Monge maps for Gromov-Wasserstein

1. Short intro to optimal transport

Gaspard Monge

Leonid Kantorovitch

[Monge, 1781], [Kantorovitch, 1942]

a measure over a set $\mathcal X$ : a function $\mu:\Sigma_{\mathcal X}\to\mathbb R$ that satisfies
1. $\mu(B)\geq0$ for all $B\in\Sigma_{\mathcal X}$
2. $\mu(\varnothing)=0$
3. countable additivity
a probability measure: $\mu(\mathcal X)=1$

\mu

\mu

A "continuous" measure $\mathrm d\mu(x)=f(x)\mathrm dx$ .
(has a density w.r.t. the Lebesgue measure $\mathrm dx$ ).

A discrete measure $\mu=\sum_{i=1}^n a_i\delta_{x_i}$ .

\mu

\mu

Introduction

measures can represent anything:
point clouds, histograms, 2D images, 3D images, densities of a fluid...

[Monge, 1781], [Kantorovitch, 1942]

\mu

\mu

A "continuous" measure $\mathrm d\mu(x)=f(x)\mathrm dx$ .
(has a density w.r.t. the Lebesgue measure $\mathrm dx$ )

A discrete measure $\mu=\sum_{i=1}^n a_i\delta_{x_i}$ .

pushforward measure $T_*\mu$ , defined as $T_*\mu(B)\coloneqq \mu(T^{-1}(B))$ for $T:\mathcal X\to\mathcal Y$

\mu

\mu

\mu

\mu

T_*\mu

T_*\mu

T

T

x

x

T(x)

T(x)

for a continuous measure:

\mu

\mu

\delta_x

\delta_x

\delta_{T(x)}

\delta_{T(x)}

T_*\mu

T_*\mu

for a discrete measure:

Introduction

\mathrm d(T_*\mu)(x)=|\!\det \mathrm d(T^{-1})(x)|\mu(T^{-1}(x))\mathrm dx

\mathrm d(T_*\mu)(x)=|\!\det \mathrm d(T^{-1})(x)|\mu(T^{-1}(x))\mathrm dx

\mu

\mu

T_*\mu

T_*\mu

[Monge, 1781], [Kantorovitch, 1942]

\mu

\mu

\nu

\nu

T

T

x

x

T(x)

T(x)

probability measures $\mu,\nu\in \mathcal P(\mathbb R^n)$
piles of sand: find strategy $T$

\displaystyle \text{OT}(\mu,\nu)= \inf_{T} \int_{\mathbb R^n} c\big(x,T(x)\big)\,\mathrm d\mu(x)\qquad\text{over }T:\mathbb R^n\to\mathbb R^n\text{ such that } T_*\mu=\nu.

\displaystyle \text{OT}(\mu,\nu)= \inf_{T} \int_{\mathbb R^n} c\big(x,T(x)\big)\,\mathrm d\mu(x)\qquad\text{over }T:\mathbb R^n\to\mathbb R^n\text{ such that } T_*\mu=\nu.

OT problem (Monge)

find the best strategy: cost function $c:\mathbb R^n\times\mathbb R^n\to\mathbb R$ (e.g. $\|x-y\|^2$ )

Optimal transport

\displaystyle \text{OT}(\mu,\nu)= \inf_{T} \int_{\mathcal X} c\big(x,T(x)\big)\,\mathrm d\mu(x)\qquad\text{over }T:\mathcal X\to\mathcal Y\text{ such that } T_*\mu=\nu.

\displaystyle \text{OT}(\mu,\nu)= \inf_{T} \int_{\mathcal X} c\big(x,T(x)\big)\,\mathrm d\mu(x)\qquad\text{over }T:\mathcal X\to\mathcal Y\text{ such that } T_*\mu=\nu.

