Unbalanced optimal transport and Hdiv minimizing geodesics

Unbalanced optimal transport and Hdiv

minimizing geodesics

Francois-Xavier Vialard

LIGM, Univ. Gustave Eiffel

Stochastic Differential Geometry and Mathematical Physics, Rennes ,2021

Unbalanced optimal transport Link to fluid dynamics Relaxation of geodesics About tightness

Contents

1 Unbalanced optimal transport

2 Link to fluid dynamics

3 Relaxation of geodesics

4 About tightness


Unbalanced optimal transport

Optimal transport applications: Imaging, machine learning, gradientflows, ...

Bottleneck in optimal transport: data has fixed total mass.

• Relax the mass constraint to extend OT distance between positivemeasures of arbitrary mass.

• Develop associated numerical algorithms.



Figure – Optimal transport between bimodal densities



Figure – Another transformation


Bibliography before (june) 2015

Taking into account locally the change of mass:

Two directions: Static and dynamic.

• Bounded Lipschitz distance...

• Static, Partial Optimal Transport [Figalli & Gigli, 2010]

• Static, Hanin 1992, Benamou and Brenier 2001.

• Dynamic, Numerics, Metamorphoses [Maas et al. , 2015]

• Dynamic, Numerics, Growth model [Lombardi & Maitre, 2013]

• Dynamic and static, [Piccoli & Rossi, 2013, Piccoli & Rossi, 2014]

• . . .

No equivalent of L2 Wasserstein distance on positive measures.


Bibliography before (june) 2015

Taking into account locally the change of mass:

Two directions: Static and dynamic.

• Bounded Lipschitz distance...

• Static, Partial Optimal Transport [Figalli & Gigli, 2010]

• Static, Hanin 1992, Benamou and Brenier 2001.

• Dynamic, Numerics, Metamorphoses [Maas et al. , 2015]

• Dynamic, Numerics, Growth model [Lombardi & Maitre, 2013]

• Dynamic and static, [Piccoli & Rossi, 2013, Piccoli & Rossi, 2014]

• . . .No equivalent of L2 Wasserstein distance on positive measures.


Bibliography after june 2015

More than 300 pages on the same model!

Starting point: Dynamic formulation

• Dynamic, Numerics, Imaging [Chizat et al. , 2015]

• Dynamic, Geometry and Static [Chizat et al. , 2015]

• Dynamic, Gradient flow [Kondratyev et al. , 2015]

• Dynamic, Gradient flow [Liero et al. , 2015b]

• Static and more [Liero et al. , 2015a]

• Optimal transport for contact forms [Rezakhanlou, 2015]

• Static relaxation of OT, machine learning [Frogner et al. , 2015]


Two possible directions

Pros and cons:

• Extend static formulation: Frogner et al.

minλKL(Proj1∗ γ, ρ1) + λKL(Proj2∗ γ, ρ2)

+

∫M2

γ(x , y)d(x , y)2 dx dy (1)

Good for numerics, but is it a distance ?

• Extend dynamic formulation: on the tangent space of a density,choose a metric on the transverse direction.Built-in metric property but does there exist a static formulation ?


An extension of Benamou-Brenier formulation

Add a source term in the constraint: (weak sense)

ρ = −∇ · (ρv) + αρ ,

where α can be understood as the growth rate.

WF2 def.= inf

(v ,α)

1

2

∫ 1

0

∫M

|v(x , t)|2ρ(x , t) dx dt

+δ2

2

∫ 1

0

∫M

α(x , t)2ρ(x , t) dx dt .

where δ is a length parameter.

Remark: very natural and not studied before.


An extension of Benamou-Brenier formulation


ρ = −∇ · (ρv) + αρ ,

where α can be understood as the growth rate.

WF2 def.= inf

(v ,α)

1

2

∫ 1

0

∫M

|v(x , t)|2ρ(x , t) dx dt

+δ2

2

∫ 1

0

∫M

α(x , t)2ρ(x , t) dx dt .

where δ is a length parameter.Remark: very natural and not studied before.


Convex reformulation


ρ = −∇ ·m + µ .

The Wasserstein-Fisher-Rao metric:

WF2 def.= inf

(v ,α)

1

2

∫ 1

0

∫M

|m(x , t)|2

ρ(x , t)dx dt +

δ2

2

∫ 1

0

∫M

µ(x , t)2

ρ(x , t)dx dt .

• Fisher-Rao metric: Hessian of the Boltzmann entropy/Kullback-Leibler divergence and reparametrization invariant.Wasserstein metric on the space of variances in 1D.

• Convex and 1-homogeneous: convex analysis (existence and more)

• Numerics: First-order splitting algorithm: Douglas-Rachford.

