UNBALANCED OPTIMAL TRANSPORT L ena c Chizat joint work with F-X. - - PowerPoint PPT Presentation

UNBALANCED OPTIMAL TRANSPORT

L´ ena¨ ıc Chizat joint work with F-X. Vialard, G. Peyr´ e & B. Schmitzer

CEREMADE Universit´ e Paris Dauphine

Mokalien 2015

Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

Motivations

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

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Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

Motivations

Image 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]);

2 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

Motivations

Image 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 .

2 / 20

Introduction Static Dynamic Examples & Numerics Conclusion

Introduction

Motivations

Image 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.

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Introduction Static Dynamic Examples & Numerics Conclusion

Outline

Static Formulation Dynamic Formulation Examples & Numerics

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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

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Introduction Static Dynamic Examples & Numerics Conclusion

From standard OT...

Ω

• ×

δx

c ( x , y )

• ×δy

Assumptions on the cost:

• lower bounded;
• l.s.c.

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Introduction Static Dynamic Examples & Numerics Conclusion

From standard OT...

Ω

• ×

δx

c ( x , y )

• ×δy

Assumptions on the cost:

• lower bounded;
• l.s.c.

Static formulation of OT:

minimize

• Ω2 c(x, y)dγ(x, y)

subject to (projx)#γ = ρ0 (projy)#γ = ρ1

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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

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Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

• ×

mxδx

c ( ( x , m

x

) , ( y , m

y

) )

• ×myδy

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Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

• ×

mxδx

c ( ( x , m

x

) , ( y , m

y

) )

• ×myδy

The cost function is

• pos. homogeneous in

(mx, my);

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Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

• ×

mxδx

c ( ( x , m

x

) , ( y , m

y

) )

• ×myδy

The cost function is

• pos. homogeneous in

(mx, my);

• subadditive in (mx, my);

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Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

• ×

mxδx

c ( ( x , m

x

) , ( y , m

y

) )

• ×myδy

The cost function is

• pos. homogeneous in

(mx, my);

• subadditive in (mx, my);
• nonnegative;

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Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

• ×

mxδx

c ( ( x , m

x

) , ( y , m

y

) )

• ×myδy

The cost function is

• pos. homogeneous in

(mx, my);

• subadditive in (mx, my);
• nonnegative;
• mx or my negative

⇒ c = +∞ ;

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Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

• ×

mxδx

c ( ( x , m

x

) , ( y , m

y

) )

• ×myδy

The cost function is

• pos. homogeneous in

(mx, my);

• subadditive in (mx, my);
• nonnegative;
• mx or my negative

⇒ c = +∞ ;

• lower semicontinuous

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Introduction Static Dynamic Examples & Numerics Conclusion

...to Unbalanced OT

Ω

• ×

mxδx

c ( ( x , m

x

) , ( y , m

y

) )

• ×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

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Introduction Static Dynamic Examples & Numerics Conclusion

Properties

Ω R+

• ×

mxδx

c1/p

• ×myδy

Cone(Ω) := (Ω × R+)/(Ω × {0})

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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 on M+(Ω).

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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 on M+(Ω).

Theorem (Duality)

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

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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 on M+(Ω).

Theorem (Duality)

For all (x, y) ∈ Ω2, c(x, ·, y, ·) is the support function of a closed convex nonempty set Q(x, y) ⊂ R2. If Q is l.s.c. in the sense of multifunctions, then C(ρ0, ρ1) = sup

φ,ψ∈C(Ω)

• Ω

φ(x)dρ0(x) +

• Ω

ψ(y)dρ1(y) subject to (φ(x), ψ(y)) ∈ Q(x, y) for all (x, y) ∈ Ω2 .

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Introduction Static Dynamic Examples & Numerics Conclusion

Outline

Static Formulation Dynamic Formulation Examples & Numerics

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt Change of variables: ω = ρv Infinitesimal cost : f (x, ρ, ω)

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt Change of variables: ω = ρv Infinitesimal cost : f (x, ρ, ω)

• homogeneous in (ρ, ω);

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt Change of variables: ω = ρv Infinitesimal cost : f (x, ρ, ω)

• homogeneous in (ρ, ω);
• subadditive in (ρ, ω);

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach: standard OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt Change of variables: ω = ρv Infinitesimal cost : f (x, ρ, ω)

• homogeneous in (ρ, ω);
• subadditive in (ρ, ω);

Standard dynamic formulation

minimize 1

• Ω

f (x, dρ dµ, dω dµ)dµ (ρ, |ω| ≪ µ) subject to ∂tρ + ∇ · ω = 0 (weakly) (projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

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

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

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

• homogeneous in (ρ, ω, ζ);

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

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

• homogeneous in (ρ, ω, ζ);
• subadditive in (ρ, ω, ζ);

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

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

• homogeneous in (ρ, ω, ζ);
• subadditive in (ρ, ω, ζ);
• ρ < 0 ⇒ f = +∞;

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

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

• homogeneous in (ρ, ω, ζ);
• subadditive in (ρ, ω, ζ);
• ρ < 0 ⇒ f = +∞;
• sign (f ) = sign (|ω| + |ζ|)

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

Variables: ω = ρv, ζ = ρα Infinitesimal 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.

