Optimal Transport in One Dimension

In this article, we briefly explore the optimal transport problem on the real line along with some examples.

Linear Cost Example

For this example, consider the cost function $c(x,y)=L(x-y)$ along with a given linear map $A:\mathbb {R} ^{d}\rightarrow \mathbb {R}$. Moreover, if let $\gamma$ be any transport plan, then by direct computation we see that

                                                                                $\int L(x-y)d\gamma =\int L(x)d\gamma -\int L(y)d\gamma =\int L(x)d\mu -\int L(y)d\nu$

which suggests that this result only depends on the marginals of $\gamma$ (wherein $\mu$ and $\nu$ are compactly supported probability measures). In fact, in such cases, every transport plan/map is optimal.

Distance Cost Example

Consider the cost function $c(x,y)=|x-y|$ along with probability measures (on $\mathbb {R}$) $\mu$ and $\nu$. Then, for any $(x,y)\in spt(\mu )\times spt(\nu )$ we see that $c(x,y)=y-x$, which then immediately puts us back in the linear cost position, so any transport map/plan is also optimal for such costs.

Book Shifting Example

Consider the cost function $c(x,y)=|x-y|$ along with $\mu ={\frac {1}{2}}\lambda _{[0,2]}$ and $\nu ={\frac {1}{2}}\lambda _{[1,3]}$ (where $\lambda$ is the one-dimensional Lebesgue measure). A (monotone) transport plan that rearranges $\mu$ to look like $\nu$ is given by $T_{0}(x)=x+1$ and its corresponding cost is

                                                                                               $M(T_{0})=\int |T_{0}(x)-x|d\mu \equiv 1$.

Furthermore, notice that the piecewise map $T_{1}$ given by $T_{1}(x)=x+2$ (for $x\leq 1$) and $T_{1}(x)=x$ (for $x>1$) satisfies $T_{1}\#\mu =\nu$, i.e. $T_{1}$ is a transport map from $\mu$ to $\nu$; moreover, the corresponding cost is

                                                                                          $M(T_{1})=\int |T_{1}(x)-x|d\mu ={\frac {1}{2}}\int _{0}^{2}2dx\equiv 1$

and so we conclude that $T_{1}$ is indeed optimal as well.

Quadratic Cost

Theorem: Let $\mu ,\nu$ be probability measures on $\mathbb {R}$ with cumulative distribution functions (CDFs) $F$ and $G$, respectively. Also, let $\pi$ be the probability measure on $\mathbb {R} ^{2}$ with the CDF $H(x,y)=\min(F(x),G(y))$. Then, $\pi \in \Gamma (\mu ,\nu )$ and is optimal (in the Kantorovich problem setting) between $\mu$ and $\nu$ for the (quadratic) cost function $c(x,y)=|x-y|^{2}$, and the corresponding cost is

                                                                                            $T_{2}(\mu ,\nu )=\int _{0}^{1}|F^{-1}(t)-G^{-1}(t)|^{2}dt$

where $F^{-1}$ and $G^{-1}$ are the pseudo-inverses of the respective CDFs.

Ideas and Remarks for the Proof

Note the proof from Cedric-Villani gives this result in arbitrary dimensions. Below is a rough outline of proof, and the full details can be found in "Topics in Optimal Transportation" (Villani, cite later). Moreover, the measure $\pi$ constructed in the theorem indeed is optimal provided that the cost function $c(x,y)$ is of the form $c(x-y)$ where $c$ is a convex, nonnegative symmetric function on $\mathbb {R}$.

One of the first major steps in proving this theorem is showing that $supp(\pi )\subset \{(x,y)\in \mathbb {R} ^{2}:F(x^{-})\leq G(y){\text{ and }}G(y^{-})\leq F(x)\}$ by considering specific cases. Upon showing this, we may conclude that $\pi$ is supported in a monotone subset of $\mathbb {R} ^{2}$ and hence also supported in the sub-differential of some lower semi-continuous convex function. From here, we make use of the Knott-Smith optimality criterion (Villani pg 66) which establishes that $\pi$ is an optimal transference plan. Then, upon showing that $\pi =(F^{-1}\times G^{-1})\#\lambda _{[0,1]}$, we see that for any nonnegative, measurable function $u$ on $\mathbb {R} ^{2}$

                                                                                         $\int _{\mathbb {R} ^{2}}u(x,y)d\pi (x,y)=\int _{0}^{1}u(F^{-1}(t),G^{-1}(t))dt$.

This then immediately yields the cost $T_{2}(\mu ,\nu )$ and completes the proof.