• Code available athttps://github.com/lchizat/optimal-transport/

https://github.com/lchizat/optimal-transport/


Convex reformulation


ρ = −∇ ·m + µ .

The Wasserstein-Fisher-Rao metric:

WF2 def.= inf

(v ,α)

1

2

∫ 1

0

∫M

|m(x , t)|2

ρ(x , t)dx dt +

δ2

2

∫ 1

0

∫M

µ(x , t)2

ρ(x , t)dx dt .

• Fisher-Rao metric: Hessian of the Boltzmann entropy/Kullback-Leibler divergence and reparametrization invariant.Wasserstein metric on the space of variances in 1D.

• Convex and 1-homogeneous: convex analysis (existence and more)

• Numerics: First-order splitting algorithm: Douglas-Rachford.

• Code available athttps://github.com/lchizat/optimal-transport/

https://github.com/lchizat/optimal-transport/


A general framework

Definition (Infinitesimal cost)

An infinitesimal cost is f : M ×R×Rd ×R→ R+ ∪ +∞ such that forall x ∈ M, f (x , ·, ·, ·) is convex, positively 1-homogeneous, lowersemicontinuous and satisfies

f (x , ρ,m, µ)

= 0 if (m, µ) = (0, 0) and ρ ≥ 0

> 0 if |m| or |µ| > 0

= +∞ if ρ < 0 .

Definition (Dynamic problem)

For (ρ,m, µ) ∈M([0, 1]×M)1+d+1, let

J(ρ,m, µ)def.=

∫ 1

0

∫M

f (x , dρdλ ,

dmdλ ,

dµdλ ) dλ(t, x) (2)

The dynamic problem is, for ρ0, ρ1 ∈M+(M),

C (ρ0, ρ1)def.= inf

(ρ,ω,ζ)∈CE10(ρ0,ρ1)

J(ρ, ω, ζ) . (3)


Existence of minimizers

Proposition (Fenchel-Rockafellar)

Let B(x) be the polar set of f (x , ·, ·, ·) for all x ∈ M and assume it is alower semicontinuous set-valued function. Then the minimum of (3) isattained and it holds

CD(ρ0, ρ1) = supϕ∈K

∫M

ϕ(1, ·)dρ1 −∫M

ϕ(0, ·) dρ0 (4)

withK

def.=ϕ ∈ C 1([0, 1]×M) : (∂tϕ,∇ϕ,ϕ) ∈ B(x), ∀(t, x) ∈ [0, 1]×M

.

WF(x , y , z) =

|y |2+δ2z2

2x if x > 0,

0 if (x , |y |, z) = (0, 0, 0)

+∞ otherwise

and the corresponding Hamilton-Jacobi equation is

∂tϕ+1

2

(|∇ϕ|2 +

ϕ2

δ2

)≤ 0 .


Existence of minimizers

Proposition (Fenchel-Rockafellar)

Let B(x) be the polar set of f (x , ·, ·, ·) for all x ∈ M and assume it is alower semicontinuous set-valued function. Then the minimum of (3) isattained and it holds

CD(ρ0, ρ1) = supϕ∈K

∫M

ϕ(1, ·)dρ1 −∫M

ϕ(0, ·) dρ0 (4)

withK

def.=ϕ ∈ C 1([0, 1]×M) : (∂tϕ,∇ϕ,ϕ) ∈ B(x), ∀(t, x) ∈ [0, 1]×M

.

WF(x , y , z) =

|y |2+δ2z2

2x if x > 0,

0 if (x , |y |, z) = (0, 0, 0)

+∞ otherwise

and the corresponding Hamilton-Jacobi equation is

∂tϕ+1

2

(|∇ϕ|2 +

ϕ2

δ2

)≤ 0 .


Numerical simulations

Figure – WFR geodesic between bimodal densities



•t = 0 t = 1t = 0.5

ρ0 ρ1

•t = 0 t = 1t = 0.5

ρ0 ρ1

Figure – Geodesics between ρ0 and ρ1 for (1st row) Hellinger, (2nd row) W2,(3rd row) partial OT, (4th row) WF.

An Interpolating Distance between Optimal Transport and Fisher-Rao, L.Chizat, B. Schmitzer, G. Peyre, and F.-X. Vialard, FoCM, 2016.



