Can we find optimal maps for Gromov-Wasserstein?

\(\mathcal X\) and \(\mathcal Y\) complete Riemannian manifolds
cost function \(c:\mathcal X\times\mathcal Y\to\mathbb R\)

\displaystyle \inf_{T_*\mu=\nu} \int_{\mathcal X} c\big(x,T(x)\big)\,\mathrm d\mu(x)

Optimal maps for Wasserstein

\displaystyle \inf_{\pi\in\Pi(\mu,\nu)} \int_{\mathcal X\times\mathcal Y} c(x,y)\,\mathrm d\pi(x,y)

Brenier/twist condition

The cost function \(c\) satisfies the twist condition if
\[\forall x_0\in\mathcal X,\quad y\mapsto \nabla_x c(x_0,y)\in T_{x_0}\mathcal X \text{ is injective.}\] If \(\mu\ll\mathcal L^n\), then the unique solution to (KP) is \[\pi^\star=\big(\text{id},c\text{-}\exp_x(\nabla f)\big)_*\mu\]where \(f:\mathcal X\to\mathbb R\) is \(c\)-convex.

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

\displaystyle\min _{\pi\in\Pi(\mu,\nu)} \iint_{\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')

GW problem

distance modulo isometries
measures living in different spaces
quadratic in \(\pi\)! \(\implies\) hard to solve!

Gromov-Wasserstein

\mu

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

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

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

The answer: well it depends on the cost

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 \[c(x,y)=\tilde c(\varphi(x),\varphi(y))\quad\forall x,y\in E\] with \(\tilde c\) a twisted cost on \(B\), then we can construct an optimal map between \(\mu\) and \(\nu\)."

more generally, for (linear) OT problem \(\displaystyle\min_\pi\int c\,\mathrm d\pi\):

Conjecture:
this result is tight: there exists cases where no map is optimal

\(\mu\ll\mathcal L\) and \(\mu,\nu\) with compact support
\(\mathcal X=\mathcal Y=\mathbb R^n\)

\implies

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

Good: OT problem (with weird cost)

Brenier, twist, etc? not always, need something more general

Brenier-like theorem for GW?

\displaystyle \min _{\pi\in\Pi(\mu,\nu)} \iint\Big|\langle x,x'\rangle-\langle y,y'\rangle\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(i) GW inner product

There is an optimal map!

\mathcal Y

\mathcal X

\mathcal Y

\mathcal X

\displaystyle\min _{\pi\in\Pi(\mu,\nu)} \iint\Big|\|x-x'\|^2-\|y-y'\|^2\Big|^{2} \,\mathrm d\pi\,\mathrm d\pi

(ii) GW squared distance

There is an optimal 2-map!

first order optimality condition

How to: sketch of proof

1. Linearize into an OT problem

relaxation

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

\(\pi\) is a transport plan

2. Apply OT conditions for existence of a map

Kantorovitch (KP)

Monge (MP)

\(\mu\in\mathcal P(\mathcal X),\, \nu\in\mathcal P(\mathcal Y)\)

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

\simeq

not linear, so we cannot use the OT conditions!

\mathcal{F}(\pi)= \underbrace{c_{\text{gw}}(\pi)}_{\substack{\text{performance}\\\text{of }\pi}}-\underbrace{\min \left\{ c_{\text{gw}}(\pi^\oplus),\, c_{\text{gw}}(\pi^\ominus)\right\}}_{\substack{\text{performance of the monotone}\\\text{rearrangements of the marginals of }\pi}}

T.D., Théo Lacombe, François-Xavier Vialard. On the existence of Monge maps for the Gromov-Wasserstein problem (FoCM 2024).

Théo Dumont
Univ. Gustave Eiffel
theo.dumont@univ-eiffel.fr

T(u,v)=\big(t_ B (u),t_ F (u,v)\big)\\\hspace{3.5cm}=\big(\tilde c\text{-}\exp_u(\nabla f(u)), \exp_v(\nabla g_u(v))\big)\\ \hspace{4.5cm}=\big(\nabla f\circ\Sigma (u), \nabla g_{u}(u,v)\big) \quad \text{(inner prod.)} \\ \hspace{4.5cm}=\dots\hspace{2.7cm}\quad \text{(squared dist.)}

\forall x_0\in\mathcal X,\ y\mapsto \nabla_x c(x_0,y)\in T_{x_0}\mathcal X \text{ is injective.}

\forall x_0, y_0,\ \text{card}\{y\mid \nabla_x c(x_0,y)=\nabla_x c(x_0,y_0)\}\leq m.

\forall y_1\neq y_2,\ x\mapsto c(x,y_1)-c(x,y_2)\text{ has} \leq 2\text{ crit. points.}

twist:

subtwist:

\(m\)-twist:

symmetric
bilinear

linear

Closed form for optimal maps

Twist conditions

Linearized OT problem

Tightness of result (ii)

Let \(M^\star\coloneqq\int x'y'^\top\,\mathrm d\pi^\star(x',y')\).

inner prod.: \(C_{\pi^\star}(x,y)=-\langle M^\star x,y\rangle\)
• always satisfies our generalized twist condition!

squared dist.: \(C_{\pi^\star}(x,y)=-\|x\|^2\|y\|^2-4\langle M^\star x,y\rangle\)
• if \(\text{rk}M^\star \geq n-1\): satisfies 2-twist!
• if \(\text{rk}M^\star \leq n-2\): satisfies our
generalized twist condition!

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

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')