UNBALANCED OPTIMAL TRANSPORT

Lenaıc Chizatjoint work with F-X. Vialard, G. Peyre & B. Schmitzer

CEREMADEUniversite Paris Dauphine

Mokalien 2015

Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

MotivationsImage matching, Machine learning, Economics, Gradient flows...

Previous work

Static relaxed marginal constraints ([Hanin, 1992], [Benamou, 2003])

Dynamic source term ([Piccoli and Rossi, 2013], [Mass et al., 2015],[Lombardi and Maitre, 2013]);

Two points of view:

• Standard optimal transport & relaxed marginal constraints ;

• Transport + variation of mass & exact marginal constraints .

Setting : Ω convex compact in Rn.

2 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

MotivationsImage matching, Machine learning, Economics, Gradient flows...

Previous work

Static relaxed marginal constraints ([Hanin, 1992], [Benamou, 2003])

Dynamic source term ([Piccoli and Rossi, 2013], [Mass et al., 2015],[Lombardi and Maitre, 2013]);

Two points of view:

• Standard optimal transport & relaxed marginal constraints ;

• Transport + variation of mass & exact marginal constraints .

Setting : Ω convex compact in Rn.

2 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

MotivationsImage matching, Machine learning, Economics, Gradient flows...

Previous work

Static relaxed marginal constraints ([Hanin, 1992], [Benamou, 2003])

Dynamic source term ([Piccoli and Rossi, 2013], [Mass et al., 2015],[Lombardi and Maitre, 2013]);

Two points of view:

• Standard optimal transport & relaxed marginal constraints ;

• Transport + variation of mass & exact marginal constraints .

Setting : Ω convex compact in Rn.

2 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

MotivationsImage matching, Machine learning, Economics, Gradient flows...

Previous work

Static relaxed marginal constraints ([Hanin, 1992], [Benamou, 2003])

Dynamic source term ([Piccoli and Rossi, 2013], [Mass et al., 2015],[Lombardi and Maitre, 2013]);

Two points of view:

• Standard optimal transport & relaxed marginal constraints ;

• Transport + variation of mass & exact marginal constraints .

Setting : Ω convex compact in Rn.

2 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Outline

Static Formulation

Dynamic Formulation

Examples & Numerics

3 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Outline

Static Formulation

Dynamic Formulation

Examples & Numerics

4 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

From standard OT...

Ω

•

×δx c(x, y)

•

×δy

Assumptions on the cost:

• lower bounded;

• l.s.c.

also linear in m.

Static formulation of OT:

minimize

∫Ω2

c(x , y)dγ(x , y)

subject to (projx)#γ = ρ0

(projy )#γ = ρ1

5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

From standard OT...

Ω

•

×δx c(x, y)

•

×δy Assumptions on the cost:

• lower bounded;

• l.s.c.

also linear in m.

Static formulation of OT:

minimize

∫Ω2

c(x , y)dγ(x , y)

subject to (projx)#γ = ρ0

(projy )#γ = ρ1

5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

From standard OT...

Ω

•

×δx c(x, y)

•

×δy Assumptions on the cost:

• lower bounded;

• l.s.c.

also linear in m.

Static formulation of OT:

minimize

∫Ω2

c(x , y)dγ(x , y)

subject to (projx)#γ = ρ0

(projy )#γ = ρ1

5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

From standard OT...

Ω

•

×mδx m.c(x, y)

•

×mδy Assumptions on the cost:

• lower bounded;

• l.s.c.

also linear in m.