•t = 0 t = 1t = 0.5

ρ0 ρ1

•t = 0 t = 1t = 0.5

ρ0 ρ1





•t = 0 t = 1t = 0.5

ρ0 ρ1

•t = 0 t = 1t = 0.5

ρ0 ρ1





•t = 0 t = 1t = 0.5

ρ0 ρ1

•t = 0 t = 1t = 0.5

ρ0 ρ1





•t = 0 t = 1t = 0.5

ρ0 ρ1

•t = 0 t = 1t = 0.5

ρ0 ρ1




A relaxed static OT formulation

Define

KL(γ, ν) =

∫dγ

dνlog

(dγ

dν

)dν + |ν| − |γ|

WF 2(ρ1, ρ2) = infγ

KL(Proj1∗ γ, ρ1) + KL(Proj2∗ γ, ρ2)

−∫M2γ(x , y) log(cos2(d(x , y)/2 ∧ π/2))dx dy

TheoremOn a Riemannian manifold (compact without boundary), the static and dynamicformulations are equal.


From dynamic to static

Group action

Mass can be moved and changed: consider m(t)δx(t).

Infinitesimal action

ρ = −∇ · (vρ) + µ ⇔

x(t) = v(t, x(t))

m(t) = µ(t, x(t))

A cone metric

WF2(x ,m) ((x , m), (x , m)) =1

2(mx2 +

m2

m) ,

Change of variable: r 2 = m...



Group action



ρ = −∇ · (vρ) + µ ⇔

x(t) = v(t, x(t))

m(t) = µ(t, x(t))

A cone metric

WF2(x ,m) ((x , m), (x , m)) =1

2(mx2 +

m2

m) ,




Group action



ρ = −∇ · (vρ) + µ ⇔

x(t) = v(t, x(t))

m(t) = µ(t, x(t))

A cone metric

WF2(x ,m) ((x , m), (x , m)) =1

2(mx2 +

m2

m) ,



Riemannian cone

Definition

Let (M, g) be a Riemannian manifold. The cone over (M, g) is theRiemannian manifold (M × R∗+, r 2g + dr 2).

r

α

For M = S1(r), radius r ≤ 1. One has sin(α) = r .


Riemannian cone

Definition

Let (M, g) be a Riemannian manifold. The cone over (M, g) is theRiemannian manifold (M × R∗+, r 2g + dr 2).

r

α

For M = S1(r), radius r ≤ 1. One has sin(α) = r .


Geometry of a cone• Change of variable: WF2 = 1

2 r 2g + 2 dr 2.

• Non complete metric space: add the vertex M × 0.• The distance:

d((x1,m1), (x2,m2))2 =

m2 + m1 − 2√

m1m2 cos

(1

2dM(x1, x2) ∧ π

). (5)

• M = R then (x ,m) 7→√

me ix/2 ∈ C local isometry.

Proposition

If (M, g) has sectional curvature greater than 1, then(M × R∗+,m g + 1

4m dm2) has non-negative sectional curvature.For X ,Y two orthornormal vector fields on M,

K (X , Y ) = (Kg (X ,Y )− 1) (6)

where K and Kg denote respectively the sectional curvatures of M × R∗+and M.


Visualize geodesics for r 2g + dr 2

Figure – Geodesics on the cone


Distance between Diracs

x

y

P1

P2

P3

1

4WF (m1δx1 ,m2δx2 )2 = m2 + m1 − 2

√m1m2 cos

(1

2dM(x1, x2) ∧ π/2

).

Proof: prove that an explicit geodesic is a critical point of the convexfunctional.

Properties: positively 1-homogeneous and convex in (m1,m2).


Idea: Generalize Otto’s Riemannian submersion

SDiff(M): Isotropy

subgroup of µ

(Densp(M),W2) µ

Diff(M)

L2(M,M)

π(ϕ) = ϕ∗(µ)

Figure – A Riemannian submersion.


Idea: Generalize Otto’s Riemannian submersion

Isotropy

subgroup of µ

(Dens(M),WFR) µ

Diff(M) n C∞(M,R∗+)

L2(M, C(M))

π(ϕ, λ) = ϕ∗(λ2µ)

Figure – Similar picture in the unbalanced case.


Generalization of Otto’s Riemannian submersionIdea of a left group action:

π :(Diff(M) n C∞(M,R∗+)

)× Dens(M) 7→ Dens(M)

π ((ϕ, λ), ρ) := ϕ∗(λ2ρ)

Group law:

(ϕ1, λ1) · (ϕ2, λ2) = (ϕ1 ϕ2, (λ1 ϕ2)λ2) (7)

Theorem

Let ρ0 ∈ Dens(M) and π0 : Diff(M) n C∞(M,R∗+) 7→ Dens(M) definedby π0(ϕ, λ) := ϕ∗(λ

2ρ0). It is a Riemannian submersion

(Diff(M) n C∞(M,R∗+), L2(M,M × R∗+))π0−→ (Dens(M),WF)

(where M × R∗+ is endowed with the cone metric).