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Introduction Static Dynamic Examples & Numerics Conclusion

A dynamic approach : unbalanced OT

Ω

• ×
• ×
• ×

ρtδx(t)

vt

αt = ∂t ρt

ρt

Variables: ω = ρv, ζ = ρα Infinitesimal 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

• Ω

f (x, dρ dµ, dω dµ, dζ dµ)dµ(t, x) subject to ∂tρ + ∇ · ω = ζ (weakly) (projt=0)#ρ = ρ0 , (projt=1)#ρ = ρ1 .

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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.

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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.

Theorem

Assume that Q is a l.s.c. multifunction. Then the minimum defining CD is attained and CD(ρ0, ρ1) = sup

ϕ∈C 1([0,1]×Ω)

• Ω

ϕ(1, x)dρ1(x) −

• Ω

ϕ(0, x)dρ0(x) subject to (∂tϕ, ∇ϕ, ϕ)(t, x) ∈ Q(x) .

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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 restricted to smooth, stable Dirac trajectories m(t)δx(t).

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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 restricted to 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 associated problem C is weakly* continuous, then C = CD (and c = cd).

Note : cd is hard to compute directly in general.

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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 restricted to 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 associated problem C is weakly* continuous, then C = CD (and c = cd).

Note : cd is hard to compute directly in general.

Example

A good candidate is the convex regularization of cp: inf

ma

0+mb 0=m0

ma

1+mb 1=m1

cp((x0, ma

0), (x1, ma 1)) + cp((x0, mb 0), (x1, mb 1))

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Introduction Static Dynamic Examples & Numerics Conclusion

Outline

Static Formulation Dynamic Formulation Examples & Numerics

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Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

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Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := min

˜ ρ0, ˜ ρ1

1 p W p

p (˜

ρ0, ˜ ρ1) + δ (|ρ0 − ˜ ρ0|TV + |ρ1 − ˜ ρ1|TV )

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Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := min

˜ ρ0, ˜ ρ1

1 p W p

p (˜

ρ0, ˜ ρ1) + δ (|ρ0 − ˜ ρ0|TV + |ρ1 − ˜ ρ1|TV )

Dynamic

min 1

• Ω

1 p ωp ρp−1 + δ|ζ| s.t. ∂tρ + ∇ · ω = ζ

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Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := min

˜ ρ0, ˜ ρ1

1 p W p

p (˜

ρ0, ˜ ρ1) + δ (|ρ0 − ˜ ρ0|TV + |ρ1 − ˜ ρ1|TV )

Dynamic

min 1

• Ω

1 p ωp ρp−1 + δ|ζ| s.t. ∂tρ + ∇ · ω = ζ

• equivalent to the “Lagrangian” formulation of partial OT: m ↔ 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 := min

˜ ρ0, ˜ ρ1

1 p W p

p (˜

ρ0, ˜ ρ1) + δ (|ρ0 − ˜ ρ0|TV + |ρ1 − ˜ ρ1|TV )

Dynamic

min 1

• 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);

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Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := min

˜ ρ0, ˜ ρ1

1 p W p

p (˜

ρ0, ˜ ρ1) + δ (|ρ0 − ˜ ρ0|TV + |ρ1 − ˜ ρ1|TV )

Dynamic

min 1

• 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;

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Introduction Static Dynamic Examples & Numerics Conclusion

Partial OT / Wasserstein-TV

Extend the results in [Piccoli and Rossi, 2013]

Static

C := min

˜ ρ0, ˜ ρ1

1 p W p

p (˜

ρ0, ˜ ρ1) + δ (|ρ0 − ˜ ρ0|TV + |ρ1 − ˜ ρ1|TV )

Dynamic

min 1

• 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

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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

min 1 4 1

• Ω

|ω|2 ρ + ζ2 ρ s.t. ∂tρ + ∇ · ω = ζ

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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

min 1 4 1

• Ω

|ω|2 ρ + ζ2 ρ s.t. ∂tρ + ∇ · ω = ζ

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

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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

min 1 4 1

• 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 − √m1eix1|2;

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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

min 1 4 1

• 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 − √m1eix1|2;
• relaxed constraints formulation : Kullback-Leibler penalization with

the cost − log(cos(|y − x| ∧ π

2 ));

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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

min 1 4 1

• 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 − √m1eix1|2;
• relaxed constraints formulation : Kullback-Leibler penalization with

the 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

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Introduction Static Dynamic Examples & Numerics Conclusion

Conclusion

In progress

• Numerics on the relaxed marginal formulation
• More applications

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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, static and dual formulations.

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Introduction Static Dynamic Examples & Numerics Conclusion

For Further Reading I

Chizat, L., Peyr´ e, G., Schmitzer, B., and Vialard, F.-X. (2015a). Unbalanced optimal transport: geometry and Kantorovich formulation. arXiv preprint arXiv:1508.05216. Chizat, L., Schmitzer, B., Peyr´ e, G., and Vialard, F.-X. (2015b). An interpolating distance between optimal transport and Fisher-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 Radon measures. Technical report, Pre-print.

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Introduction Static Dynamic Examples & Numerics Conclusion

For Further Reading II

Liero, M., Mielke, A., and Savar´ e, G. (2015). Optimal Entropy-Transport problems and a new Hellinger-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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