Static formulation of OT:

minimize

∫Ω2

c(dγ

dλ, x , y)dλ(x , y) (γ λ)

subject to (projx)#γ = ρ0

(projy )#γ = ρ1

5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

•

×mxδx c((x,mx ), (y,my ))

•

×myδy

The cost function is

• pos. homogeneous in(mx ,my );

• subadditive in (mx ,my );

• nonnegative;

• mx or my negative⇒ c = +∞ ;

• lower semicontinuous

Static formulation of Unbalanced OT

C (ρ0, ρ1) := minimize

∫Ω2

c((x ,dγ0

dγ), (y ,

dγ1

dγ))dγ(x , y)

subject to (πx)#γ0 = ρ0

(πy )#γ1 = ρ1

6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

•

×mxδx c((x,mx ), (y,my ))

•

×myδy

The cost function is

• pos. homogeneous in(mx ,my );

• subadditive in (mx ,my );

• nonnegative;

• mx or my negative⇒ c = +∞ ;

• lower semicontinuous

Static formulation of Unbalanced OT

C (ρ0, ρ1) := minimize

∫Ω2

c((x ,dγ0

dγ), (y ,

dγ1

dγ))dγ(x , y)

subject to (πx)#γ0 = ρ0

(πy )#γ1 = ρ1

6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

•

×mxδx c((x,mx ), (y,my ))

•

×myδy

The cost function is

• pos. homogeneous in(mx ,my );

• subadditive in (mx ,my );

• nonnegative;

• mx or my negative⇒ c = +∞ ;

• lower semicontinuous

Static formulation of Unbalanced OT

C (ρ0, ρ1) := minimize

∫Ω2

c((x ,dγ0

dγ), (y ,

dγ1

dγ))dγ(x , y)

subject to (πx)#γ0 = ρ0

(πy )#γ1 = ρ1

6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

•

×mxδx c((x,mx ), (y,my ))

•

×myδy

The cost function is

• pos. homogeneous in(mx ,my );

• subadditive in (mx ,my );

• nonnegative;

• mx or my negative⇒ c = +∞ ;

• lower semicontinuous

Static formulation of Unbalanced OT

C (ρ0, ρ1) := minimize

∫Ω2

c((x ,dγ0

dγ), (y ,

dγ1

dγ))dγ(x , y)

subject to (πx)#γ0 = ρ0

(πy )#γ1 = ρ1

6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

•

×mxδx c((x,mx ), (y,my ))

•

×myδy

The cost function is

• pos. homogeneous in(mx ,my );

• subadditive in (mx ,my );

• nonnegative;

• mx or my negative⇒ c = +∞ ;

• lower semicontinuous

Static formulation of Unbalanced OT

C (ρ0, ρ1) := minimize

∫Ω2

c((x ,dγ0

dγ), (y ,

dγ1

dγ))dγ(x , y)

subject to (πx)#γ0 = ρ0

(πy )#γ1 = ρ1

6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

•

×mxδx c((x,mx ), (y,my ))

•

×myδy

The cost function is

• pos. homogeneous in(mx ,my );

• subadditive in (mx ,my );

• nonnegative;

• mx or my negative⇒ c = +∞ ;

• lower semicontinuous

Static formulation of Unbalanced OT

C (ρ0, ρ1) := minimize

∫Ω2

c((x ,dγ0

dγ), (y ,

dγ1

dγ))dγ(x , y)

subject to (πx)#γ0 = ρ0

(πy )#γ1 = ρ1

6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

•

×mxδx c((x,mx ), (y,my ))

•

×myδy

The cost function is

• pos. homogeneous in(mx ,my );

• subadditive in (mx ,my );

• nonnegative;

• mx or my negative⇒ c = +∞ ;

• lower semicontinuous

Static formulation of Unbalanced OT

C (ρ0, ρ1) := minimize

∫Ω2

c((x ,dγ0

dγ), (y ,

dγ1

dγ))dγ(x , y)

subject to (πx)#γ0 = ρ0

(πy )#γ1 = ρ1

6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Properties

Ω

R+•

×mxδx

c1/p

•

×myδy

Cone(Ω) := (Ω× R+)/(Ω× 0)

Theorem (Metric property)

If c1/p is a metric on Cone(Ω)then C 1/p is a metric onM+(Ω).

