0
$\begingroup$

Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-discrete optimal transport problem defined by

$$\inf_{(X,Y)}~ \mathbb E\big[|Y-X|^2\big],\quad \quad \quad \quad (\ast)$$

where the inf is taken overall the pairs of random variables $(X,Y)$ such that $X\sim\mu$ and $Y\sim\nu$. Under which conditions on $\rho$, $(p_i)_{1\le i\le n}$ and $(y_i)_{1\le i\le n}$, it holds $\mathbb E[Y^*|X^*]=X^*$? Here $(X^*,Y^*)$ denotes the optimizer of $(\ast)$.

Any thoughts or references are highly appreciated!

Improve this question
$\endgroup$
2
  • $\begingroup$ Isn't this impossible, since $Y$ is discrete and $X$ is continuous? Or should it be the other way around? $\endgroup$
    – Steve
    Commented Aug 23, 2019 at 7:18
  • $\begingroup$ The optimizer may not be unique. $\endgroup$
    – MOMO
    Commented Nov 14, 2019 at 10:36

0

Reset to default

You must log in to answer this question.