OT problem (Monge)

[Monge, 1781], [Kantorovitch, 1942]

\mu

\mu

\nu

\nu

T

T

$\mathcal X$ and $\mathcal Y$ Polish spaces
$\mu\in\mathcal P(\mathcal X),\, \nu\in\mathcal P(\mathcal Y)$
cost function $c:\mathcal X\times\mathcal Y\to\mathbb R$

Optimal transport

\mu

\mu

\nu

\nu

T

T

x

x

graph of $T$ : $\big\{(x,T(x))\mid x\in\mathcal X\big\}\subset \mathcal X\times\mathcal Y$

T(x)

T(x)

\displaystyle \text{OT}(\mu,\nu)= \inf_{T} \int_{\mathcal X} c\big(x,T(x)\big)\,\mathrm d\mu(x)\qquad\text{over }T:\mathcal X\to\mathcal Y\text{ such that } T_*\mu=\nu.

\displaystyle \text{OT}(\mu,\nu)= \inf_{T} \int_{\mathcal X} c\big(x,T(x)\big)\,\mathrm d\mu(x)\qquad\text{over }T:\mathcal X\to\mathcal Y\text{ such that } T_*\mu=\nu.

OT problem (Monge)

\displaystyle \text{OT}_K(\mu,\nu)= \inf_{\pi} \int_{\mathcal X\times\mathcal Y} c(x,y)\,\mathrm d\pi(x,y)\quad\text{over }\pi\in\mathcal P(\mathcal X\times\mathcal Y)\text{ of marginals }\mu \text{ and } \nu.

\displaystyle \text{OT}_K(\mu,\nu)= \inf_{\pi} \int_{\mathcal X\times\mathcal Y} c(x,y)\,\mathrm d\pi(x,y)\quad\text{over }\pi\in\mathcal P(\mathcal X\times\mathcal Y)\text{ of marginals }\mu \text{ and } \nu.

OT problem (Kantorovitch)

\delta_x

\delta_x

\frac12\delta_{y_1}

\frac12\delta_{y_1}

\frac12\delta_{y_2}

\frac12\delta_{y_2}

not feasible by a map!

$\pi$ is induced by a transport map $T$

$\pi$ is a transport plan

relaxation

\mu

\mu

\nu

\nu

\pi

\pi

\mu

\mu

\nu

\nu

\pi

\pi

[Monge, 1781], [Kantorovitch, 1942]

\mu

\mu

\nu

\nu

T

T

$\mathcal X$ and $\mathcal Y$ Polish spaces
$\mu\in\mathcal P(\mathcal X),\, \nu\in\mathcal P(\mathcal Y)$
cost function $c:\mathcal X\times\mathcal Y\to\mathbb R$

Optimal transport

\displaystyle \text{OT}_K(\mu,\nu)= \inf_{\pi} \int_{\mathcal X\times\mathcal Y} c(x,y)\,\mathrm d\pi(x,y)\quad\text{over }\pi\in\mathcal P(\mathcal X\times\mathcal Y)\text{ of marginals }\mu \text{ and } \nu.

\displaystyle \text{OT}_K(\mu,\nu)= \inf_{\pi} \int_{\mathcal X\times\mathcal Y} c(x,y)\,\mathrm d\pi(x,y)\quad\text{over }\pi\in\mathcal P(\mathcal X\times\mathcal Y)\text{ of marginals }\mu \text{ and } \nu.

OT problem (Kantorovitch)

[Monge, 1781], [Kantorovitch, 1942]

\mu

\mu

\nu

\nu

T

T

$\mathcal X$ and $\mathcal Y$ Polish spaces
$\mu\in\mathcal P(\mathcal X),\, \nu\in\mathcal P(\mathcal Y)$
cost function $c:\mathcal X\times\mathcal Y\to\mathbb R$

the set of transport plans is non-empty (always $\mu\otimes\nu$ ), so existence of minimizers!
linear program in $\pi$ : easy to solve!
if $c(x,y)=\|x-y\|^p_2$ : $p$ -Wasserstein distance (sometimes Earth Mover Distance)

Optimal transport

Brenier's theorem

When $\mathcal X=\mathcal Y=\mathbb R^n$ and $c(x,y)=\|x-y\|^2$ , if $\mu\ll\mathrm dx$ , then there is a unique solution to (KP), and it is induced by a map $T=\nabla f$ with $f:\mathbb R^n\to\mathbb R$ convex.

relaxation

$\pi$ is induced by a transport map $T$

$\pi$ is a transport plan

Monge (maps)

Kantorovitch (plans)

\mu

\mu

\nu

\nu

\pi

\pi

\mu

\mu

\nu

\nu

\pi

\pi

[Brenier, 1987]

Can we say that the solution of (KP) is a map?

