Multi- to one-dimensional transportation

Abstract

Fix probability densities $f$ and $g$ on open sets $X\subset {\mathbf{R}}^m$ and $Y\subset {\mathbf{R}}^n$ with $m\ge n\ge 1$. Consider transporting $f$ onto $g$ so as to minimize the cost $-s(x,y)$. We give a non-degeneracy condition (a) on $s\in C^{1,1}$ which ensures the set of $x$ paired with [$g$-a.e.] $y\in Y$ lie in a codimension $n$ submanifold of $X$. Specializing to the case $m>n=1$, we discover a nestedness criteria relating $s$ to $(f,g)$ which allows us to construct a unique optimal solution in the form of a map $F:X⟶\overline{Y}$. When $s\in C^2\cap W^{3,1}$ and $\log f$ and $\log g$ are bounded, the Kantorovich dual potentials $(u,v)$ satisfy $v\in C_{loc}^{1,1}(Y)$, and the normal velocity $V$ of $F^{-1}(y)$ with respect to changes in $y$ is given by $V(x)=v"(f(x))-s_{yy}(x,f(x))$. Positivity (b) of $V$ locally implies a Lipschitz bound on $f$; moreover, $v\in C^2$ if $F^{-1}(y)$ intersects $\partial X\in C^1$ transversally (c). On subsets where (a)-(c) can be be quantified, for each integer $r\ge 1$ the norms of $u,v\in C^{r+1,1}$ and $F\in C^{r,1}$ are controlled by these bounds, $||\log f,\log g,\partial X|{|}_{C^{r-1,1}},||\partial X|{|}_{C^{1,1}}$, $||s|{|}_{C^{r+1,1}}$, and the smallness of $F^{-1}(y)$. We give examples showing regularity extends from $X$ to part of $\bar{X}$, but not from $Y$ to $\bar{Y}$. We also show that when $s$ remains nested for all $(f,g)$, the problem in ${\mathbf{R}}^m\times \mathbf{R}$ reduces to a supermodular problem in $\mathbf{R}\times \mathbf{R}$.

References in corpus (1)

• Multidimensional matching

Cited by in corpus (3)

• The boundary method for semi-discrete optimal transport partitions and Wasserstein distance computation
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