A new Hellinger-Kantorovich distance between positive measures and optimal Entropy-Transport problems

Transcription

1 A new Hellinger-Kantorovich distance between positive measures and optimal Entropy-Transport problems Giuseppe Savaré savare Dipartimento di Matematica, Università di Pavia Nonlocal Nonlinear Partial Differential Equations and Applications Anacapri, September 14, 2015 In collaboration with M. Liero e A. Mielke

2 2 Outline 1 The Kantorovich-Wasserstein distance: four equivalent definitions 2 A new distance between positive measures of arbitrary mass 3 Optimal Entropy-Transport problems and homogenous marginal projections

3 3 Outline 1 The Kantorovich-Wasserstein distance: four equivalent definitions 2 A new distance between positive measures of arbitrary mass 3 Optimal Entropy-Transport problems and homogenous marginal projections

4 4 The Benamou-Brenier dynamic characterization µ C 0 ([0, 1]; M(R d )); v : R d (0, 1) R d is a Borel vector field satisfying 1 v t(x) 2 dµ t(x) dt < 0 Continuity equation governed by the field v tµ t + (v tµ t) = 0 in the sense of distributions of R d (0, 1). (CE) Benamou-Brenier dynamic formulation: { W 2 1 (µ 0, µ 1) = min v t 2 dµ t dt : µ C([0, 1]; M(R d )), 0 tµ t + (v tµ t) = 0, µ t=i = µ i }. (BB) The Borel vector field v realizing the minimum is uniquely determined µ L 1 -a.e. in R d (0, 1) and v t(x) 2 dµ t(x) = µ t 2 W 2 (µ t, µ t+h ) W = lim h 0 h 2

5 4 The Benamou-Brenier dynamic characterization µ C 0 ([0, 1]; M(R d )); v : R d (0, 1) R d is a Borel vector field satisfying 1 v t(x) 2 dµ t(x) dt < 0 Continuity equation governed by the field v tµ t + (v tµ t) = 0 in the sense of distributions of R d (0, 1). (CE) Benamou-Brenier dynamic formulation: { W 2 1 (µ 0, µ 1) = min v t 2 dµ t dt : µ C([0, 1]; M(R d )), 0 tµ t + (v tµ t) = 0, µ t=i = µ i }. (BB) The Borel vector field v realizing the minimum is uniquely determined µ L 1 -a.e. in R d (0, 1) and v t(x) 2 dµ t(x) = µ t 2 W 2 (µ t, µ t+h ) W = lim h 0 h 2

6 Duality with the quadratic Hamilton-Jacobi equation If ζ is a regular subsolution to the Hamilton-Jacobi equation tζ t Dζt 2 0 (HJ) and then ζ 1 dµ 1 tµ t + (v tµ t) = 0 ζ 0 dµ v t 2 dµ t dt. HJ-duality (Otto-Villani, Bobkov-Ledoux) 1 { 2 W2 (µ 0, µ 1) = sup ζ 1 dµ 1 ζ 0 dµ 0 : ζ C 1 ([0, 1]; Lip b (R d )) tζ t + 1 } 2 Dζt 2 0.

7 Duality with the quadratic Hamilton-Jacobi equation If ζ is a regular subsolution to the Hamilton-Jacobi equation tζ t Dζt 2 0 (HJ) and then ζ 1 dµ 1 tµ t + (v tµ t) = 0 ζ 0 dµ v t 2 dµ t dt. HJ-duality (Otto-Villani, Bobkov-Ledoux) 1 { 2 W2 (µ 0, µ 1) = sup ζ 1 dµ 1 ζ 0 dµ 0 : ζ C 1 ([0, 1]; Lip b (R d )) tζ t + 1 } 2 Dζt 2 0.

8 6 Hopf-Lax formula and the dual Kantorovich formulation Given ζ 0 Lip b (R d ) the viscosity solution (or the maximal subsolution) of the Hamilton Jacobi equation tζ t Dζt 2 = 0 (HJ) is given by the Hopf-Lax semigroup Q tζ 0(x) := inf y ζ0(y) + 1 2t x y 2 (Hopf-Lax) Dual Kantorovich formulation 1 { 2 W2 (µ 0, µ 1) = sup ζ 1 dµ 1 { = sup ζ 1 dµ 1 ζ 0 dµ 0 : ζ 1 = Q 1ζ 0 } ζ 0 dµ 0 : ζ 1(y) ζ 0(x) 1 x y 2} 2

9 6 Hopf-Lax formula and the dual Kantorovich formulation Given ζ 0 Lip b (R d ) the viscosity solution (or the maximal subsolution) of the Hamilton Jacobi equation tζ t Dζt 2 = 0 (HJ) is given by the Hopf-Lax semigroup Q tζ 0(x) := inf y ζ0(y) + 1 2t x y 2 (Hopf-Lax) Dual Kantorovich formulation 1 { 2 W2 (µ 0, µ 1) = sup ζ 1 dµ 1 { = sup ζ 1 dµ 1 ζ 0 dµ 0 : ζ 1 = Q 1ζ 0 } ζ 0 dµ 0 : ζ 1(y) ζ 0(x) 1 x y 2} 2

