Existence of optimal maps

for the Gromov-Wasserstein problem

Théo Dumont

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

\mu

\nu

\pi

1. Optimal maps for OT

Gaspard Monge

Leonid Kantorovitch

Yann Brenier

\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.

Optimal Transport

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.

OT problem (Kantorovitch)

\delta_x

\frac12\delta_{y_1}

\frac12\delta_{y_2}

not feasible by a map!

\(\pi\) is induced by a transport map \(T\)

\(\pi\) is a transport plan

relaxation

\mu

\nu

\pi

\mu

\nu

\pi

[Monge, 1781], [Kantorovitch, 1942]

\mu

\nu

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

Optimal Transport

Brenier's theorem

When \(\mathcal X=\mathcal Y=\mathbb R^n\) and \(c(x,y)=\|x-y\|^2\), if \(\mu\ll\mathcal L\), then there is a unique solution to (KP), and it is of the form \[\pi^\star=(\text{id},\nabla f)_*\mu\]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

\nu

\pi

\mu

\nu

\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 \(c\)?

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 \(\mu \ll \mathcal{L}^n\), 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 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 \(c=d^2/2\).

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 \(\mu \ll \mathcal{L}^n\), then (KP) admits a unique solution and it is supported on the union of a graph and an anti-graph:
\[\pi^\star=(\text{id},G)_*\bar \mu+(H,\text{id})_*(\nu- G_*\bar\mu).\]

Map solutions of OT

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 X

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

\(m\)-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 \(\mu \ll \mathcal{L}^n\), 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 \(S\subset \mathcal X\times \mathcal Y\).

Map solutions of OT

\mathcal Y

\mathcal X

[Moameni, 2016]

\mathcal Y

\mathcal X

\mathcal Y

\mathcal X

\mathcal Y

\mathcal X

twist

map

\(\implies\)

subtwist

map/anti-map

\(\implies\)

\(m\)-twist

\(m\)-map

\(\implies\)

Map solutions of OT

(for simplicity, when \(\mu\ll\mathcal L^n\) and \(\mu,\nu\) have compact support)

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

for linear OT problem:

2. Optimal maps for Gromov-Wasserstein

Karl-Theodor Sturm

Facundo Mémoli

Mikhaïl Gromov

The Gromov-Wasserstein problem

[Sturm, 2012]

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

\mu

\nu

c(x,y)

Wasserstein:

\mu

\nu

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

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)

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

\nu

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

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

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

Optimal maps for GW

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

\(\mu\ll\mathcal L\) and \(\mu,\nu\) with compact support

There is an optimal map!

\mathcal Y

\mathcal X

\mathcal Y

\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

(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

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

\(\mathcal X=\mathcal Y=\mathbb R^n\)

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

quadratic

Optimal maps for GW

symmetric
bilinear

\implies

\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')

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