Existence of optimal maps

for the Gromov-Wasserstein problem

Théo Dumont

D., Lacombe, Vialard. On the Existence of Monge maps for the Gromov-Wasserstein problem, FoCM 2024

\mu
\nu
\pi

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

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

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

Introduction

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

[Monge, 1781], [Kantorovitch, 1942]

\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
T_*\mu
T
x
T(x)

for a continuous measure:

\mu
\delta_x
\delta_{T(x)}
T_*\mu

for a discrete measure:

Introduction

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

[Monge, 1781], [Kantorovitch, 1942]

\mu
\nu
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.

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.

OT problem (Monge)

[Monge, 1781], [Kantorovitch, 1942]

\mu
\nu
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
\nu
T
x

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

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.

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
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.

OT problem (Kantorovitch)

[Monge, 1781], [Kantorovitch, 1942]

\mu
\nu
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
\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 \(c\)?

?

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 X
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

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 X
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 \(\mu \ll \mathrm dx\), 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 X
x

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

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 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 X
x

[Moameni, 2016]

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 X
x

[Moameni, 2016]

Map solutions of OT

\mathcal Y
\mathcal X
\mathcal Y
\mathcal X
\mathcal Y
\mathcal X

twist

map

\(\implies\)

subtwist

map/anti-map

\(\implies\)

\(m\)-twist

\(m\)-map

\(\implies\)

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

\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
x
\mu
y
\nu
c(x,y)

Wasserstein:

\mu
\nu
x
x'
y'
y
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

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

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

?

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

\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)

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

:(

[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_*\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_*\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_*\nu

A more general twist condition

[D., Lacombe & Vialard, 2023]

A more general twist condition

A more general twist condition

\mathcal Y
\mathcal X
\mathcal Y
\mathcal X
\mathcal Y
\mathcal X

twist

map

\(\implies\)

subtwist

map/anti-map

\(\implies\)

\(m\)-twist

\(m\)-map

\(\implies\)

(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)

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

(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 \(M^\star=\int x'y'^\top\,\mathrm d\pi(x',y')\)

\(\implies\)

\(\implies\)

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

  1. up to a SVD, suppose that \(M^\star\) is a diagonal matrix of singular values: \[M^\star=\begin{pmatrix}\sigma_1 & & & & & \\ & \ddots & & & & &\\& & \sigma_h & & &\\ & & & 0 & & \\ & & & & \ddots &\\ & & & & & 0 \end{pmatrix}\]
     
  2. 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 \(\mathbb R^h\)
     
  3. 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

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

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

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

\(\implies\)

\(\implies\)

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 \(M^\star=\int x'y'^\top\,\mathrm d\pi(x',y')\)

\(\implies\)

\(\implies\)

sometimes satisfies our general condition,
sometimes satisfies 2-twist

\(\implies\)

there exists an optimal 2-map

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

linearize

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

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

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

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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

(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

Monge maps for Gromov-Wasserstein

By Théo Dumont

Monge maps for Gromov-Wasserstein

Talk about the existence of Monge maps for the Gromov-Wasserstein problem (https://arxiv.org/abs/2210.11945).

  • 279

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