O’Neill’s formula: sectional curvature of (Dens(M),WF).


Generalization of Otto’s Riemannian submersionIdea of a left group action:

π :(Diff(M) n C∞(M,R∗+)

)× Dens(M) 7→ Dens(M)

π ((ϕ, λ), ρ) := ϕ∗(λ2ρ)

Group law:

(ϕ1, λ1) · (ϕ2, λ2) = (ϕ1 ϕ2, (λ1 ϕ2)λ2) (7)

Theorem

Let ρ0 ∈ Dens(M) and π0 : Diff(M) n C∞(M,R∗+) 7→ Dens(M) definedby π0(ϕ, λ) := ϕ∗(λ

2ρ0). It is a Riemannian submersion

(Diff(M) n C∞(M,R∗+), L2(M,M × R∗+))π0−→ (Dens(M),WF)

(where M × R∗+ is endowed with the cone metric).

O’Neill’s formula: sectional curvature of (Dens(M),WF).


Consequences

Monge formulation

WF (ρ0, ρ1) = inf(ϕ,λ)

‖(ϕ, λ)− (Id, 1)‖L2(ρ0) : ϕ∗(λ

2ρ0) = ρ1

(8)

Under existence and smoothness of the minimizer, there exists a functionp ∈ C∞(M,R) such that

(ϕ(x), λ(x)) = expC(M)x

(1

2∇p(x), p(x)

), (9)

Equivalent to Monge-Ampere equation

With zdef.= log(1 + p) one has

(1 + |∇z |2)e2zρ0 = det(Dϕ)ρ1 ϕ (10)

and

ϕ(x) = expM(x,1)

(arctan

(1

2|∇z |

)∇z(x)

|∇z(x)|

).


Yet another equivalent formulation

Liero, Mielke, Savare, Invent. Math.

WFR is Wasserstein 2 on P(M × R+) with second moment constraint.

WFR(µ, ν) = minµ,ν

W2(µ, ν) (11)

with the constraint: ∫R+

r 2 dµ(x , r) = µ(x) , (12)

and ∫R+

r 2 dν(x , r) = ν(x) , (13)

Here r 2 = m.


Contents




4 About tightness


Incompressible Euler and optimal transport

Optimal transport appears in the projection onto SDiff in Brenier’s work.

What is the corresponding fluid dynamic equation?


Arnold’s remark on incompressible Euler

Sur la geometrie differentielle des groupes de Lie de dimension infinie etses applications a l’hydrodynamique des fluides parfaits, Ann. Inst.Fourier, 1966.

TheoremThe incompressible Euler equation is the geodesic flow of the(right-invariant) L2 Riemannian metric on SDiff(M) (volume preservingdiffeomorphisms).

• An intrinsic point of view by Ebin and Marsden, Groups ofdiffeomorphisms and the motion of an incompressible fluid, Ann. ofMath., 1970. Short time existence results for smooth initialconditions.

• An extrinsic point of view by Brenier, relaxation of the variationalproblem, optimal transport, polar factorization.


Arnold’s remark on incompressible Euler

Sur la geometrie differentielle des groupes de Lie de dimension infinie etses applications a l’hydrodynamique des fluides parfaits, Ann. Inst.Fourier, 1966.

TheoremThe incompressible Euler equation is the geodesic flow of the(right-invariant) L2 Riemannian metric on SDiff(M) (volume preservingdiffeomorphisms).

• An intrinsic point of view by Ebin and Marsden, Groups ofdiffeomorphisms and the motion of an incompressible fluid, Ann. ofMath., 1970. Short time existence results for smooth initialconditions.

• An extrinsic point of view by Brenier, relaxation of the variationalproblem, optimal transport, polar factorization.


Arnold’s remark continuedRewritten in terms of the flow ϕ, the action reads∫ 1

0

∫M

|∂tϕ(t, x)|2 dx dt , (14)

under the constraint

ϕ(t) ∈ SDiff(M) for all t ∈ [0, 1] . (15)

Riemannian submanifold point of view:

Let M → Rd be isometrically embedded: A smooth curve c(t) ∈ M is ageodesic if and only if c ⊥ TcM.

Incompressible Euler in Lagrangian form:ϕ = −∇p ϕϕ(t) ∈ SDiff(M) .

(16)


Arnold’s remark continuedRewritten in terms of the flow ϕ, the action reads∫ 1

0

∫M

|∂tϕ(t, x)|2 dx dt , (14)

under the constraint

ϕ(t) ∈ SDiff(M) for all t ∈ [0, 1] . (15)

Riemannian submanifold point of view:

Let M → Rd be isometrically embedded: A smooth curve c(t) ∈ M is ageodesic if and only if c ⊥ TcM.

