Regularity of Optimal Transport Maps and the Monge-Ampére Equation

When considering the Monge Problem, it is natural to ask about the regularity of optimal transport maps (when they exist). In particular, we can consider the Monge Problem variant

$\inf _{T}\left\{F(T):=\int _{X}|x-T(x)|^{2}d\mu \right\}$

where we are taking the infimum over all transport plans for our associated measure. This allows us to reformulate the problem into a boundary value problem for a specific partial differential equation. From there, one can ask about the regularity of solutions to the PDE, which are associated with optimal transport plans.

The Monge Ampère Equation

The Monge Ampère Equation is a nonlinear second-order elliptic partial differential equation. When we consider the kind of Monge Problem discussed earlier, we can show that the optimal transport plan, $T$ must satisfy the equation

$g(y)={\frac {f(T^{-1}(y))}{\det(DT(T^{-1}(y)))}}$

where $g(y)$ and $f(y)$ are the respective target and starting measures for the transport problem. Note that these measures are assumed to be absolutely continuous to the Lebesgue measure. The relevant Monge Problem has a quadratic cost problem, which can be shown to imply that $T=\nabla u$, where $u$ is a convex function. If we require $u$ to be strictly convex, a change of variables gives us the Monge Ampère equation

$\det(D^{2}u(x))={\frac {f(x)}{g(\nabla u(x))}}$

From here, we can ask about regularity of solutions of to the PDE, which in turns gives us regularity on $T=\nabla u$. For example, we have the following theorem.

Theorem. If $f$ and $g$ are $C^{0,\alpha }(\Omega )$ and are both bounded from above and from below on the whole $\Omega$ by positive constants and $\Omega$ is a convex open set, then the unique Brenier solution of $u$ of the Monge Ampère equation belongs to $C^{2,\alpha }(\Omega )\cap C^{1,\alpha }{\overline {\Omega }}$, and $u$ satisfies the equation in the classical sense.

Here, a Brenier solution simply implies that $T$ is a transport plan from $f$ to $g$. Note that the support of $g$ is very relevant when trying to show regularity of our transport plan. Without any conditions on the support of $g$, $T$ may have singularities. On the other hand, conditions on the support of $g$ can eliminate such singularities. For example, Caffarelli has shown that if $f,g$ are smooth and strictly positive on their support, and the support of $g$ is convex, the optimal transport plan will be smooth in the support of $f$. Moreover, if both supports are smooth and uniformly convex, one can show that the optimal transport plan is a smooth diffeomorphism between the support of $f$ and the support of $g$.

Existence and Uniqueness on Riemannian Manifolds

Parts of the previous section can be summarized into the following theorem.

Theorem. Let $\mu$ and $\nu$ be two compactly supported probability measures on $R^{n}$. If $\mu$ is absolutely continuous with respect to the Lebesgue measure, then:

• There exists a unique solution $T$ to the Monge problem.
• The optimal map $T$ is characterized by the structure $T(x)=\nabla u(x)$, for some convex function $u:R^{n}\to R$.

Furthermore, if $\mu (dx)=f(x)dx$, and $\nu (dy)=g(y)dy$,

$|\mathrm {det} (\nabla T(x))|={\frac {f(x)}{g(T(x))}}$ for $\mu$-a.e. $x\in R^{n}$

Note that many of these notions, such as Lebesgue measure and gradients, all have well-defined generalizations on all Riemannian manifolds. In particular, the above theorem was able to be extended to compact Riemannian manifolds. To state the theorem for Riemannian manifolds, we need the following definitions. Let $c:X\times Y\to R$ be an arbitrary fucntion. Then:

A function is $\psi :X\to R\cup \{+\infty \}$ is c-convex if

$\psi (x)=\sup _{y\in Y}\left[\psi ^{c}(y)-c(x,y)\right]$ for all $x\in X$, where

$\psi ^{c}(y)=\inf _{x\in X}\left[\psi (x)+c(x,y)\right]$ for all $y\in Y$.

Moreover, for a $c$-convex function $\psi (x)$, we can definite its $c$-subdifferential at $x$ as

$\partial ^{c}\psi (x):=\{y\in Y\,|\,\psi (x)=\psi ^{c}(y)-c(x,y)\}$

Remark: observe that if we take both $X$ and $Y$ to be Euclidean space with $-c(x,y)$ as the usual inner product, then $\psi ^{c}(y)_{=}-\sup _{x\in X}\left[x\cdot y-\psi (x)\right]=-\psi ^{*}(y)$ where $\psi ^{*}$ denotes the conjugate of $\psi$. Consequently, if $\psi$ is c-convex, then replacing $\psi ^{c}(y)$ with $-\psi ^{*}(y)$ in the equality above for $\psi (x)$ yields that $\psi (x)=\psi ^{**}(x)$; ergo, $\psi$ is convex (assuming it is proper). Similarly, the $c$-subdifferential coincides with the usual subdifferential.

