Optimal Transport Map – Existence and Uniqueness with Strict Convexity

calculus-of-variationsconvex-analysisoc.optimization-and-controloptimal-transportationreference-request

Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable maps $T:X \to Y$ such that $T_\sharp \mu = \nu$. Let
$$
\mathbb M (T) := \int_X c(x, T(x)) \mathrm d \mu(x) \quad \forall T \in \mathcal T.
$$

Then we are interested in the Monge's transportation problem
$$
\mathrm{MP} : \quad\inf_{T \in \mathcal T} \mathbb M (T).
$$

The existence and uniqueness of the solution of $\mathrm{MP}$ is guaranteed if $c$ is strictly convex and the supports of $\mu, \nu$ are compact [1]. We can remove the assumption of compact supports by stronger conditions on $c$ [2].

Let $\Pi (\mu, \nu)$ be the set of Borel probability measures on $X \times Y$ with marginals $\mu$ on $X$ and $\nu$ on $Y$. Recently, I have seen a strong theorem from this lecture note, i.e.,

Theorem 3.14. Assume

  1. $h:\mathbb R^d \to [0, \infty)$ is strictly convex and $c(x, y) := h(x-y)$ for all $(x, y) \in X \times Y$.
  2. $\int_{X \times Y} c \mathrm{d} \gamma <\infty$ for some $\gamma \in \Pi(\mu, \nu)$,
  3. $\mu\left(\left\{x \in X: \int_Y c(x, y) \mathrm{d} \nu(y)<\infty\right\}\right)>0$,
  4. $\nu\left(\left\{y \in Y: \int_X c(x, y) \mathrm{d} \mu(x)<\infty\right\}\right)>0$,
  5. $\mu$ is absolutely continuous with respect to Lebesgue measure.

Then $\mathrm{MP}$ has a unique (up to $\mu$-a.e. solution).

The conditions 2, 3, 4 are very mild and just to ensure the dual of the dual of the corresponding Kantorovich's problem has a solution in a form of a pair of $c$-conjugates. Theorem 3.14. is striking because it does not require the supports of $\mu, \nu$ to be compact nor any condition on $c$ besides strict convexity.

Could you elaborate if there are some references of Theorem 3.14.?

[1] Caffarelli, Luis A. "Allocation maps with general cost functions." Partial differential equations and applications. Routledge, 2017. 29-35.

[2] Gangbo, Wilfrid, and Robert J. McCann. "The geometry of optimal transportation." Acta Mathematica 177.2 (1996): 113-161.

Best Answer

First a comment: you write (before stating your Theorem 3.14) that "The existence and uniqueness of the solution of the Monge Problem is guaranteed if $c$ is strictly convex and the supports of $\mu,\nu$ are compact", but this is absolutely not true: you really need some conditions on the starting point, i-e that $\mu$ does not charge "small sets" in some sense (this is exactly assumption 5 in your theorem 3.14, but this can be relaxed). If $\mu=\delta_x$ is a Dirac delta there exists no transport map $T$ from $\mu$ to $\nu$, unless $\nu=\delta_{T(x)}$, so clearly the Monge problem is ill-posed in general.

Next, my real answer: this precise statement is often called the Brenier-McCann theorem. You can find an extremely general version in [1], Theorem 10.38. Note in particular that this is stated without any compactness assumptions or behaviour at infinity, and that the strict convexity $c(x,y)=h(|x-y|)$ is not needed (only the so-called and weaker twist condition). This result is usually credited to Brenier, Rachev and Rüschendorf for the quadratic cost in Euclidean spaces, and then R. McCann extended to Riemannian manifolds.

[1] Villani, C. (2009). Optimal transport: old and new (Vol. 338, p. 23). Berlin: springer.

Related Question