Theorem (Duality)

For all (x , y) ∈ Ω2, c(x , ·, y , ·) is the support function of a closedconvex nonempty set Q(x , y) ⊂ R2. If Q is l.s.c. in the sense ofmultifunctions, then

C (ρ0, ρ1) = supφ,ψ∈C(Ω)

∫Ωφ(x)dρ0(x) +

∫Ωψ(y)dρ1(y)

subject to (φ(x), ψ(y)) ∈ Q(x , y) for all (x , y) ∈ Ω2 .

7 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Properties

Ω

R+•

×mxδx

c1/p

•

×myδy

Cone(Ω) := (Ω× R+)/(Ω× 0)

Theorem (Metric property)

If c1/p is a metric on Cone(Ω)then C 1/p is a metric onM+(Ω).

Theorem (Duality)

For all (x , y) ∈ Ω2, c(x , ·, y , ·) is the support function of a closedconvex nonempty set Q(x , y) ⊂ R2. If Q is l.s.c. in the sense ofmultifunctions, then

C (ρ0, ρ1) = supφ,ψ∈C(Ω)

∫Ωφ(x)dρ0(x) +

∫Ωψ(y)dρ1(y)

subject to (φ(x), ψ(y)) ∈ Q(x , y) for all (x , y) ∈ Ω2 .

7 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Properties

Ω

R+•

×mxδx

c1/p

•

×myδy

Cone(Ω) := (Ω× R+)/(Ω× 0)

Theorem (Metric property)

If c1/p is a metric on Cone(Ω)then C 1/p is a metric onM+(Ω).

Theorem (Duality)

For all (x , y) ∈ Ω2, c(x , ·, y , ·) is the support function of a closedconvex nonempty set Q(x , y) ⊂ R2.

If Q is l.s.c. in the sense ofmultifunctions, then

C (ρ0, ρ1) = supφ,ψ∈C(Ω)

∫Ωφ(x)dρ0(x) +

∫Ωψ(y)dρ1(y)

subject to (φ(x), ψ(y)) ∈ Q(x , y) for all (x , y) ∈ Ω2 .

7 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Properties

Ω

R+•

×mxδx

c1/p

•

×myδy

Cone(Ω) := (Ω× R+)/(Ω× 0)

Theorem (Metric property)

If c1/p is a metric on Cone(Ω)then C 1/p is a metric onM+(Ω).

Theorem (Duality)

For all (x , y) ∈ Ω2, c(x , ·, y , ·) is the support function of a closedconvex nonempty set Q(x , y) ⊂ R2. If Q is l.s.c. in the sense ofmultifunctions, then

C (ρ0, ρ1) = supφ,ψ∈C(Ω)

∫Ωφ(x)dρ0(x) +

∫Ωψ(y)dρ1(y)

subject to (φ(x), ψ(y)) ∈ Q(x , y) for all (x , y) ∈ Ω2 .

7 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Outline

Static Formulation

Dynamic Formulation

Examples & Numerics

8 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

•

×

•

ו

×ρtδx(t)

vt

Change of variables: ω = ρvInfinitesimal cost : f (x , ρ, ω)

• homogeneous in (ρ, ω);

• subadditive in (ρ, ω);

Standard dynamic formulation

minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ)dµ (ρ, |ω| µ)

subject to ∂tρ+∇ · ω = 0 (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

•

×

•

ו

×ρtδx(t)

vt

Change of variables: ω = ρvInfinitesimal cost : f (x , ρ, ω)

• homogeneous in (ρ, ω);

• subadditive in (ρ, ω);

Standard dynamic formulation

minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ)dµ (ρ, |ω| µ)

subject to ∂tρ+∇ · ω = 0 (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

•

×

•

ו

×ρtδx(t)

vt

Change of variables: ω = ρvInfinitesimal cost : f (x , ρ, ω)

• homogeneous in (ρ, ω);

• subadditive in (ρ, ω);

