Entropic Regularization of Wasserstein Barycenters

Talks given at the "Optimal Transport Workshop" in Toulouse, updated for a talk at "Workshop on Computational Information Geometry for Image and Signal Processing" in Edinburg.

Transcript

1. Numerical Optimal Transport and Applications Gabriel Peyré www.numerical-tours.com Joint works with: Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, Justin Solomon
2. Imaging: Statistical Image Models Source image (X) Style image (Y) Source image after color transfer J. Rabin Wasserstein Regularization Colors distribution: each pixel point in R3
3. Imaging: Statistical Image Models Input image Modified image Source image (X) Style image (Y) Source image after color transfer J. Rabin Wasserstein Regularization Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Sliced Wasserstein projection of X to style image color statistics Y Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Style image (Y) Sliced Wasserstein projection of X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization Colors distribution: each pixel point in R3
4. Other applications: Imaging: Statistical Image Models Input image Modified image Source image (X) Style image (Y) Source image after color transfer J. Rabin Wasserstein Regularization Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Sliced Wasserstein projection of X to style image color statistics Y Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Style image (Y) Sliced Wasserstein projection of X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization Colors distribution: each pixel point in R3 Texture synthesis, segmentation, . . . Classification, clustering . . . Surface processing, reflectance modeling . . .
5. Machine Learning: Bag of Features Image descriptors: Gradient distribution Histogram of features
6. Machine Learning: Bag of Features Image descriptors: Text descriptors: Gradient distribution Histogram of features
7. Overview • Optimal Transport • Regularized Transport • Wasserstein Barycenters • Heat Kernel Approximation
8. Optimal Transport Discrete densities: Histograms: µ = P i p i xi xi 2 Rd ⌃N def. = p 2 RN + ; P i pi = 1
9. Optimal Transport Discrete densities: Histograms: Couplings: µ = P i p i xi xi 2 Rd q p ⇡ C(p, q) def. = ⇡ 2 (R +)N⇥N ; ⇡1 = p, ⇡T 1 = q ⌃N def. = p 2 RN + ; P i pi = 1
10. Optimal Transport Discrete densities: Histograms: Couplings: µ = P i p i xi xi 2 Rd Ground cost: c 2 (R +)N⇥N . p-Wasserstein transport: ci,j = || xi xj ||p q p ⇡ C(p, q) def. = ⇡ 2 (R +)N⇥N ; ⇡1 = p, ⇡T 1 = q ⌃N def. = p 2 RN + ; P i pi = 1
11. Optimal Transport Discrete densities: Histograms: Couplings: µ = P i p i xi xi 2 Rd Ground cost: c 2 (R +)N⇥N . p-Wasserstein transport: ci,j = || xi xj ||p q p ⇡ C(p, q) def. = ⇡ 2 (R +)N⇥N ; ⇡1 = p, ⇡T 1 = q W(p, q) def. = min nP i,j ⇡i,jci,j ; ⇡ 2 C(p, q)o Optimal transport distance: ⌃N def. = p 2 RN + ; P i pi = 1
12. Algorithms Arbitrary discrete measures: ⇡? 2 argmin ⇡2C(p,q) hc, ⇡i µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj
13. Algorithms Arbitrary discrete measures: ⇡? 2 argmin ⇡2C(p,q) hc, ⇡i µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj ! Linear program interior points (polynomial) transportation simplex