10 The primal formulation: optimal transport Transport plans Plan(µ 0, µ 1): µ M(R d R d ) with fixed marginals µ 0 and µ 1, i.e. µ(a R d ) = µ 0(A), µ(r d B) = µ 1(B). Optimal transport formulation of the Kantorovich-Wasserstein distance { } W 2 (µ 0, µ 1) = min x y 2 dµ(x, y) : µ Plan(µ 0, µ 1) Kantorovich duality: { 1 } min 2 x y 2 dµ(x, y) : µ Plan(µ 0, µ 1) { = sup ζ 1 dµ 1 ζ 0 dµ 0 : ζ 1(y) ζ 0(x) 1 x y 2} 2

11 8 Four equivalent formulations Benamou Brenier Dynamic superposition = Hamilton Jacobi Geodesic interpolation Hopf-Lax Optimal Transport If µ 0 L d with finite quadratic moment: Linear programming Kantorovich duality Uniqueness of optimal plans: µ = (i r) µ 0, r(x) := x Dζ 0(x), W 2 (µ 0, µ 1) = r(x) x 2 dµ 0(x) x Dζ 0(x) is monotone, ζ 0 is an optimal Kantorovich potential r t(x) := (1 t)x + tr(x) are the characteristics of the continuity equation and of the HJ equation. µ t = (r t) µ 0, v t(r t(x)) = r(x) x, ζ t(r t(x)) = ζ 0(x) + t 2 r(x) x 2. v t = Dζ t

12 8 Four equivalent formulations Benamou Brenier Dynamic superposition = Hamilton Jacobi Geodesic interpolation Hopf-Lax Optimal Transport If µ 0 L d with finite quadratic moment: Linear programming Kantorovich duality Uniqueness of optimal plans: µ = (i r) µ 0, r(x) := x Dζ 0(x), W 2 (µ 0, µ 1) = r(x) x 2 dµ 0(x) x Dζ 0(x) is monotone, ζ 0 is an optimal Kantorovich potential r t(x) := (1 t)x + tr(x) are the characteristics of the continuity equation and of the HJ equation. µ t = (r t) µ 0, v t(r t(x)) = r(x) x, ζ t(r t(x)) = ζ 0(x) + t 2 r(x) x 2. v t = Dζ t

13 9 Outline 1 The Kantorovich-Wasserstein distance: four equivalent definitions 2 A new distance between positive measures of arbitrary mass 3 Optimal Entropy-Transport problems and homogenous marginal projections

14 10 Starting point: a dynamic formulation of HK Let µ C 0 ([0, 1]; M(R d )), (v, w) : R d (0, 1) R d+1 be a Borel vector field satisfying 1 ( ) v t(x) 2 + wt 2 (x) dµ t(x) dt <. 0 X Continuity equation with reaction governed by the field (v, w) if tµ t + (v tµ t) = w tµ t in D (R d (0, 1)) (CER) The Hellinger-Kantorovich distance via dynamic interpolation { HK 2 1 (µ 0, µ 1) = min 0 ( v t wt 2) dµ t dt : µ C([0, 1]; M(R d )), tµ t + (v tµ t) = w tµ t, µ t=i = µ i }. HK is a convex and subadditive functional (cf. Dolbeault-Nazaret-S.). A similar approach has been independently proposed by Kondratiev, Monsaingeon, Vorotnikov and Chizat, Peyré, Vialard, Schmitzer.

15 10 Starting point: a dynamic formulation of HK Let µ C 0 ([0, 1]; M(R d )), (v, w) : R d (0, 1) R d+1 be a Borel vector field satisfying 1 ( ) v t(x) 2 + wt 2 (x) dµ t(x) dt <. 0 X Continuity equation with reaction governed by the field (v, w) if tµ t + (v tµ t) = w tµ t in D (R d (0, 1)) (CER) The Hellinger-Kantorovich distance via dynamic interpolation { HK 2 1 (µ 0, µ 1) = min 0 ( v t wt 2) dµ t dt : µ C([0, 1]; M(R d )), tµ t + (v tµ t) = w tµ t, µ t=i = µ i }. HK is a convex and subadditive functional (cf. Dolbeault-Nazaret-S.). A similar approach has been independently proposed by Kondratiev, Monsaingeon, Vorotnikov and Chizat, Peyré, Vialard, Schmitzer.