Standard dynamic formulation

minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ)dµ (ρ, |ω| µ)

subject to ∂tρ+∇ · ω = 0 (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

•

×

•

ו

×ρtδx(t)

vt

Change of variables: ω = ρvInfinitesimal cost : f (x , ρ, ω)

• homogeneous in (ρ, ω);

• subadditive in (ρ, ω);

Standard dynamic formulation

minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ)dµ (ρ, |ω| µ)

subject to ∂tρ+∇ · ω = 0 (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

•

×

•

ו

×ρtδx(t)

vt

Change of variables: ω = ρvInfinitesimal cost : f (x , ρ, ω)

• homogeneous in (ρ, ω);

• subadditive in (ρ, ω);

Standard dynamic formulation

minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ)dµ (ρ, |ω| µ)

subject to ∂tρ+∇ · ω = 0 (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)

• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

•

×

•

ו

×ρtδx(t)

vt

αt =∂tρtρt

Variables: ω = ρv , ζ = ραInfinitesimal cost : f (x , ρ, ω, ζ)

• homogeneous in (ρ, ω, ζ);

• subadditive in (ρ, ω, ζ);

• ρ < 0⇒ f = +∞;

• sign (f ) = sign (|ω|+ |ζ|)• mult. dependancy in x , l.s.c.

Unbalanced dynamic formulation

CD(ρ0, ρ1) := minimize

∫ 1

0

∫Ωf (x ,

dρ

dµ,dω

dµ,dζ

dµ)dµ(t, x)

subject to ∂tρ+∇ · ω = ζ (weakly)

(projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Existence of minimizers & duality

For all x ∈ Ω, f (x , ·) is the support function of Q(x), a closed,convex, non-empty set.

TheoremAssume that Q is a l.s.c. multifunction. Then the minimumdefining CD is attained and

CD(ρ0, ρ1) = supϕ∈C1([0,1]×Ω)

∫Ωϕ(1, x)dρ1(x)−

∫Ωϕ(0, x)dρ0(x)

subject to (∂tϕ,∇ϕ,ϕ)(t, x) ∈ Q(x) .

11 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Existence of minimizers & duality

For all x ∈ Ω, f (x , ·) is the support function of Q(x), a closed,convex, non-empty set.

TheoremAssume that Q is a l.s.c. multifunction. Then the minimumdefining CD is attained and

CD(ρ0, ρ1) = supϕ∈C1([0,1]×Ω)

∫Ωϕ(1, x)dρ1(x)−

∫Ωϕ(0, x)dρ0(x)

subject to (∂tϕ,∇ϕ,ϕ)(t, x) ∈ Q(x) .

11 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Dynamic to Static : “Benamou-Brenier” formula

Costs between points in Cone(Ω):

Dirac-based cost cd : CD(m0δx0 ,m1δx1)

Path-based cost cp : infimum of the dynamic functional restrictedto smooth, stable Dirac trajectories m(t)δx(t).

Theorem (C. et al., 2015)

Let c be a cost function satisfying cd ≤ c ≤ cp. If the associatedproblem C is weakly* continuous, then C = CD (and c = cd).

Note : cd is hard to compute directly in general.

ExampleA good candidate is the convex regularization of cp:

infma

0+mb0 =m0

ma1+mb

1 =m1

cp((x0,ma0), (x1,m

a1)) + cp((x0,m

b0), (x1,m

b1))

12 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Dynamic to Static : “Benamou-Brenier” formula

Costs between points in Cone(Ω):

Dirac-based cost cd : CD(m0δx0 ,m1δx1)

Path-based cost cp : infimum of the dynamic functional restrictedto smooth, stable Dirac trajectories m(t)δx(t).

Theorem (C. et al., 2015)

Let c be a cost function satisfying cd ≤ c ≤ cp. If the associatedproblem C is weakly* continuous, then C = CD (and c = cd).

Note : cd is hard to compute directly in general.