?

generalizations to complete Riemannian manifolds $\mathcal X$ and $\mathcal Y$ and other cost functions $c$ ?

?

Optimal transport

2. Optimal maps for OT

Yann Brenier

Robert McCann

Cédric Villani

Map solutions of OT

Twist condition

We say that $c$ satisfies the twist condition if
$\text{for all }x_0\in\mathcal X,\quad y\mapsto \nabla_x c(x_0,y)\in T_{x_0}\mathcal X \text{ is injective.}$

Suppose this is satisfied. If $\mu \ll \mathrm dx$ , then (KP) admits a unique solution and it is supported on the graph of a map which is the gradient of a $c$ -convex function $f:\mathcal X\to\mathbb{R}$ :
$\pi^\star=(\text{id},c\text{-}\exp_x(\nabla f))_*\mu.$

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

x

x

T(x)

T(x)

[Gangbo, 1996], [Villani, 2008], [McCann and Guillen, 2011]

Examples:
- $\|x-y\|^2$ in $\mathbb R^n$
- $\langle x,y\rangle$ in $\mathbb R^n$
- $\langle x,y\rangle$ on $\mathbb S^{n-1}$

$c$ - $\exp_x(p)$ is the unique $y$ satisfying $\nabla_xc(x,y)+p=0$ .
usual Riemannian exp when $c=d^2/2$ .

Map solutions of OT

Subtwist condition

We say that $c$ satisfies the subtwist condition if
$\text{for all }y_1\neq y_2,\quad x\mapsto c(x,y_1)-c(x,y_2)\text{ has at most 2 critical points.}$

Suppose this is satisfied. If $\mu \ll \mathrm dx$ , then (KP) admits a unique solution and it is supported on the union of a graph and an anti-graph:
$\pi^\star=(\text{id},G)_*\bar \mu+(H,\text{id})_*(\nu- G_*\bar\mu).$

Examples:
- $\|x-y\|^2$ in $\mathbb R^n$
- $\langle x,y\rangle$ in $\mathbb R^n$
- $\langle x,y\rangle$ on $\mathbb S^{n-1}$

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

x

x

[Ahmad et al., 2011], [Chiappori et al., 2010]

Map solutions of OT

$m$ -twist condition

We say that $c$ satisfies the $m$ -twist condition if
$\text{for all }x_0, y_0,\quad \text{card}\{y\mid \nabla_x c(x_0,y)=\nabla_x c(x_0,y_0)\}\leq m.$

Suppose this is satisfied and $c$ is bounded. If $\mu \ll \mathrm dx$ , then optimals plans of (KP) are supported on the graphs of $m$ maps:
$\pi^\star=\sum_{i=1}^m\alpha_i (\text{id},T_i)_* \mu.$

in the sense $\pi^\star(S)=\sum_i \int_{\mathcal X}\alpha_i(x)\chi_S(x,T_i(x))\,\mathrm d\mu$ for any Borel $S\subset \mathcal X\times \mathcal Y$ .

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

x

x

[Moameni, 2016]

Map solutions of OT

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

twist

map

$\implies$

subtwist

map/anti-map

$\implies$

$m$ -twist

$m$ -map

$\implies$

(for simplicity, when $\mu\ll\mathrm dx$ and $\mu,\nu$ have compact support)

\min_\pi\int c(x,y)\,\mathrm d\pi(x,y)

\min_\pi\int c(x,y)\,\mathrm d\pi(x,y)

for linear OT problem:

Map solutions of OT

3. Optimal maps for Gromov-Wasserstein

Karl-Theodor Sturm

Facundo Mémoli

Mikhaïl Gromov

[Sturm, 2012]

|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')|^2

|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')|^2

x

x

\mu

\mu

y

y

\nu

\nu

c(x,y)

c(x,y)