16 HK is always finite and dominated by the Hellinger distance Hellinger distance: Hell 2 (µ 0, µ 1) = ( ϱ 0 ϱ 1) 2 dµ, µ i = ϱ iµ µ. Estimate of the pure reaction part, v 0. Given a curve ϱ t = r 2 t : (0, 1) (0, ) joining ϱ 0 = r 2 0, ϱ 1 = r 2 1 0, if ϱ t = 2r tṙ t = r 2 t w t we get w t = 2ṙ t/r t { 1 min w 2 t ϱ t dt = 1 0 ṙ t 2 dt } wt 2 ϱ t dt : ϱ t=i = ϱ i, ϱ t = ϱ tw t = ( ) 2 r 1 r 0 Thus, writing µ i = ϱ iµ = ri 2 µ with, e.g. µ := µ 1 + µ 2, and considering the curve µ t := ((1 t)r 0 + tr 1) 2 µ we obtain HK 2 (µ 0, µ 1) Hell 2 (r1 ) 2 (µ 0, µ 1) = r 0 dµ µ 0(R d ) + µ 1(R d ).

17 11 HK is always finite and dominated by the Hellinger distance Hellinger distance: Hell 2 (µ 0, µ 1) = ( ϱ 0 ϱ 1) 2 dµ, µ i = ϱ iµ µ. Estimate of the pure reaction part, v 0. Given a curve ϱ t = r 2 t : (0, 1) (0, ) joining ϱ 0 = r 2 0, ϱ 1 = r 2 1 0, if ϱ t = 2r tṙ t = r 2 t w t we get w t = 2ṙ t/r t { 1 min w 2 t ϱ t dt = 1 0 ṙ t 2 dt } wt 2 ϱ t dt : ϱ t=i = ϱ i, ϱ t = ϱ tw t = ( ) 2 r 1 r 0 Thus, writing µ i = ϱ iµ = ri 2 µ with, e.g. µ := µ 1 + µ 2, and considering the curve µ t := ((1 t)r 0 + tr 1) 2 µ we obtain HK 2 (µ 0, µ 1) Hell 2 (r1 ) 2 (µ 0, µ 1) = r 0 dµ µ 0(R d ) + µ 1(R d ).

18 11 HK is always finite and dominated by the Hellinger distance Hellinger distance: Hell 2 (µ 0, µ 1) = ( ϱ 0 ϱ 1) 2 dµ, µ i = ϱ iµ µ. Estimate of the pure reaction part, v 0. Given a curve ϱ t = r 2 t : (0, 1) (0, ) joining ϱ 0 = r 2 0, ϱ 1 = r 2 1 0, if ϱ t = 2r tṙ t = r 2 t w t we get w t = 2ṙ t/r t { 1 min w 2 t ϱ t dt = 1 0 ṙ t 2 dt } wt 2 ϱ t dt : ϱ t=i = ϱ i, ϱ t = ϱ tw t = ( ) 2 r 1 r 0 Thus, writing µ i = ϱ iµ = ri 2 µ with, e.g. µ := µ 1 + µ 2, and considering the curve µ t := ((1 t)r 0 + tr 1) 2 µ we obtain HK 2 (µ 0, µ 1) Hell 2 (r1 ) 2 (µ 0, µ 1) = r 0 dµ µ 0(R d ) + µ 1(R d ).

19 12 The distance between two Dirac masses Suppose that µ i = r 2 i δ xi. We have HK 2 (r 2 0δ x0, r 2 1δ x1 ) = r r 2 1 2r 0r 1 cos( x 1 x 0 π/2 ) Truncation effect: when x 0 x 1 π/2 we have Cone distance: HK 2 (r 2 0δ x0, r 2 1δ x1 ) = r r 2 1. d 2 C((x 0, r 0), (x 1, r 1)) = r r 2 1 2r 0r 1 cos( x 1 x 0 π) Cone space: identify all the points (x, 0) with the vertex o. C := ( R d [0, ) ) { /, (x, r ) (x, r x = x, r = r 0, ) r = r = 0 C \ {o} can be considered as a smooth Riemannian manifold with metric g(dx, dr) := r 2 dx 2 + dr 2

20 12 The distance between two Dirac masses Suppose that µ i = r 2 i δ xi. We have HK 2 (r 2 0δ x0, r 2 1δ x1 ) = r r 2 1 2r 0r 1 cos( x 1 x 0 π/2 ) Truncation effect: when x 0 x 1 π/2 we have Cone distance: HK 2 (r 2 0δ x0, r 2 1δ x1 ) = r r 2 1. d 2 C((x 0, r 0), (x 1, r 1)) = r r 2 1 2r 0r 1 cos( x 1 x 0 π) Cone space: identify all the points (x, 0) with the vertex o. C := ( R d [0, ) ) { /, (x, r ) (x, r x = x, r = r 0, ) r = r = 0 C \ {o} can be considered as a smooth Riemannian manifold with metric g(dx, dr) := r 2 dx 2 + dr 2

21 Duality with the conical Hamilton-Jacobi equation If tξ t Dξt 2 + 2ξ 2 t (x) 0 (CHJ) and then ξ 1 dµ 1 tµ t + (v tµ t) = w tµ t ξ 0 dµ ( v t ) 4 w2 t dµ t dt. HK in duality with subsolutions to the conical Hamilton-Jacobi equations 1 { 2 HK2 (µ 0, µ 1) = sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ C 1 ([0, 1]; Lip b (R d )) tξ t + 1 } 2 Dξt 2 + 2ξt 2 0.