We now state the desired theorem:

Theorem. Let $(M,g)$ be a Riemannian manifold, take $\mu$ and $\nu$ two compactly supported measures in on $M$, and consider the optimal transport problem from $\mu$ to $\nu$ with cost $c(x,y)=d(x,y)^{2}/2$, where $d(x,y)$ denotes the Riemannian distance on $M$ If $\mu$ is absolutely continuous with respect to the volume measure, then:

• There exists a unique solution $T$ to the Monge problem.
• $T$ is characterized by the structure $T(x)=\exp _{x}(\nabla \psi (x))\in \partial ^{c}\psi (x)$ for some $c$-convex function $\psi :M\to R$
• For $\mu _{0}$-a.e. $x\in M$, there exists a unique minimizing geodesic from $x$ to $T(x)$, which is given by $t\to \exp _{x}(t\nabla \psi (x))\in [0,1]$

Furthermore, if $\mu (dx)=f(x)\mathrm {vol} (dx)$, and $\nu (dy)=g(y)\mathrm {vol} (dy)$,

$|\det(\nabla T(x))|={\frac {f(x)}{g(T(x)}}$ for $\mu$-a.e. $x\in M$.

Note that although the determinant of $\nabla T(x)$ is not intrinsically defined, but the absolute value of that same determinant is intrinsically defined.

Regularity on Compact Riemannian Manifolds

Once again, we will make use of the Monge Ampère Equation,

$|\det(\nabla T(x))|={\frac {f(x)}{g(T(x)}}$

to make claims about regularity. Recall that we want the condition $T(x)=\exp _{x}(\nabla \psi (x))\in \partial ^{c}\psi (x)$. It can be shown that this is equivalent to

$\nabla \psi (x)+\nabla _{x}c(x,T(x))=0$

By differentiating the above identity with respect to $x$ and writing everything in charts, we get the equation

$\det(D^{2}\psi (x))+D_{x}^{2}c(x,\exp _{x}(\nabla \psi (x)))={\frac {f(x)\mathrm {vol} _{x}}{g((T(x))\mathrm {vol} _{T(x)}|\det(d_{\nabla \psi (x)}\exp _{x}|}}$

, which is very similar to the Monge Ampère Equation we derived for transport plans in $R^{n}$. We simply have a perturbation of $D_{x}^{2}c(x,\exp _{x}(\nabla \psi (x)))$. This perturbation can obstruct smoothness, which is where the MTW condition comes in. One can try to take the second derivative of the previous equation in order to make an a priori estimate on the second derivatives of $\psi$. Doing so requires a condition on the sign of a term that is now called the Ma-Trundinger-Wang tensor (or the MTW tensor for short). The tensor is defined as the following:

$\vartheta _{(x,y)}(\xi ,\eta ):={\frac {3}{2}}\sum _{ijklrs}(c_{ij,r}c^{r,s}c_{s,kl}-c_{ij,kl})\xi ^{i}\xi ^{j}\eta ^{k}\eta ^{l},\,\xi \in T_{x}M,\eta \in T_{y}M$.

The MTW Condition is

$\vartheta _{(x,y)}(\xi ,\eta )\geq 0$ whenever $\sum _{ij}c_{i,j}\xi ^{i}\eta ^{j}=0$.

From here, one can prove the Riemannian analogue to one of the regularity theorems we mentioned for the Monge problem in $R^{n}$. The theorem is as follows:

Theorem. Let $(M,g)$ be a Riemannian manifold. Assume the MTW condition holds, that $f$ and $g$ are smooth and bounded away from zero and infinity on their respective supports $\Omega$ and $\Omega '$, and that the cost function $c=d^{2}/2$ is smooth on the set ${\overline {\Omega }}\times {\overline {\Omega '}}$. Finally, suppose that:

• $\Omega$ and $\Omega '$ are smooth;
• $(\exp _{x})^{-1}(\Omega ')\subset T_{x}M$ is uniformly convex for all $x\in \Omega$;
• $(\exp _{y})^{-1}(\Omega )\subset T_{y}M$ is uniformly convex for all $y\in \Omega '$.

Then $\psi \in C^{\infty }({\overline {\Omega }})$, and $T:{\overline {\Omega }}\to {\overline {\Omega '}}$ is a smooth diffeomorphism.