Wasserstein:

\mu

\mu

\nu

\nu

x

x

x'

x'

y'

y'

y

y

c_{\mathcal X}(x,x')

c_{\mathcal X}(x,x')

c_{\mathcal Y}(y,y')

c_{\mathcal Y}(y,y')

Gromov-Wasserstein:

cost function
$c:\mathcal X\times\mathcal Y\to\mathbb R$

cost functions
$c_{\mathcal X}:\mathcal X\times\mathcal X\to\mathbb R$
$c_{\mathcal Y}:\mathcal Y\times\mathcal Y\to\mathbb R$

The Gromov-Wasserstein problem

[Sturm, 2012]

\displaystyle \text{GW}^2(\mu,\nu)=\min _{\pi} \int_{\mathcal X\times\mathcal Y}\int_{\mathcal X\times\mathcal Y}\Big|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')\Big|^{2} \,\mathrm d\pi(x,y)\,\mathrm d\pi(x',y')\quad\text{over }\pi\in\Pi(\mu,\nu)

\displaystyle \text{GW}^2(\mu,\nu)=\min _{\pi} \int_{\mathcal X\times\mathcal Y}\int_{\mathcal X\times\mathcal Y}\Big|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')\Big|^{2} \,\mathrm d\pi(x,y)\,\mathrm d\pi(x',y')\quad\text{over }\pi\in\Pi(\mu,\nu)

GW problem

distance modulo isometries: not really transport but rather correspondence
applications: invariance by isometries + measures living in different spaces
quadratic in $\pi$ ! $\implies$ hard to solve

?

optimal plans = maps?

\mu

\mu

\nu

\nu

x

x

x'

x'

y'

y'

y

y

c_{\mathcal X}(x,x')

c_{\mathcal X}(x,x')

c_{\mathcal Y}(y,y')

c_{\mathcal Y}(y,y')

|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')|^2

|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')|^2

[Alvares-Melis et al., 2019], [Vayer, 2020], [Sturm, 2012], [D., Lacombe & Vialard, 2023]

$\mu\ll\mathrm dx$ and $\mu,\nu$ with compact support

There is an optimal map!

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

There is an optimal 2-map!

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(i) Inner product case, $c_{\mathcal X}=c_{\mathcal Y}=\langle\cdot,\cdot\rangle$

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(ii) Squared distance case, $c_{\mathcal X}=c_{\mathcal Y}=\|\cdot-\cdot\|^2$

$\mathcal X=\mathcal Y=\mathbb R^n$

?

Can we simply apply the twist conditions? No....

\int c(x,y)\,\mathrm d\pi(x,y)

\int c(x,y)\,\mathrm d\pi(x,y)

for linear OT problems:

Optimal maps for GW

\text{GW}=\min_\pi Q(\pi)=\min_\pi F(\pi,\pi)

\text{GW}=\min_\pi Q(\pi)=\min_\pi F(\pi,\pi)

quadratic

Optimal maps for GW

symmetric
bilinear

\implies

\implies

\min_\pi\int C_{\pi^\star}(x,y)\,\mathrm d\pi(x,y)

\min_\pi\int C_{\pi^\star}(x,y)\,\mathrm d\pi(x,y)

\text{with } C_{\pi^\star}(x,y)=\int \big|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')\big|^{2}\,\mathrm d\pi^\star(x',y')

\text{with } C_{\pi^\star}(x,y)=\int \big|c_{\mathcal X}(x,x')-c_{\mathcal Y}(y,y')\big|^{2}\,\mathrm d\pi^\star(x',y')

Idea: relax into linear problem and try to apply twist conditions

[D., Lacombe & Vialard, 2023]

First order optimality condition:

$\pi^\star$ minimizes $\pi\mapsto F(\pi,\pi)$ $\pi^\star$ minimizes $\pi\mapsto 2F(\pi,\pi^\star)$

Good news: we now have a OT problem with cost $C_{\pi^\star}$ !