22 Duality with the conical Hamilton-Jacobi equation If tξ t Dξt 2 + 2ξ 2 t (x) 0 (CHJ) and then ξ 1 dµ 1 tµ t + (v tµ t) = w tµ t ξ 0 dµ ( v t ) 4 w2 t dµ t dt. HK in duality with subsolutions to the conical Hamilton-Jacobi equations 1 { 2 HK2 (µ 0, µ 1) = sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ C 1 ([0, 1]; Lip b (R d )) tξ t + 1 } 2 Dξt 2 + 2ξt 2 0.

23 14 Conical Hopf-Lax representation formula Given ξ 0 Lip b (R d ) the viscosity solution (or the maximal subsolution) of the conical Hamilton Jacobi equation tξ t Dξt 2 + 2ξ 2 t = 0 (CHJ) is given by the conical Hopf-Lax semigroup (cf. Barron-Jensen-Liu for different representation formulae) P tξ(x) := inf y [ ] 1 1 cos2 ( y x π/2 ) 2t 1 + 2tξ(x) (CHL) where x y π/2 := x y π/2. Conical Hopf-Lax representation for HK 1 { } 2 HK2 (µ 0, µ 1) = sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ 1 = P 1ξ 0

24 14 Conical Hopf-Lax representation formula Given ξ 0 Lip b (R d ) the viscosity solution (or the maximal subsolution) of the conical Hamilton Jacobi equation tξ t Dξt 2 + 2ξ 2 t = 0 (CHJ) is given by the conical Hopf-Lax semigroup (cf. Barron-Jensen-Liu for different representation formulae) P tξ(x) := inf y [ ] 1 1 cos2 ( y x π/2 ) 2t 1 + 2tξ(x) (CHL) where x y π/2 := x y π/2. Conical Hopf-Lax representation for HK 1 { } 2 HK2 (µ 0, µ 1) = sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ 1 = P 1ξ 0

25 Conical lift of the Hopf-Lax formula Formally, if ξ is a solution of tξ t Dξt 2 + 2ξ 2 t 0 (CHJ) then ζ t(x, r) := ξ t(x)r 2 is a solution of tζ t D Cξ t 2 0 (HJ) since 1 2 D Cζ 2 = 1 2 g (D xζ, rζ) = 1 ( 1 2 r 2 Dxζ 2 + ( rζ) 2) = The Hopf-Lax semigroup in C ( 1 2 Dξt 2 r 2 + 2ξ 2 t ) r 2 yields Qt C ζ(x, r) = min ζ(y, s) + 1 y,s 2t d2 C((x, r), (y, s)) = min y,s ξ(y)s t ( r 2 + s 2 2rs cos( x y π) ) Q C t ζ(x, r) = ξ t(x)r 2, ξ t = P tξ.

26 Conical lift of the Hopf-Lax formula Formally, if ξ is a solution of tξ t Dξt 2 + 2ξ 2 t 0 (CHJ) then ζ t(x, r) := ξ t(x)r 2 is a solution of tζ t D Cξ t 2 0 (HJ) since 1 2 D Cζ 2 = 1 2 g (D xζ, rζ) = 1 ( 1 2 r 2 Dxζ 2 + ( rζ) 2) = The Hopf-Lax semigroup in C ( 1 2 Dξt 2 r 2 + 2ξ 2 t ) r 2 yields Qt C ζ(x, r) = min ζ(y, s) + 1 y,s 2t d2 C((x, r), (y, s)) = min y,s ξ(y)s t ( r 2 + s 2 2rs cos( x y π) ) Q C t ζ(x, r) = ξ t(x)r 2, ξ t = P tξ.

27 Conical Hopf-Lax and dual Kantorovich formulation Setting ψ i := 2ξ i in the conical Hopf-Lax formula [ ] 1 P 1ξ(x) := inf 1 cos2 ( y x π/2 ) y ξ(x) (CHL) Dual Kantorovich formulation (I) { HK 2 (µ 0, µ 1) = sup ψ 1 dµ 1 ψ 0 dµ 0 : ψ 0 > 1, ψ 1 < 1 } (1 ψ 1(y))(1 + ψ 0(x)) cos 2 ( y x π/2 ) Change of variable: φ 1 := log(1 ψ 1), φ 0 := log(1 + ψ 0) (1 ψ 1(y))(1 + ψ 0(x)) cos 2 ( y x π/2 ) φ 1(y) φ 0(x) c(x, y), ( ) ( ) c(x, y) = log cos 2 ( y x π/2 ) = log 1 + tan 2 ( y x π/2 ) Dual Kantorovich formulation (II) { HK 2 (µ 0, µ 1) = sup (1 e φ 1 ) dµ 1 (e φ 0 1) dµ 0 : } φ 1(y) φ 0(x) c(x, y)