Incompressible Euler in Lagrangian form:ϕ = −∇p ϕϕ(t) ∈ SDiff(M) .

(16)


A geometric picture: Otto’s Riemannian submersion

SDiff(M): Isotropy

subgroup of µ

(Densp(M),W2) µ

Diff(M)

L2(M,M)

π(ϕ) = ϕ∗(µ)

Figure – A Riemannian submersion: SDiff(M) as a Riemannian submanifold ofL2(M,M): Incompressible Euler equation on SDiff(M)


Riemannian submersionLet (M, gM) and (N, gN) be two Riemannian manifolds and f : M 7→ N adifferentiable mapping.

DefinitionThe map f is a Riemannian submersion if f is a submersion and for anyx ∈ M, the map dfx : Ker(dfx)⊥ 7→ Tf (x)N is an isometry.

• Vertx := Ker(df (x)) is the vertical space.

• Horxdef.= Ker(df (x))⊥ is the horizontal space.

• Geodesics on N can be lifted ”horizontally” to geodesics on M.

Theorem (O’Neill’s formula)

Let X ,Y be two orthonormal vector fields on M with horizontal lifts Xand Y , then

KN(X ,Y ) = KM(X , Y ) +3

4‖ vert([X , Y ])‖2

M . (17)


Consequences

SDiff(M)

Id

g1

(Densp(M),W2) µ

Diff(M)

L2(M,M)

π(ϕ) = ϕ∗(µ)

Figure – A Riemannian submersion: SDiff(M) as a Riemannian submanifold ofL2(M,M): Incompressible Euler equation on SDiff(M)


Consequences

SDiff(M)

Id

g1

(Densp(M),W2) µ π(g1) = µ1

Diff(M)

L2(M,M)

π(ϕ) = ϕ∗(µ)

Figure – A pre polar factorization


Consequences

SDiff(M)

Id

g1

g0

(Densp(M),W2) µ π(g1) = µ1

Diff(M)

L2(M,M)

π(ϕ) = ϕ∗(µ)

Figure – Polar factorization: g0 = arg ming∈SDiff ‖g1 − g‖L2


The Riemannian submersion for WFR

Isotropy

subgroup of µ

(Dens(M),WFR) µ

Diff(M) n C∞(M,R∗+)

L2(M, C(M))

π(ϕ, λ) = ϕ∗(λ2µ)

Figure – The same picture in our case: what is the corresponding equation toEuler?


The isotropy subgroup for unbalanced optimal transport

Recall that

π−10 (ρ0) = (ϕ, λ) ∈ Diff(M) n C∞(M,R∗+) : ϕ∗(λ

2ρ0) = ρ0

π−10 (ρ0) = (ϕ,

√Jac(ϕ)) ∈ Diff(M) n C∞(M,R∗+) : ϕ ∈ Diff(M) .

The induced metric is

G (v , div v) =

∫M

|v |2 dµ+1

4

∫M

| div v |2 dµ . (18)

The Hdiv right-invariant metric on the group of diffeomorphisms.


The isotropy subgroup for unbalanced optimal transport

Recall that

π−10 (ρ0) = (ϕ, λ) ∈ Diff(M) n C∞(M,R∗+) : ϕ∗(λ

2ρ0) = ρ0

π−10 (ρ0) = (ϕ,

√Jac(ϕ)) ∈ Diff(M) n C∞(M,R∗+) : ϕ ∈ Diff(M) .

The induced metric is

G (v , div v) =

∫M

|v |2 dµ+1

4

∫M

| div v |2 dµ . (18)

The Hdiv right-invariant metric on the group of diffeomorphisms.


Degenerate distances

Theorem (Michor and Mumford, 2005)

If V = L2, then dist is degenerate, that is dist(ϕ0, ϕ1) = 0.If V = Hdiv, then the distance is NOT degenerate.

Theorem (Case of Sobolev metrics H s)

If d = 1, dist is not degenerate iif s > 1/2. [Bauer et al., 2012]If d ≥ 2, dist is not degenerate iif s ≥ 1. [Jerrard et al., 2018].

Main point of the talk: closer look at the critical case: (Diff,Hdiv).Embed Hdiv flow isometrically in L2 + generalized flow a la Brenier


Degenerate distances

Theorem (Michor and Mumford, 2005)

If V = L2, then dist is degenerate, that is dist(ϕ0, ϕ1) = 0.If V = Hdiv, then the distance is NOT degenerate.

Theorem (Case of Sobolev metrics H s)

If d = 1, dist is not degenerate iif s > 1/2. [Bauer et al., 2012]If d ≥ 2, dist is not degenerate iif s ≥ 1. [Jerrard et al., 2018].