ExampleA good candidate is the convex regularization of cp:

infma

0+mb0 =m0

ma1+mb

1 =m1

cp((x0,ma0), (x1,m

a1)) + cp((x0,m

b0), (x1,m

b1))

12 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Dynamic to Static : “Benamou-Brenier” formula

Costs between points in Cone(Ω):

Dirac-based cost cd : CD(m0δx0 ,m1δx1)

Path-based cost cp : infimum of the dynamic functional restrictedto smooth, stable Dirac trajectories m(t)δx(t).

Theorem (C. et al., 2015)

Let c be a cost function satisfying cd ≤ c ≤ cp. If the associatedproblem C is weakly* continuous, then C = CD (and c = cd).

Note : cd is hard to compute directly in general.

ExampleA good candidate is the convex regularization of cp:

infma

0+mb0 =m0

ma1+mb

1 =m1

cp((x0,ma0), (x1,m

a1)) + cp((x0,m

b0), (x1,m

b1))

12 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Outline

Static Formulation

Dynamic Formulation

Examples & Numerics

13 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := minρ0,ρ1

1

pW p

p (ρ0, ρ1)

+ δ (|ρ0 − ρ0|TV + |ρ1 − ρ1|TV )

Dynamic

min

∫ 1

0

∫Ω

1

p

ωp

ρp−1+ δ|ζ|

s.t. ∂tρ+∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m↔ δ;

• C 1/p defines a metric on M+(Ω);

• geodesics are not absolutely continuous;

• dual formula : add the contraint “bounded by δ”.

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Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := minρ0,ρ1

1

pW p

p (ρ0, ρ1)

+ δ (|ρ0 − ρ0|TV + |ρ1 − ρ1|TV )

Dynamic

min

∫ 1

0

∫Ω

1

p

ωp

ρp−1+ δ|ζ|

s.t. ∂tρ+∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m↔ δ;

• C 1/p defines a metric on M+(Ω);

• geodesics are not absolutely continuous;

• dual formula : add the contraint “bounded by δ”.

14 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := minρ0,ρ1

1

pW p

p (ρ0, ρ1)

+ δ (|ρ0 − ρ0|TV + |ρ1 − ρ1|TV )

Dynamic

min

∫ 1

0

∫Ω

1

p

ωp

ρp−1+ δ|ζ|

s.t. ∂tρ+∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m↔ δ;

• C 1/p defines a metric on M+(Ω);

• geodesics are not absolutely continuous;

• dual formula : add the contraint “bounded by δ”.

14 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := minρ0,ρ1

1

pW p

p (ρ0, ρ1)

+ δ (|ρ0 − ρ0|TV + |ρ1 − ρ1|TV )

Dynamic

min

∫ 1

0

∫Ω

1

p

ωp

ρp−1+ δ|ζ|

s.t. ∂tρ+∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m↔ δ;

• C 1/p defines a metric on M+(Ω);

• geodesics are not absolutely continuous;

• dual formula : add the contraint “bounded by δ”.

14 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := minρ0,ρ1

1

pW p

p (ρ0, ρ1)

+ δ (|ρ0 − ρ0|TV + |ρ1 − ρ1|TV )

Dynamic

min

∫ 1

0

∫Ω

1

p

ωp

ρp−1+ δ|ζ|

s.t. ∂tρ+∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m↔ δ;

• C 1/p defines a metric on M+(Ω);

• geodesics are not absolutely continuous;

• dual formula : add the contraint “bounded by δ”.

14 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := minρ0,ρ1

1

pW p

p (ρ0, ρ1)

+ δ (|ρ0 − ρ0|TV + |ρ1 − ρ1|TV )

Dynamic

min

∫ 1

0

∫Ω

1

p

ωp

ρp−1+ δ|ζ|

s.t. ∂tρ+∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m↔ δ;

• C 1/p defines a metric on M+(Ω);

• geodesics are not absolutely continuous;

• dual formula : add the contraint “bounded by δ”.