:)

twist conditions for $C_{\pi^\star}$ ? not always, need something more general

:(

[D., Lacombe & Vialard, 2023]

"Let $\mu,\nu\in\mathcal P(E)$ .

A more general twist condition

[D., Lacombe & Vialard, 2023]

"Let $\mu,\nu\in\mathcal P(E)$ . If we can send $\mu$ and $\nu$ in a space $B$ by a map $\varphi:E\to B$ ,

\varphi_*\mu

\varphi_*\mu

\varphi_*\nu

\varphi_*\nu

A more general twist condition

[D., Lacombe & Vialard, 2023]

"Let $\mu,\nu\in\mathcal P(E)$ . If we can send $\mu$ and $\nu$ in a space $B$ by a map $\varphi:E\to B$ , such that $c(x,y)=\tilde c(\varphi(x),\varphi(y))\quad\text{for all }x,y\in E$ with $\tilde c$ a twisted cost on $B$ ,

\varphi_*\mu

\varphi_*\mu

\varphi_*\nu

\varphi_*\nu

A more general twist condition

[D., Lacombe & Vialard, 2023]

"Let $\mu,\nu\in\mathcal P(E)$ . If we can send $\mu$ and $\nu$ in a space $B$ by a map $\varphi:E\to B$ , such that $c(x,y)=\tilde c(\varphi(x),\varphi(y))\quad\text{for all }x,y\in E$ with $\tilde c$ a twisted cost on $B$ , then we can construct an optimal map between $\mu$ and $\nu$ ."

\varphi_*\mu

\varphi_*\mu

\varphi_*\nu

\varphi_*\nu

A more general twist condition

[D., Lacombe & Vialard, 2023]

A more general twist condition

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

\mathcal Y

\mathcal Y

\mathcal X

\mathcal X

twist

map

$\implies$

subtwist

map/anti-map

$\implies$

$m$ -twist

$m$ -map

$\implies$

(for simplicity, when $\mu\ll\mathrm dx$ and $\mu,\nu$ have compact support)

A more general twist condition

our general condition

\min_\pi\int c(x,y)\,\mathrm d\pi(x,y)

\min_\pi\int c(x,y)\,\mathrm d\pi(x,y)

for linear OT problem:

[D., Lacombe & Vialard, 2023]

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(i) Inner product case, $c_{\mathcal X}=c_{\mathcal Y}=\langle\cdot,\cdot\rangle$

OT problem with cost
$C_{\pi^\star}(x,y)=-\langle M^\star x,y\rangle$
where $M^\star=\int x'y'^\top\,\mathrm d\pi(x',y')$

$\implies$

satisfies our general condition

$\implies$

there exists an optimal map

$\mu\ll\mathrm dx$ and $\mu,\nu$ with compact support

$\mathcal X=\mathcal Y=\mathbb R^n$

linearize

up to a SVD, suppose that $M^\star$ is a diagonal matrix of singular values: $M^\star=\begin{pmatrix}\sigma_1 & & & & & \\ & \ddots & & & & &\\& & \sigma_h & & &\\ & & & 0 & & \\ & & & & \ddots &\\ & & & & & 0 \end{pmatrix}$
rephrase the cost: $c(x,y)=-\langle M^\star x,y\rangle=-\sum_{i=1}^h\sigma_i x_iy_i=\tilde c(p(x),p(y))$ with $p$ the orthogonal projection on $\mathbb R^h$
check if $\tilde c$ is twisted: it is!

+ some structure!
$T(u,v)=(\nabla f\circ M^\star(u), \nabla g_u(v))$

Does it satisfy our general condition?