Main point of the talk: closer look at the critical case: (Diff,Hdiv).Embed Hdiv flow isometrically in L2 + generalized flow a la Brenier


CH embeds in incompressible Euler (IE)

Interpret CH as incompressible Euler

Simple idea: replaceminx

f (x) + g(x) , (19)

withminx,y

f (x) + g(y) , (20)

under the constraint x = y .

Introduce div(v) = λ, obtain

infv(t),α

∫ 1

0

∫M

|v(t, x)|2 + δ2α(t, x)2 dx dt . (21)

under the constraint that div(v) = α.


CH embeds in incompressible Euler (IE)

Interpret CH as incompressible Euler

Simple idea: replaceminx

f (x) + g(x) , (19)

withminx,y

f (x) + g(y) , (20)

under the constraint x = y .

Introduce div(v) = λ, obtain

infv(t),α

∫ 1

0

∫M

|v(t, x)|2 + δ2α(t, x)2 dx dt . (21)

under the constraint that div(v) = α.


Rewrite in Lagrangian coordinates

Compute flows of (v , α) from (Id, 1) : M ×M 7→ M × R>0.

Aut(M×R>0) = (ϕ, λ) |ϕ ∈ Diff(M), λ ∈ C (M,R>0) ⊂ Diff(M×R>0)(22)

It is a group.

infϕ,λ

∫ 1

0

∫M

(|∂t(ϕ)(t, x)|2 + δ2(∂t log(λ)(t, x)2)2)λ(t, x)dx dt , (23)

under the constraint that Jac(ϕ) = λ or equivalently

ϕ∗(λ Leb) = Leb . (24)

Also equivalent to

(ϕ, λ)∗[(1/r 3)dr d Leb] = (1/r 3)dr d Leb . (25)


An isometric embedding in L2(M ,M × R>0)

Up to a change of variable, λ = r 2, we get

infϕ,r

∫ 1

0

∫M

r(t, x)2|∂t(ϕ)(t, x)|2 + 4δ2(|∂tr(t, x)|2)dx dt , (26)

under the constraint Jac(ϕ) = r 2.If δ = 1/2, then M × R>0 with (M, g), then

r 2g + dr 2 (27)

is a cone metric.If M = S1 and δ = 1/2, then the metric is std Euclidean on R2


An isometric embedding in L2(M ,M × R>0)

Up to a change of variable, λ = r 2, we get

infϕ,r

∫ 1

0

∫M

r(t, x)2|∂t(ϕ)(t, x)|2 + 4δ2(|∂tr(t, x)|2)dx dt , (26)

under the constraint Jac(ϕ) = r 2.If δ = 1/2, then M × R>0 with (M, g), then

r 2g + dr 2 (27)

is a cone metric.If M = S1 and δ = 1/2, then the metric is std Euclidean on R2


A new geometric picture

Autvol(C(M))

(Dens(M),WFR) vol

Aut(C(M))

L2(M, C(M))

π(ϕ, λ) = ϕ∗(λ2 vol)

Aut(C(M))

Diff(C(M))

L2(C(M))

(Dens(C(M)),W2) ν = r−3 dvol dr

Diff ν(C(M))

Autvol(C(M))

π(ψ) = ψ∗(ν)

Figure – Generalization of Otto’s riemannian submersion. (with Tho. Gallouet)


CH as IE

Theorem

Solutions to the Hdiv Euler-Arnold equations are particular solutions tothe incompressible Euler equation on M × R>0 × S1 for the metricr 2g(x) + ( dr)2 + r−2(3+d)(dy)2, where (x , r , y) ∈ M × R>0 × S1.

Embedding CH equations into incompressible Euler. (arxiv 2018,Natale, FXV)

On the universality of the incompressible Euler equation on compactmanifolds, (arxiv 2017, T. Tao)


Results

TheoremLet ϕ be the flow of a smooth solution to the Camassa-Holm equation

then Ψ(θ, r)def.= (ϕ(θ),

√Jac(ϕ(θ))r) is the flow of a solution to the

incompressible Euler equation for the density 1r4 r dr dθ.

Case where M = S1, M(ϕ) = [(θ, r) 7→ r√∂xϕ(θ)e iϕ(θ)] then the CH

equation is∂tu − 1

4∂txxu u + 3∂xu u − 12∂xxu ∂xu − 1

4∂xxxu u = 0

∂tϕ(t, x) = u(t, ϕ(t, x)) .(28)

The Euler equation on the cone, C(M) = R2 \ 0 for the densityρ = 1

r4 Leb is v +∇vv = −∇p ,

∇ · (ρv) = 0 .(29)

where v(θ, r)def.=(u(θ), r2∂xu(θ)

).


