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... 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]); 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 R n . 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... • • × δ y Ω δ x × y ) x c ( , 5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion From standard OT... • • Assumptions on the cost: × δ y Ω • lower bounded; δ x × y ) x c ( , • l.s.c. 5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion From standard OT... • • Assumptions on the cost: × δ y Ω • lower bounded; δ x × y ) x c ( , • l.s.c. Static formulation of OT: � minimize Ω 2 c ( x , y ) d γ ( x , y ) subject to (proj x ) # γ = ρ 0 (proj y ) # γ = ρ 1 5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion From standard OT... • • Assumptions on the cost: × m δ y Ω • lower bounded; m δ x × ) y ( x c , m . • l.s.c. also linear in m. Static formulation of OT: � Ω 2 c ( d γ minimize d λ, x , y ) d λ ( x , y ) ( γ ≪ λ ) subject to (proj x ) # γ = ρ 0 (proj y ) # γ = ρ 1 5 / 20

Introduction Static Dynamic Examples & Numerics Conclusion ...to Unbalanced OT • • × m y δ y Ω ) ) m x δ x × m ( y y ) , m , ( x x c ( , 6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion ...to Unbalanced OT The cost function is • • pos. homogeneous in ( m x , m y ); • × m y δ y Ω ) ) m x δ x × m ( y y ) , m , ( x x c ( , 6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion ...to Unbalanced OT The cost function is • • pos. homogeneous in ( m x , m y ); • subadditive in ( m x , m y ); • × m y δ y Ω ) ) m x δ x × m ( y y ) , m , ( x x c ( , 6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion ...to Unbalanced OT The cost function is • • pos. homogeneous in ( m x , m y ); • subadditive in ( m x , m y ); • • nonnegative; × m y δ y Ω ) ) m x δ x × m ( y y ) , m , ( x x c ( , 6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion ...to Unbalanced OT The cost function is • • pos. homogeneous in ( m x , m y ); • subadditive in ( m x , m y ); • • nonnegative; × m y δ y Ω ) ) • m x or m y negative m x δ x × m ( y y ) , m , ( x x c ( , ⇒ c = + ∞ ; 6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion ...to Unbalanced OT The cost function is • • pos. homogeneous in ( m x , m y ); • subadditive in ( m x , m y ); • • nonnegative; × m y δ y Ω ) ) • m x or m y negative m x δ x × m ( y y ) , m , ( x x c ( , ⇒ c = + ∞ ; • lower semicontinuous 6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion ...to Unbalanced OT The cost function is • • pos. homogeneous in ( m x , m y ); • subadditive in ( m x , m y ); • • nonnegative; × m y δ y Ω ) ) • m x or m y negative m x δ x × m ( y y ) , m , ( x x c ( , ⇒ c = + ∞ ; • lower semicontinuous Static formulation of Unbalanced OT � Ω 2 c (( x , d γ 0 d γ ) , ( y , d γ 1 C ( ρ 0 , ρ 1 ) := minimize d γ )) d γ ( x , y ) subject to ( π x ) # γ 0 = ρ 0 ( π y ) # γ 1 = ρ 1 6 / 20

Introduction Static Dynamic Examples & Numerics Conclusion Properties • R + c 1 / p Cone(Ω) := (Ω × R + ) / (Ω × { 0 } ) • × m y δ y m x δ x × Ω 7 / 20

Introduction Static Dynamic Examples & Numerics Conclusion Properties • R + c 1 / p Cone(Ω) := (Ω × R + ) / (Ω × { 0 } ) • × m y δ y Theorem (Metric property) m x δ x × Ω If c 1 / p is a metric on Cone (Ω) then C 1 / p is a metric on M + (Ω) . 7 / 20

Introduction Static Dynamic Examples & Numerics Conclusion Properties • R + c 1 / p Cone(Ω) := (Ω × R + ) / (Ω × { 0 } ) • × m y δ y Theorem (Metric property) m x δ x × Ω If c 1 / p is a metric on Cone (Ω) then C 1 / p is a metric on M + (Ω) . Theorem (Duality) For all ( x , y ) ∈ Ω 2 , c ( x , · , y , · ) is the support function of a closed convex nonempty set Q ( x , y ) ⊂ R 2 . 7 / 20

Introduction Static Dynamic Examples & Numerics Conclusion Properties • R + c 1 / p Cone(Ω) := (Ω × R + ) / (Ω × { 0 } ) • × m y δ y Theorem (Metric property) m x δ x × Ω If c 1 / p is a metric on Cone (Ω) then C 1 / p is a metric on M + (Ω) . Theorem (Duality) For all ( x , y ) ∈ Ω 2 , c ( x , · , y , · ) is the support function of a closed convex nonempty set Q ( x , y ) ⊂ R 2 . If Q is l.s.c. in the sense of multifunctions, then � � C ( ρ 0 , ρ 1 ) = sup φ ( x ) d ρ 0 ( x ) + ψ ( y ) d ρ 1 ( y ) φ,ψ ∈ C (Ω) Ω Ω subject to ( φ ( x ) , ψ ( y )) ∈ Q ( x , y ) for all ( x , y ) ∈ Ω 2 . 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 v t × ρ t δ x ( t ) 9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion A dynamic approach: standard OT • • Change of variables: ω = ρ v • × Infinitesimal cost : f ( x , ρ, ω ) × Ω v t × ρ t δ x ( t ) 9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion A dynamic approach: standard OT • • Change of variables: ω = ρ v • × Infinitesimal cost : f ( x , ρ, ω ) × Ω v t • homogeneous in ( ρ, ω ); × ρ t δ x ( t ) 9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion A dynamic approach: standard OT • • Change of variables: ω = ρ v • × Infinitesimal cost : f ( x , ρ, ω ) × Ω v t • homogeneous in ( ρ, ω ); × ρ t δ x ( t ) • subadditive in ( ρ, ω ); 9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion A dynamic approach: standard OT • • Change of variables: ω = ρ v • × Infinitesimal cost : f ( x , ρ, ω ) × Ω v t • homogeneous in ( ρ, ω ); × ρ t δ x ( t ) • subadditive in ( ρ, ω ); Standard dynamic formulation � 1 � f ( x , d ρ d µ, d ω minimize d µ ) d µ ( ρ, | ω | ≪ µ ) 0 Ω subject to ∂ t ρ + ∇ · ω = 0 (weakly) (proj t =0 ) # ρ = ρ 0 , (proj t =1 ) # ρ = ρ 1 . 9 / 20

Introduction Static Dynamic Examples & Numerics Conclusion A dynamic approach : unbalanced OT • α t = ∂ t ρ t ρ t • • × × Ω v t × ρ t δ x ( t ) 10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion A dynamic approach : unbalanced OT • Variables: ω = ρ v , ζ = ρα Infinitesimal cost : f ( x , ρ, ω, ζ ) α t = ∂ t ρ t ρ t • • × × Ω v t × ρ t δ x ( t ) 10 / 20

Introduction Static Dynamic Examples & Numerics Conclusion A dynamic approach : unbalanced OT • Variables: ω = ρ v , ζ = ρα Infinitesimal cost : f ( x , ρ, ω, ζ ) • homogeneous in ( ρ, ω, ζ ); α t = ∂ t ρ t ρ t • • × × Ω v t × ρ t δ x ( t ) 10 / 20