28 Conical Hopf-Lax and dual Kantorovich formulation Setting ψ i := 2ξ i in the conical Hopf-Lax formula [ ] 1 P 1ξ(x) := inf 1 cos2 ( y x π/2 ) y ξ(x) (CHL) Dual Kantorovich formulation (I) { HK 2 (µ 0, µ 1) = sup ψ 1 dµ 1 ψ 0 dµ 0 : ψ 0 > 1, ψ 1 < 1 } (1 ψ 1(y))(1 + ψ 0(x)) cos 2 ( y x π/2 ) Change of variable: φ 1 := log(1 ψ 1), φ 0 := log(1 + ψ 0) (1 ψ 1(y))(1 + ψ 0(x)) cos 2 ( y x π/2 ) φ 1(y) φ 0(x) c(x, y), ( ) ( ) c(x, y) = log cos 2 ( y x π/2 ) = log 1 + tan 2 ( y x π/2 ) Dual Kantorovich formulation (II) { HK 2 (µ 0, µ 1) = sup (1 e φ 1 ) dµ 1 (e φ 0 1) dµ 0 : } φ 1(y) φ 0(x) c(x, y)

29 Conical Hopf-Lax and dual Kantorovich formulation Setting ψ i := 2ξ i in the conical Hopf-Lax formula [ ] 1 P 1ξ(x) := inf 1 cos2 ( y x π/2 ) y ξ(x) (CHL) Dual Kantorovich formulation (I) { HK 2 (µ 0, µ 1) = sup ψ 1 dµ 1 ψ 0 dµ 0 : ψ 0 > 1, ψ 1 < 1 } (1 ψ 1(y))(1 + ψ 0(x)) cos 2 ( y x π/2 ) Change of variable: φ 1 := log(1 ψ 1), φ 0 := log(1 + ψ 0) (1 ψ 1(y))(1 + ψ 0(x)) cos 2 ( y x π/2 ) φ 1(y) φ 0(x) c(x, y), ( ) ( ) c(x, y) = log cos 2 ( y x π/2 ) = log 1 + tan 2 ( y x π/2 ) Dual Kantorovich formulation (II) { HK 2 (µ 0, µ 1) = sup (1 e φ 1 ) dµ 1 (e φ 0 1) dµ 0 : } φ 1(y) φ 0(x) c(x, y)

30 Primal formulation: Logarithmic Entropy-Transport problem We introduce the Legendre conjugate of G(φ) := e φ 1 and define the Entropy functional LE(s) := s log s (s 1) ( dγ ) E (γ µ) := LE(σ) dµ = log dγ + γ(r d ) µ(r d ) γ = σµ µ. dµ When γ is a plan in M(R d R d ) with marginals γ i we find (recall that c(x, y) = log(cos 2 ( x y π/2 ))) Logarithmic Entropy-Transport (LET) formulation LET(µ 0, µ 1) = ( min E (γ 0 µ 0) + E (γ 1 µ 1) + γ M(R d R d ) ) c(x, y) dγ(x, y) HK 2 (µ 0, µ 1) = LET(µ 0, µ 1)

31 Primal formulation: Logarithmic Entropy-Transport problem We introduce the Legendre conjugate of G(φ) := e φ 1 and define the Entropy functional LE(s) := s log s (s 1) ( dγ ) E (γ µ) := LE(σ) dµ = log dγ + γ(r d ) µ(r d ) γ = σµ µ. dµ When γ is a plan in M(R d R d ) with marginals γ i we find (recall that c(x, y) = log(cos 2 ( x y π/2 ))) Logarithmic Entropy-Transport (LET) formulation LET(µ 0, µ 1) = ( min E (γ 0 µ 0) + E (γ 1 µ 1) + γ M(R d R d ) ) c(x, y) dγ(x, y) HK 2 (µ 0, µ 1) = LET(µ 0, µ 1)

32 Primal formulation: Logarithmic Entropy-Transport problem We introduce the Legendre conjugate of G(φ) := e φ 1 and define the Entropy functional LE(s) := s log s (s 1) ( dγ ) E (γ µ) := LE(σ) dµ = log dγ + γ(r d ) µ(r d ) γ = σµ µ. dµ When γ is a plan in M(R d R d ) with marginals γ i we find (recall that c(x, y) = log(cos 2 ( x y π/2 ))) Logarithmic Entropy-Transport (LET) formulation LET(µ 0, µ 1) = ( min E (γ 0 µ 0) + E (γ 1 µ 1) + γ M(R d R d ) ) c(x, y) dγ(x, y) HK 2 (µ 0, µ 1) = LET(µ 0, µ 1)