14 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := minρ0,ρ1

1

pW p

p (ρ0, ρ1)

+ δ (|ρ0 − ρ0|TV + |ρ1 − ρ1|TV )

Dynamic

min

∫ 1

0

∫Ω

1

p

ωp

ρp−1+ δ|ζ|

s.t. ∂tρ+∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m↔ δ;

• C 1/p defines a metric on M+(Ω);

• geodesics are not absolutely continuous;

• dual formula : add the contraint “bounded by δ”.

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Introduction Static Dynamic Examples & Numerics Conclusion

Wasserstein-Fisher-Rao : WF

Static

WF 2 := min|ρ0|TV + |ρ1|TV

−2

∫Ω2

cos

(|y − x |

2∧ π

2

)d√γ0γ1(x , y)

s.t. projxγ0 = ρ0 , projyγ1 = ρ1

Dynamic

min1

4

∫ 1

0

∫Ω

|ω|2

ρ+ζ2

ρ

s.t. ∂tρ+∇ · ω = ζ

• WF defines a Riemannian-like metric on M+(Ω) (curvature);

• static cost in 1D : |√m0eix0 −√m1e

ix1 |2;

• relaxed constraints formulation : Kullback-Leibler penalization withthe cost − log(cos(|y − x | ∧ π

2 ));

• [Liero et al., 2015, Kondratyev et al., 2015, Chizat et al., 2015b] .

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Introduction Static Dynamic Examples & Numerics Conclusion

Wasserstein-Fisher-Rao : WF

Static

WF 2 := min|ρ0|TV + |ρ1|TV

−2

∫Ω2

cos

(|y − x |

2∧ π

2

)d√γ0γ1(x , y)

s.t. projxγ0 = ρ0 , projyγ1 = ρ1

Dynamic

min1

4

∫ 1

0

∫Ω

|ω|2

ρ+ζ2

ρ

s.t. ∂tρ+∇ · ω = ζ

• WF defines a Riemannian-like metric on M+(Ω) (curvature);

• static cost in 1D : |√m0eix0 −√m1e

ix1 |2;

• relaxed constraints formulation : Kullback-Leibler penalization withthe cost − log(cos(|y − x | ∧ π

2 ));

• [Liero et al., 2015, Kondratyev et al., 2015, Chizat et al., 2015b] .

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Introduction Static Dynamic Examples & Numerics Conclusion

Wasserstein-Fisher-Rao : WF

Static

WF 2 := min|ρ0|TV + |ρ1|TV

−2

∫Ω2

cos

(|y − x |

2∧ π

2

)d√γ0γ1(x , y)

s.t. projxγ0 = ρ0 , projyγ1 = ρ1

Dynamic

min1

4

∫ 1

0

∫Ω

|ω|2

ρ+ζ2

ρ

s.t. ∂tρ+∇ · ω = ζ

• WF defines a Riemannian-like metric on M+(Ω) (curvature);

• static cost in 1D : |√m0eix0 −√m1e

ix1 |2;

• relaxed constraints formulation : Kullback-Leibler penalization withthe cost − log(cos(|y − x | ∧ π

2 ));

• [Liero et al., 2015, Kondratyev et al., 2015, Chizat et al., 2015b] .

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Introduction Static Dynamic Examples & Numerics Conclusion

Wasserstein-Fisher-Rao : WF

Static

WF 2 := min|ρ0|TV + |ρ1|TV

−2

∫Ω2

cos

(|y − x |

2∧ π

2

)d√γ0γ1(x , y)

s.t. projxγ0 = ρ0 , projyγ1 = ρ1

Dynamic

min1

4

∫ 1

0

∫Ω

|ω|2

ρ+ζ2

ρ

s.t. ∂tρ+∇ · ω = ζ

• WF defines a Riemannian-like metric on M+(Ω) (curvature);

• static cost in 1D : |√m0eix0 −√m1e

ix1 |2;

• relaxed constraints formulation : Kullback-Leibler penalization withthe cost − log(cos(|y − x | ∧ π

2 ));

• [Liero et al., 2015, Kondratyev et al., 2015, Chizat et al., 2015b] .