Optimal maps for GW

[D., Lacombe & Vialard, 2023]

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(i) Inner product case, $c_{\mathcal X}=c_{\mathcal Y}=\langle\cdot,\cdot\rangle$

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(ii) Squared distance case, $c_{\mathcal X}=c_{\mathcal Y}=\|\cdot-\cdot\|^2$

OT problem with cost
$C_{\pi^\star}(x,y)=-\langle M^\star x,y\rangle$
where $M^\star=\int x'y'^\top\,\mathrm d\pi(x',y')$

$\implies$

satisfies our general condition

$\implies$

there exists an optimal map

OT problem with cost
$C_{\pi^\star}(x,y)=-\|x\|^2\|y\|^2-4\langle M^\star x,y\rangle$
where $M^\star=\int x'y'^\top\,\mathrm d\pi(x',y')$

$\implies$

sometimes satisfies our general condition,
sometimes satisfies 2-twist

$\implies$

there exists an optimal 2-map

$\mathcal X=\mathcal Y=\mathbb R^n$

linearize

+ if $\text{rk}(M^\star)\leq n-2$ , there exists an optimal map!

$\mu\ll\mathrm dx$ and $\mu,\nu$ with compact support

Summary

[D., Lacombe & Vialard, 2023]

There is an optimal map!

There is an optimal 2-map!

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(i) Inner product case, $c_{\mathcal X}=c_{\mathcal Y}=\langle\cdot,\cdot\rangle$

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(ii) Squared distance case, $c_{\mathcal X}=c_{\mathcal Y}=\|\cdot-\cdot\|^2$

Conjecture (computational):
this result is tight: there exists cases where no optimal plan is a map

Additional study of 1D case:

non-optimality of monotone rearrangements in general (additional counter-example)
optimality of monotone rearrangements in specific cases

$\mathcal X=\mathcal Y=\mathbb R^n$

$\mu\ll\mathrm dx$ and $\mu,\nu$ with compact support

Ahmad, N., Kim, H. K., and McCann, R. J. (2011). Optimal transportation, topology and uniqueness.

Alvarez-Melis, D., Jegelka, S., and Jaakkola, T. S. (2019). Towards optimal transport with global invariances.

Beinert, R., Heiss, C., and Steidl, G. (2022). On assignment problems related to gromov-wasserstein distances on the real line.

Brenier, Y. (1987). Décomposition polaire et réarrangement monotone des champs de vecteurs

Dumont, T., Lacombe, T., and Vialard, F.-X. (2023). On the Existence of Monge maps for the Gromov-Wasserstein problem.

Fontbona, J., Guérin, H., and Méléard, S. (2010). Measurability of optimal transportation and strong coupling of martingale measures.

Gangbo, W., & McCann, R. J. (1996). The geometry of optimal transportation.

Kantorovich, L. (1942). On the translocation of masses.

McCann, R. J. and Guillen, N. (2011). Five lectures on optimal transportation: geometry, regularity and applications.

Mémoli, F. (2011). Gromov–wasserstein distances and the metric approach to object matching.

Moameni, A. (2016). A characterization for solutions of the monge-kantorovich mass transport problem.

Séjourné, T., Vialard, F.-X., and Peyré, G. (2021). The unbalanced gromov wasserstein distance: Conic formulation and relaxation.

Sturm, K.-T. (2020). The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces.

Vayer, T. (2020). A contribution to optimal transport on incomparable spaces

Villani, C. (2008). Optimal transport: old and new, volume 338.

slides available at https://slides.com/theodumont/monge-gw

•

References

[D., Lacombe & Vialard, 2023]

There always is an optimal 2-map!

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

\displaystyle\text{GW}^2= \min _{\pi} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(ii) Squared distance case, $c_{\mathcal X}=c_{\mathcal Y}=\|\cdot-\cdot\|^2$

Conjecture (computational):
this result is tight: there exists cases where no optimal plan is a map

$\mu\ll\mathcal L$ and $\mu,\nu$ with compact support

$\mathcal X=\mathcal Y=\mathbb R^n$

?

Can we say better? i.e.
"There always exists an optimal map"?

how to exhibit such cases? not so easy, in practice maps are very often optimal.
in practice, the monotone increasing $\pi^\oplus_{\text{mon}}$ and decreasing $\pi^\ominus_{\text{mon}}$ rearrangements are very often optimal
move away from measures of optimal plans $\pi^\oplus_{\text{mon}}$ and $\pi^\ominus_{\text{mon}}$ by gradient descent

Sharpness

Existence of optimal maps

for the Gromov-Wasserstein problem