33 18 Outline 1 The Kantorovich-Wasserstein distance: four equivalent definitions 2 A new distance between positive measures of arbitrary mass 3 Optimal Entropy-Transport problems and homogenous marginal projections

34 19 General Entropy-Transport problems Given two convex entropy functions F i : [0, ) [0, ] we define the Entropy functionals F i (also called Csiszàr F -divergence) F i(γ i µ i) := F i(σ i) dµ i + γi (R d ) γ i = σ iµ i + γ i c is a general nonnegative lower semicontinuous cost. Optimal Entropy Transport problem ET(µ 0, µ 1) := min F γ 0(γ 0 µ 0) + F 1(γ 1 µ 1) + c(x, y) dγ(x, y). (ET) Typically F i(1) = 0 so that (ET) can be considered as a sort of relaxation of the Optimal Transport problem, where the constraints i-marginal of γ=µ i has been substituted by the penalizing entropies F i. General duality and optimality theory in arbitrary topological Hausdorff spaces.

35 19 General Entropy-Transport problems Given two convex entropy functions F i : [0, ) [0, ] we define the Entropy functionals F i (also called Csiszàr F -divergence) F i(γ i µ i) := F i(σ i) dµ i + γi (R d ) γ i = σ iµ i + γ i c is a general nonnegative lower semicontinuous cost. Optimal Entropy Transport problem ET(µ 0, µ 1) := min F γ 0(γ 0 µ 0) + F 1(γ 1 µ 1) + c(x, y) dγ(x, y). (ET) Typically F i(1) = 0 so that (ET) can be considered as a sort of relaxation of the Optimal Transport problem, where the constraints i-marginal of γ=µ i has been substituted by the penalizing entropies F i. General duality and optimality theory in arbitrary topological Hausdorff spaces.

36 Four equivalent formulations for HK Dynamic formulation Duality = Conical Hamilton Jacobi? Conical Hopf-Lax Optimal Entropy-Transport Convex duality Kantorovich duality { HK 2 1 ( (µ 0, µ 1) = min v t wt 2) dµ t dt : µ C([0, 1]; M(R d )), } tµ t + (v tµ t) = w tµ t, µ t=i = µ i (CER) { = 2 sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ C 1 ([0, 1]; Lip b (R d )) tξ t + 1 } 2 Dξt 2 + 2ξt 2 0 { } = 2 sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ 1 = P 1ξ 0 = min E (γ γ 0 µ 0) + E (γ 1 µ 1) + c(x, y) dγ(x, y). (CHJ) (CHL) (LET)

37 Four equivalent formulations for HK Dynamic formulation Duality = Conical Hamilton Jacobi? Conical Hopf-Lax Optimal Entropy-Transport Convex duality Kantorovich duality { HK 2 1 ( (µ 0, µ 1) = min v t wt 2) dµ t dt : µ C([0, 1]; M(R d )), } tµ t + (v tµ t) = w tµ t, µ t=i = µ i (CER) { = 2 sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ C 1 ([0, 1]; Lip b (R d )) tξ t + 1 } 2 Dξt 2 + 2ξt 2 0 { } = 2 sup ξ 1 dµ 1 ξ 0 dµ 0 : ξ 1 = P 1ξ 0 = min E (γ γ 0 µ 0) + E (γ 1 µ 1) + c(x, y) dγ(x, y). (CHJ) (CHL) (LET)

38 Two basic questions What is the connection between - the Logarithmic Entropy LE(s) = s log s (s 1), - the cost c(x, y) = log(cos 2 ( x y π/2 ) - the conic distance d 2 C = r r 2 1 2r 0r 1 cos( x y π)? Why LET is a squared distance?

39 21 Two basic questions What is the connection between - the Logarithmic Entropy LE(s) = s log s (s 1), - the cost c(x, y) = log(cos 2 ( x y π/2 ) - the conic distance d 2 C = r r 2 1 2r 0r 1 cos( x y π)? Why LET is a squared distance?

40 22 Reverse and homogeneous formulation Reverse entropies R i(r) = rf i(1/r), R(r) := r 1 log r = rle(1/r). F (γ µ) = R(µ γ) Reverse formulation HK 2 (µ 0, µ 1) = min γ R(µ 0 γ 0) + R(µ 1 γ 1) + = min γ ( R(r 0(x)) + R(r 1(y)) + c(x, y) Homogeneous formulation by scaling invariance c(x, y) dγ(x, y) ) dγ(x, y) HK 2 (µ 0, µ 1) = min H(r γ 0(x), r 1(y), c(x, y)) dγ(x, y) where ( ) H(r 0, r 1, c) := inf ϑ R(r 0/ϑ) + R(r 1/ϑ) + c ϑ>0 A simple calculation yields H(r 2 0, r 2 1, c) = r r 2 1 2r 0r 1e c/2 = r r 2 1 2r 0r 1 cos( x y π/2 ).

