $\Gamma(\mu,\nu)$ the set of measures with prescribed marginals $\mu$ and $\nu$.

What is known about $$\sup_{\pi \in \Gamma(\mu_1,\mu_2)}\inf_{\nu \in \Gamma(\pi,\mu_3)} \int_{[0,1]^k\times[0,1]^k} \lvert x-y\rvert² d\nu(x,y) \ ?$$

Further assume that one discretises such measures (by $n$ points, say). Is there any numerical scheme to solve this discretised problem, alike the Hungarian algorithm in the classical case ?

I can figure out that one will be looking for a $n\times n \times n $ object containing only zeros and ones, with sum conditions but I am starting to be stuck at this point. My browsing of the literature did not yield much so far.

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  • $\begingroup$ You are looking for a coupling $\pi$ which maximizes the Wasserstein distance to a given measure $\mu_3$. A special case of that is the question mathoverflow.net/q/264231/106046 If you are only interested in numerics, I think by discretising the problem and using the dual formulation for the inner problem, you end up with a linear program. $\endgroup$
    – Steve
    Commented Feb 24, 2020 at 11:52
  • $\begingroup$ thank you for the fast reply. I am digging into this $\endgroup$
    – Gilles Mordant
    Commented Feb 25, 2020 at 8:14

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