Results

TheoremLet ϕ be the flow of a smooth solution to the Camassa-Holm equation

then Ψ(θ, r)def.= (ϕ(θ),

√Jac(ϕ(θ))r) is the flow of a solution to the

incompressible Euler equation for the density 1r4 r dr dθ.

Case where M = S1, M(ϕ) = [(θ, r) 7→ r√∂xϕ(θ)e iϕ(θ)] then the CH

equation is∂tu − 1

4∂txxu u + 3∂xu u − 12∂xxu ∂xu − 1

4∂xxxu u = 0

∂tϕ(t, x) = u(t, ϕ(t, x)) .(28)

The Euler equation on the cone, C(M) = R2 \ 0 for the densityρ = 1

r4 Leb is v +∇vv = −∇p ,

∇ · (ρv) = 0 .(29)

where v(θ, r)def.=(u(θ), r2∂xu(θ)

).


CH as incompressible Euler

Proposition

Solutions to the Hdiv Euler-Arnold equations are solutions to theincompressible Euler equation on M × R>0 × S1 for the metricr 2g(x) + ( dr)2 + r−2(3+d)(dy)2, where (x , r , y) ∈ M × R>0 × S1.

Generalization to similar equations:∂tm + umx + 2mux = −gρ∂xρ

m = u − ∂xxu

∂tρ+ ∂x(ρu) = 0 ,

(30)

where ρ(t = 0) is a positive density and g is a positive constant.

Theorem

The Hdiv2 equations on M can be isom. embedded in the Hdiv geodesicflow on M × S1.Solutions to the Hdiv2 equations can be mapped to particular solutions ofthe Hdiv(M × S1) geodesic flow.


Contents




4 About tightness


Generalized flows a la Brenier

Allow particles to split.

Similar to Young measures.

Embed

infϕ

∫ 1

0

∫M

|∂tϕ(t, x)|2 dx dt (31)

under the constraint ϕ∗(Leb) = Leb intoLet µ ∈ P(C ([0, 1],M)),

infµ〈µ, ‖ω‖2

L2〉 (32)

under the constraint [e0,1]∗µ = δx,ϕ(x) and marginal constraints[et ]∗µ = Leb.

1 Short time smooth solutions of inc. Euler are unique minimizers(Brenier).

2 Tightness by Schnirelman (≥ 3D, but not in 2D).


Relaxation of Hdiv

With A. Natale.

Optimization set: P(Ω) where Ω = C ([0, 1],M ×R>0) and minimize〈µ, ‖ω‖2

L2〉.Marginal constraints:

∫r

r 2[et ]∗(µ) = Leb.

Particular boundary constraints.

1 Existence of generalized solutions.

2 Smooth solutions for short times are unique minimizers (up todilation).

3 Uniqueness of the pressure.

4 If M = S1, then every rotation by θ ≤ π is a minimizer.


What boundary conditions ?

Deterministic problem z = (x , 1)→ (ϕ(x), λ(x)).

Strong constraint

∀f ∈ Cb((M × R>0)2,R)∫Ω

f (z(0), z(1))dµ =

∫M

f ((x , 1), (ϕ(x), λ(x))) dx (33)

Marginal constraints not stable w.r.t. narrow convergence.Paths with unbounded jacobian can be charged/mass can escape.

Homogeneous constraints

∀f ∈ C ((M × R>0)2,R) positively 2-homogeneous,

f ((x1, δr1), (x2, δr2)) = δ2f ((x1, r1), (x2, r2)) (34)


What boundary conditions ?

Deterministic problem z = (x , 1)→ (ϕ(x), λ(x)).

Strong constraint

∀f ∈ Cb((M × R>0)2,R)∫Ω

f (z(0), z(1))dµ =

∫M

f ((x , 1), (ϕ(x), λ(x))) dx (33)

Marginal constraints not stable w.r.t. narrow convergence.Paths with unbounded jacobian can be charged/mass can escape.

Homogeneous constraints

∀f ∈ C ((M × R>0)2,R) positively 2-homogeneous,

f ((x1, δr1), (x2, δr2)) = δ2f ((x1, r1), (x2, r2)) (34)


Invariance property: dilations

θ : Ω 7→ R.

On Ω, consider divθ : ω = (x , r)→ (x , r/θ(ω)).

Define

dilθ :M(Ω) 7→ M(Ω)

µ→ [divθ]∗(θ2µ) .

then

Lemma

Let θ be pos. 1-homogeneous, θ(δ · ω) = δθ(ω). Consider µ ∈ P(Ω) andassume C = (

∫Ωθ2 dµ)1/2 <∞ then, dilθ/C (µ) ∈ P(Ω),

1 〈f , dilθ/C (µ)〉 = 〈f , µ〉 for all 2-homogeneous f .

