View source for 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

:<math> \inf_{T} \left \{ F(T) := \int_{X} |x - T(x)|^2 d \mu \right \} </math>

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 [[Optimal Transport and the Monge Ampère equation | Monge Ampère Equation]]<ref name="Santambrogio" /> 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, <math> T </math> must satisfy the equation

:<math>  g(y) = \frac{f(T^{-1}(y))}{\det (DT(T^{-1}(y)))} </math>

where <math> g(y) </math> and <math> f(y) </math> 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 <math> T = \nabla u </math>, where <math> u </math> is a convex function. If we require <math> u </math> to be strictly convex, a change of variables gives us the Monge Ampère equation

:<math> \det (D^2 u(x)) = \frac{f(x)}{g(\nabla u(x))} </math>

From here, we can ask about regularity of solutions of to the PDE, which in turns gives us regularity on <math> T = \nabla u </math>. For example, we have the following theorem.
: '''Theorem.'''<ref name=Santambrogio /> If <math> f </math> and <math> g </math> are <math> C^{0, \alpha}(\Omega) </math> and are both bounded from above and from below on the whole <math> \Omega </math> by positive constants and <math> \Omega </math> is a convex open set, then the unique Brenier solution of <math> u </math> of the Monge Ampère equation belongs to <math> C^{2, \alpha} (\Omega) \cap C^{1, \alpha} {\overline{\Omega}} </math>, and <math> u </math> satisfies the equation in the classical sense.
Here, a Brenier solution simply implies that <math> T </math> is a transport plan from <math> f </math> to <math> g </math>. Note that the support of <math> g </math> is very relevant when trying to show regularity of our transport plan. Without any conditions on the support of <math> g </math>, <math> T</math> may have singularities. On the other hand, conditions on the support of <math> g</math> can eliminate such singularities. For example, Caffarelli has shown that if <math> f,g </math> are smooth and strictly positive on their support, and the support of <math> g </math> is convex, the optimal transport plan will be smooth in the support of <math> f </math>. 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 <math> f </math> and the support of <math> g </math>.

== Existence and Uniqueness on Riemannian Manifolds ==
Parts of the previous section can be summarized into the following theorem.
: '''Theorem.'''<ref name="Regularity"/> Let <math> \mu </math> and <math> \nu </math> be two compactly supported probability measures on <math> R^n </math>. If <math> \mu </math> is absolutely continuous with respect to the Lebesgue measure, then:
:* There exists a unique solution <math> T </math> to the Monge problem.
:* The optimal map <math> T </math> is characterized by the structure <math> T(x) = \nabla u(x) </math>, for some convex function <math> u: R^n \to R </math>.
:Furthermore, if <math> \mu(dx) = f(x)dx </math>, and <math> \nu(dy) = g(y) dy </math>, 
:<math> | \mathrm{det} (\nabla T(x)) | = \frac{f(x)}{g(T(x))} </math> for <math>\mu</math>-a.e. <math>x \in R^n</math>
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<ref name="Regularity"/>. To state the theorem for Riemannian manifolds, we need the following definitions. Let <math> c:X \times Y \to R </math> be an arbitrary fucntion. Then:

A function is <math> \psi: X \to R \cup \{+\infty\} </math> is '''c-convex''' if
:<math> \psi(x) = \sup_{y \in Y} \left[\psi^c(y) - c(x,y)\right] </math> for all <math> x \in X </math>, where
:<math> \psi^c(y) = \inf_{x \in X} \left[\psi(x) + c(x, y) \right] </math> for all <math> y \in Y </math>.
The '''<math>c</math>-subdifferential at <math>x</math>''' of a <math>c</math>-convex function <math>\psi</math> is
:<math> \partial^c \psi(x) := \{ y \in Y \, | \, \psi(x) = \psi^c(y) - c(x, y) \} </math>

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

We now state the desired theorem:

:'''Theorem.'''<ref name="Regularity"/> Let <math> (M, g) </math> be a Riemannian manifold, take <math> \mu </math> and <math> \nu </math> two compactly supported measures in on <math> M </math>, and consider the optimal transport problem from <math> \mu </math> to <math> \nu </math> with cost <math> c(x,y) = d(x,y)^2/2 </math>, where <math> d(x,y) </math> denotes the Riemannian distance on <math> M </math> If <math> \mu </math> is absolutely continuous with respect to the volume measure, then:
:* There exists a unique solution <math> T </math> to the Monge problem.
:* <math> T </math> is characterized by the structure <math> T(x) = \exp_x (\nabla \psi(x)) \in \partial^c \psi(x) </math> for some <math>c</math>-convex function <math>\psi: M \to R</math>
:* For <math>\mu_0</math>-a.e. <math> x \in M </math>, there exists a unique minimizing geodesic from <math> x </math> to <math> T(x) </math>, which is given by <math> t \to \exp_x(t\nabla \psi(x)) \in [0,1] </math>
:Furthermore, if <math> \mu(dx) = f(x) \mathrm{vol}(dx) </math>, and <math> \nu(dy) = g(y) \mathrm{vol}(dy) </math>,
:<math> |\det(\nabla T(x))| = \frac{f(x)}{g(T(x)} </math> for <math>\mu</math>-a.e. <math>x \in M</math>.
Note that although the determinant of <math> \nabla T(x) </math> 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,
:<math> |\det(\nabla T(x))| = \frac{f(x)}{g(T(x)} </math>
to make claims about regularity. Recall that we want the condition <math> T(x) = \exp_x (\nabla \psi(x)) \in \partial^c \psi(x) </math>. It can be shown that this is equivalent to 
:<math> \nabla \psi(x) + \nabla_x c(x, T(x)) = 0 </math>
By differentiating the above identity with respect to <math> x </math> and writing everything in charts, we get the equation
:<math> \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 |} </math>
, which is very similar to the Monge Ampère Equation we derived for transport plans in <math> R^n </math>. We simply have a perturbation of <math> D_x^2 c(x, \exp_x (\nabla \psi(x))) </math>. 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 <math> \psi </math>. 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:
:<math> \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_xM, \eta \in T_yM </math>.

The MTW Condition is
:<math> \vartheta_{(x,y)}(\xi, \eta) \ge 0 </math> whenever <math> \sum_{ij} c_{i,j}\xi^i\eta^j = 0 </math>.

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

:'''Theorem.'''<ref name="Regularity"/> Let <math> (M, g) </math> be a Riemannian manifold. Assume the MTW condition holds, that <math> f </math> and <math> g </math> are smooth and bounded away from zero and infinity on their respective supports <math> \Omega </math> and <math> \Omega' </math>, and that the cost function <math> c=d^2/2 </math> is smooth on the set <math> \overline{\Omega} \times \overline{\Omega'} </math>. Finally, suppose that:
:* <math> \Omega </math> and <math> \Omega' </math> are smooth;
:* <math> (\exp_x)^{-1}(\Omega') \subset T_xM </math> is uniformly convex for all <math> x \in \Omega </math>;
:* <math> (\exp_y)^{-1}(\Omega) \subset T_yM </math> is uniformly convex for all <math> y \in \Omega' </math>.
:Then <math> \psi \in C^{\infty} (\overline{\Omega}) </math>, and <math> T: \overline{\Omega} \to \overline{\Omega'} </math> is a smooth diffeomorphism.

== References ==

<references>
<ref name="Santambrogio">[https://link.springer.com/book/10.1007/978-3-319-20828-2 F. Santambrogio, ''Optimal Transport for Applied Mathematicians'', p. 54-57]</ref>
<ref name="Regularity">[http://www.numdam.org/item/AST_2010__332__341_0.pdf A. Figalli, ''Regularity of optimal transport maps '']</ref>
</references>