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Introduction Static Dynamic Examples & Numerics Conclusion

Wasserstein-Fisher-Rao : WF

Static

WF 2 := min|ρ0|TV + |ρ1|TV

−2

∫Ω2

cos

(|y − x |

2∧ π

2

)d√γ0γ1(x , y)

s.t. projxγ0 = ρ0 , projyγ1 = ρ1

Dynamic

min1

4

∫ 1

0

∫Ω

|ω|2

ρ+ζ2

ρ

s.t. ∂tρ+∇ · ω = ζ

• WF defines a Riemannian-like metric on M+(Ω) (curvature);

• static cost in 1D : |√m0eix0 −√m1e

ix1 |2;

• relaxed constraints formulation : Kullback-Leibler penalization withthe cost − log(cos(|y − x | ∧ π

2 ));

• [Liero et al., 2015, Kondratyev et al., 2015, Chizat et al., 2015b] .

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Introduction Static Dynamic Examples & Numerics Conclusion

Wasserstein-Fisher-Rao : WF

Static

WF 2 := min|ρ0|TV + |ρ1|TV

−2

∫Ω2

cos

(|y − x |

2∧ π

2

)d√γ0γ1(x , y)

s.t. projxγ0 = ρ0 , projyγ1 = ρ1

Dynamic

min1

4

∫ 1

0

∫Ω

|ω|2

ρ+ζ2

ρ

s.t. ∂tρ+∇ · ω = ζ

• WF defines a Riemannian-like metric on M+(Ω) (curvature);

• static cost in 1D : |√m0eix0 −√m1e

ix1 |2;

• relaxed constraints formulation : Kullback-Leibler penalization withthe cost − log(cos(|y − x | ∧ π

2 ));

• [Liero et al., 2015, Kondratyev et al., 2015, Chizat et al., 2015b] .

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Introduction Static Dynamic Examples & Numerics Conclusion

Numerics

Proximal splitting algorithms on the dynamic formulation :https://github.com/lchizat/optimal-transport

Figure: FR Figure: W2 Figure: W2−TV Figure: W2 − FR

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Introduction Static Dynamic Examples & Numerics Conclusion

Conclusion

In progress

• Numerics on the relaxed marginal formulation

• More applications

Take home message

A unified framework for unbalanced OT allowing dynamic, staticand dual formulations.

18 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Conclusion

In progress

• Numerics on the relaxed marginal formulation

• More applications

Take home message

A unified framework for unbalanced OT allowing dynamic, staticand dual formulations.

18 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Conclusion

In progress

• Numerics on the relaxed marginal formulation

• More applications

Take home message

A unified framework for unbalanced OT allowing dynamic, staticand dual formulations.

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Introduction Static Dynamic Examples & Numerics Conclusion

For Further Reading I

Chizat, L., Peyre, G., Schmitzer, B., and Vialard, F.-X.(2015a).Unbalanced optimal transport: geometry and Kantorovichformulation.arXiv preprint arXiv:1508.05216.

Chizat, L., Schmitzer, B., Peyre, G., and Vialard, F.-X.(2015b).An interpolating distance between optimal transport andFisher-Rao.http://arxiv.org/abs/1506.06430.

Kondratyev, S., Monsaingeon, L., and Vorotnikov, D. (2015).A new optimal transport distance on the space of finite Radonmeasures.Technical report, Pre-print.

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Introduction Static Dynamic Examples & Numerics Conclusion

For Further Reading II

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

Piccoli, B. and Rossi, F. (2013).On properties of the Generalized Wasserstein distance.arXiv:1304.7014. d

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