2 Supp(dilθ/C (µ)) ⊂ θ(ω) = C.


First results

Theorem

Existence of generalized flows for Hdiv.

Sketch of proof.

Rescale a minimizing sequence, dilθ/Cn(µn) with

θ(ω) = (∫ 1

0r 2 dt + r(0)2 + r(1)2)1/2, in fact Cn =

√3.

By Lemma,Supp(dilθ/C (µn)) ⊂ θ(ω) =

√3 (35)

Uniform integrability of homogeneous constraints.

Lemma (Decomposition result)

Any µ satisfying the deterministic homogeneous coupling constraint canbe decomposed into µ = µ0 + µ with µ charging paths ω s.t.ω(0) = ω(1) = 0 and µ0(ω | 0 /∈ ω(0), ω(1), ) = 0.


Other results

1 Smooth solutions for short times are unique minimizers (up todilation).

2 Uniqueness of the pressure.

3 If M = S1, then every rotation by θ ≤ π is a minimizer.


Contents




4 About tightness


Generalized energy diameter bounded by π2

Lemma (A mixture)

Let ϕ ∈ Diff(M) then

µ =1

2δ(id,s(t))∗ Leb +

1

2δ(ϕ(t),c(t))∗ Leb (36)

where s(t) =√

2 sin(πt) and

c(t) =

√2 cos(πt) if t < 1/2√2 Jac(ϕ(x))| cos(πt)| otherwise.

ϕ(t) =

id if t < 1/2

ϕ(x)otherwise..

Then, its cost is π2.

Consequence: no tightness in 1D.


Scaling argumentAt small scales, cost driven by the L2 norm of div: Hunter-Saxton limit.

Figure – Peakons for the Hunter-Saxton equations

Figure – Approximation using lots of peakon-antipeakon collision


Result on the torus

Proposition

On R2/Z2, for (x , y)→ (x + π, y + π) the singular flow is a limit ofdeterministic flows and it is a minimizer. (Whereas constant speedrotation is not).


Approximation sequence


















Questions

For the generalized geodesics1 Tightness in general for d ≥ 2.

2 What about d = 1: partial answer in a recent paper.

3 Regularity of the pressure.

For the unbalanced optimal transport1 Regularity theory is not known.

2 Quasi-Linear time algorithms in 1D open (keyword: slicedWasserstein)

3 Some extensions to matrix valued OT have been proposed inconnection with the Bures-Wasserstein metric.

4 Extension of the Schrodinger problem to unbalanced OT completelyopen.


References I

Chizat, L., Schmitzer, B., Peyre, G., & Vialard, F.-X. 2015.An Interpolating Distance between Optimal Transport andFisher-Rao.ArXiv e-prints, June.

Figalli, A., & Gigli, N. 2010.A new transportation distance between non-negative measures, withapplications to gradients flows with Dirichlet boundary conditions.Journal de mathematiques pures et appliquees, 94(2), 107–130.

Frogner, C., Zhang, C., Mobahi, H., Araya-Polo, M., & Poggio, T.2015.Learning with a Wasserstein Loss.Preprint 1506.05439. Arxiv.

Kondratyev, S., Monsaingeon, L., & Vorotnikov, D. 2015.A new optimal trasnport distance on the space of finite Radonmeasures.Tech. rept. Pre-print.


References II

Liero, M., Mielke, A., & Savare, G. 2015a.Optimal Entropy-Transport problems and a newHellinger-Kantorovich distance between positive measures.ArXiv e-prints, Aug.

Liero, M., Mielke, A., & Savare, G. 2015b.Optimal transport in competition with reaction: theHellinger-Kantorovich distance and geodesic curves.ArXiv e-prints, Aug.

Lombardi, D., & Maitre, E. 2013.Eulerian models and algorithms for unbalanced optimal transport.<hal-00976501v3>.

Maas, J., Rumpf, M., Schonlieb, C., & Simon, S. 2015.A generalized model for optimal transport of images includingdissipation and density modulation.arXiv:1504.01988.


References III

Piccoli, B., & Rossi, F. 2013.On properties of the Generalized Wasserstein distance.arXiv:1304.7014.

Piccoli, B., & Rossi, F. 2014.Generalized Wasserstein distance and its application to transportequations with source.Archive for Rational Mechanics and Analysis, 211(1), 335–358.

Rezakhanlou, F. 2015.Optimal Transport Problems For Contact Structures.

Documents

Unbalanced optimal transport and Hdiv minimizing geodesics