41 22 Reverse and homogeneous formulation Reverse entropies R i(r) = rf i(1/r), R(r) := r 1 log r = rle(1/r). F (γ µ) = R(µ γ) Reverse formulation HK 2 (µ 0, µ 1) = min γ R(µ 0 γ 0) + R(µ 1 γ 1) + = min γ ( R(r 0(x)) + R(r 1(y)) + c(x, y) Homogeneous formulation by scaling invariance c(x, y) dγ(x, y) ) dγ(x, y) HK 2 (µ 0, µ 1) = min H(r γ 0(x), r 1(y), c(x, y)) dγ(x, y) where ( ) H(r 0, r 1, c) := inf ϑ R(r 0/ϑ) + R(r 1/ϑ) + c ϑ>0 A simple calculation yields H(r 2 0, r 2 1, c) = r r 2 1 2r 0r 1e c/2 = r r 2 1 2r 0r 1 cos( x y π/2 ).

42 Homgeneous marginals and Kantorovich-Wasserstein distance on C Any measure µ M(R d ) can be easily lifted to a measure in the cone C by α := µ δ 1. Conversely, any measure α in the cone C can be projected on R d by taking the homogeneous marginal: µ = hα = π x ( r2 α ), ξ(x) dµ(x) = ξ(x)r 2 dα(x, r). If α M(C C) with marginals α i, we set h i α = hα i = h(π i α). C HK and W dc via homogeneous marginals { } HK 2 (µ 0, µ 1) = min d 2 C(z 0, z 1) dα(z 0, z 1) : h i α = µ i α M(C C) } = min {W 2 dc (α 0, α 1) : α i M(C), hα i = µ i

43 Homgeneous marginals and Kantorovich-Wasserstein distance on C Any measure µ M(R d ) can be easily lifted to a measure in the cone C by α := µ δ 1. Conversely, any measure α in the cone C can be projected on R d by taking the homogeneous marginal: µ = hα = π x ( r2 α ), ξ(x) dµ(x) = ξ(x)r 2 dα(x, r). If α M(C C) with marginals α i, we set h i α = hα i = h(π i α). C HK and W dc via homogeneous marginals { } HK 2 (µ 0, µ 1) = min d 2 C(z 0, z 1) dα(z 0, z 1) : h i α = µ i α M(C C) } = min {W 2 dc (α 0, α 1) : α i M(C), hα i = µ i

44 A list of useful properties HK can be defined starting from any metric space (X, d). (M(X), HK) is a complete and separable metric space if X is complete and separable; the induced topology coincides with the topology of weak convergence (no bounds on moments are required). If X is compact then bounded sets in M(X) are relatively compact. HK(µ 0, µ 1) Hell(µ 0, µ 1) W(µ 0, µ 1). HK nd Hell, nhk d/n W. If (X, d) is length (resp. geodesic) then (M(X), HK) is length (resp. geodesic). If X = R d and µ 0 L d then there exists a unique geodesic connecting µ 0 to µ 1 and a unique optimal plan γ minimizing LET(µ 0, µ 1). (M(X), HK) has nonnegative curvature (in the sense of Aleksandrov) if and only if X has locally curvature 1. The Heat flow in any Riemannian manifold with nonnegative Ricci curvature (or, more generally, in any RCD(0, ) space) is contracting. Same result for the Fokker-Planck equation in R d generated by a convex potential.

45 A list of useful properties HK can be defined starting from any metric space (X, d). (M(X), HK) is a complete and separable metric space if X is complete and separable; the induced topology coincides with the topology of weak convergence (no bounds on moments are required). If X is compact then bounded sets in M(X) are relatively compact. HK(µ 0, µ 1) Hell(µ 0, µ 1) W(µ 0, µ 1). HK nd Hell, nhk d/n W. If (X, d) is length (resp. geodesic) then (M(X), HK) is length (resp. geodesic). If X = R d and µ 0 L d then there exists a unique geodesic connecting µ 0 to µ 1 and a unique optimal plan γ minimizing LET(µ 0, µ 1). (M(X), HK) has nonnegative curvature (in the sense of Aleksandrov) if and only if X has locally curvature 1. The Heat flow in any Riemannian manifold with nonnegative Ricci curvature (or, more generally, in any RCD(0, ) space) is contracting. Same result for the Fokker-Planck equation in R d generated by a convex potential.

46 A list of useful properties HK can be defined starting from any metric space (X, d). (M(X), HK) is a complete and separable metric space if X is complete and separable; the induced topology coincides with the topology of weak convergence (no bounds on moments are required). If X is compact then bounded sets in M(X) are relatively compact. HK(µ 0, µ 1) Hell(µ 0, µ 1) W(µ 0, µ 1). HK nd Hell, nhk d/n W. If (X, d) is length (resp. geodesic) then (M(X), HK) is length (resp. geodesic). If X = R d and µ 0 L d then there exists a unique geodesic connecting µ 0 to µ 1 and a unique optimal plan γ minimizing LET(µ 0, µ 1). (M(X), HK) has nonnegative curvature (in the sense of Aleksandrov) if and only if X has locally curvature 1. The Heat flow in any Riemannian manifold with nonnegative Ricci curvature (or, more generally, in any RCD(0, ) space) is contracting. Same result for the Fokker-Planck equation in R d generated by a convex potential.

47 A list of useful properties HK can be defined starting from any metric space (X, d). (M(X), HK) is a complete and separable metric space if X is complete and separable; the induced topology coincides with the topology of weak convergence (no bounds on moments are required). If X is compact then bounded sets in M(X) are relatively compact. HK(µ 0, µ 1) Hell(µ 0, µ 1) W(µ 0, µ 1). HK nd Hell, nhk d/n W. If (X, d) is length (resp. geodesic) then (M(X), HK) is length (resp. geodesic). If X = R d and µ 0 L d then there exists a unique geodesic connecting µ 0 to µ 1 and a unique optimal plan γ minimizing LET(µ 0, µ 1). (M(X), HK) has nonnegative curvature (in the sense of Aleksandrov) if and only if X has locally curvature 1. The Heat flow in any Riemannian manifold with nonnegative Ricci curvature (or, more generally, in any RCD(0, ) space) is contracting. Same result for the Fokker-Planck equation in R d generated by a convex potential.

48 A list of useful properties HK can be defined starting from any metric space (X, d). (M(X), HK) is a complete and separable metric space if X is complete and separable; the induced topology coincides with the topology of weak convergence (no bounds on moments are required). If X is compact then bounded sets in M(X) are relatively compact. HK(µ 0, µ 1) Hell(µ 0, µ 1) W(µ 0, µ 1). HK nd Hell, nhk d/n W. If (X, d) is length (resp. geodesic) then (M(X), HK) is length (resp. geodesic). If X = R d and µ 0 L d then there exists a unique geodesic connecting µ 0 to µ 1 and a unique optimal plan γ minimizing LET(µ 0, µ 1). (M(X), HK) has nonnegative curvature (in the sense of Aleksandrov) if and only if X has locally curvature 1. The Heat flow in any Riemannian manifold with nonnegative Ricci curvature (or, more generally, in any RCD(0, ) space) is contracting. Same result for the Fokker-Planck equation in R d generated by a convex potential.

49 24 A list of useful properties HK can be defined starting from any metric space (X, d). (M(X), HK) is a complete and separable metric space if X is complete and separable; the induced topology coincides with the topology of weak convergence (no bounds on moments are required). If X is compact then bounded sets in M(X) are relatively compact. HK(µ 0, µ 1) Hell(µ 0, µ 1) W(µ 0, µ 1). HK nd Hell, nhk d/n W. If (X, d) is length (resp. geodesic) then (M(X), HK) is length (resp. geodesic). If X = R d and µ 0 L d then there exists a unique geodesic connecting µ 0 to µ 1 and a unique optimal plan γ minimizing LET(µ 0, µ 1). (M(X), HK) has nonnegative curvature (in the sense of Aleksandrov) if and only if X has locally curvature 1. The Heat flow in any Riemannian manifold with nonnegative Ricci curvature (or, more generally, in any RCD(0, ) space) is contracting. Same result for the Fokker-Planck equation in R d generated by a convex potential.

50 24 A list of useful properties HK can be defined starting from any metric space (X, d). (M(X), HK) is a complete and separable metric space if X is complete and separable; the induced topology coincides with the topology of weak convergence (no bounds on moments are required). If X is compact then bounded sets in M(X) are relatively compact. HK(µ 0, µ 1) Hell(µ 0, µ 1) W(µ 0, µ 1). HK nd Hell, nhk d/n W. If (X, d) is length (resp. geodesic) then (M(X), HK) is length (resp. geodesic). If X = R d and µ 0 L d then there exists a unique geodesic connecting µ 0 to µ 1 and a unique optimal plan γ minimizing LET(µ 0, µ 1). (M(X), HK) has nonnegative curvature (in the sense of Aleksandrov) if and only if X has locally curvature 1. The Heat flow in any Riemannian manifold with nonnegative Ricci curvature (or, more generally, in any RCD(0, ) space) is contracting. Same result for the Fokker-Planck equation in R d generated by a convex potential.

51 Open problems and future directions Geodesic convexity of entropy functionals and applications to Reaction-diffusion equations Find more explicit representations, e.g. in R or for finite combinations of Dirac masses Study optimal Entropy-Transport problems for different entropies: they provide interesting Transport versions of well known distances in statistics and information theory, as the Jensen-Shannon divergence, the triangular discrimination and the total variation distance. Apply these techniques to other dynamic distances (cf. Dolbeault-Nazaret-S., Mielke, Maas,...)