14. Algorithms Arbitrary discrete measures: ⇡? 2 argmin ⇡2C(p,q) hc, ⇡i Point clouds: N = P, pi = qj = 1/N. W(p, q) = min 2 PermN P i ci, (i) ! Hungarian/auction algorithms, complexity O(N3). µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj ! Linear program interior points (polynomial) transportation simplex µ ⌫
15. Algorithms Arbitrary discrete measures: ⇡? 2 argmin ⇡2C(p,q) hc, ⇡i Point clouds: N = P, pi = qj = 1/N. W(p, q) = min 2 PermN P i ci, (i) 1-D and convex cost: ci,j = | xi xj |p , p > 1. ! Hungarian/auction algorithms, complexity O(N3). µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj ! Linear program interior points (polynomial) transportation simplex µ ⌫ sorting the values, O(N log(N)) operations. µ ⌫
16. Algorithms Arbitrary discrete measures: ⇡? 2 argmin ⇡2C(p,q) hc, ⇡i Point clouds: N = P, pi = qj = 1/N. W(p, q) = min 2 PermN P i ci, (i) 1-D and convex cost: ci,j = | xi xj |p , p > 1. ! Hungarian/auction algorithms, complexity O(N3). µ = P N i =1 p i xi , ⌫ = P P j =1 q j yj ! Linear program interior points (polynomial) transportation simplex µ ⌫ sorting the values, O(N log(N)) operations. µ ⌫ Need for fast approximate algorithms for generic c .
17. Overview • Optimal Transport • Regularized Transport • Wasserstein Barycenters • Heat Kernel Approximation • Wasserstein Gradient Flows
18. Entropy Regularized Transport (minus) Entropy: E ( ⇡ ) def. = X i,j ⇡i,j(log( ⇡i,j) 1) + ◆R+ ( ⇡i,j)
19. Entropy Regularized Transport (minus) Entropy: Regularized distance: E ( ⇡ ) def. = X i,j ⇡i,j(log( ⇡i,j) 1) + ◆R+ ( ⇡i,j) W (p, q) def. = min {h⇡, ci + E(⇡) ; ⇡ 2 C(p, q)} ⇡ def. = argmin {h⇡, ci + E(⇡) ; ⇡ 2 C(p, q)} [Schrodinger 1931] Used in economy [Galichon Salani´ e 2008] and machine learning [Cuturi 2013]
20. Entropy Regularized Transport (minus) Entropy: Regularized distance: ⇡ c E ( ⇡ ) def. = X i,j ⇡i,j(log( ⇡i,j) 1) + ◆R+ ( ⇡i,j) W (p, q) def. = min {h⇡, ci + E(⇡) ; ⇡ 2 C(p, q)} ⇡ def. = argmin {h⇡, ci + E(⇡) ; ⇡ 2 C(p, q)} [Schrodinger 1931] Used in economy [Galichon Salani´ e 2008] and machine learning [Cuturi 2013]
21. The Impact of Regularization Proposition: ⇡ !0 ! argmin ⇡2S E(⇡) W (p, q) !0 ! W(p, q) S def. = argmin {h⇡, ci ; ⇡ 2 C(p, q)}
22. The Impact of Regularization Proposition: ⇡ !+1 ! pqT ⇡ !0 ! argmin ⇡2S E(⇡) W (p, q) !0 ! W(p, q) 1 W (p, q) !+1 ! E(p) + E(q) S def. = argmin {h⇡, ci ; ⇡ 2 C(p, q)}
23. The Impact of Regularization Proposition: ⇡ !+1 ! pqT ⇡ !0 ! argmin ⇡2S E(⇡) W (p, q) !0 ! W(p, q) 1 W (p, q) !+1 ! E(p) + E(q) S def. = argmin {h⇡, ci ; ⇡ 2 C(p, q)} EMD Entrop ⇡ p q
24. Kullback-Leibler Projections KL( ⇡|⇠ ) def. = P i,j ⇡i,j log ⇣ ⇡i,j ⇠i,j ⌘ + ⇠i,j ⇡i,j KL divergence:
25. Kullback-Leibler Projections KL( ⇡|⇠ ) def. = P i,j ⇡i,j log ⇣ ⇡i,j ⇠i,j ⌘ + ⇠i,j ⇡i,j KL divergence: where ⇠ = e c One has: h⇡, ci + E(⇡) = KL(⇡|⇠) + C
26. Kullback-Leibler Projections W (p, q) = min {KL(⇡|⇠) ; ⇡ 2 C(p, q)} ⇡ = ProjC(p,q)( ⇠ ) def. = argmin { KL( ⇡|⇠ ) ; ⇡ 2 C ( p, q ) } Proposition: KL( ⇡|⇠ ) def. = P i,j ⇡i,j log ⇣ ⇡i,j ⇠i,j ⌘ + ⇠i,j ⇡i,j KL divergence: where ⇠ = e c One has: h⇡, ci + E(⇡) = KL(⇡|⇠) + C
27. Kullback-Leibler Projections W (p, q) = min {KL(⇡|⇠) ; ⇡ 2 C(p, q)} Constraint splitting: q p ⇡ C(p, q) = C1 \ C2 ⇢ C1 = ⇡ 2 (R +)N⇥N ; ⇡1 = p , C2 = ⇡ 2 (R +)N⇥N ; ⇡T 1 = q . ⇡ = ProjC(p,q)( ⇠ ) def. = argmin { KL( ⇡|⇠ ) ; ⇡ 2 C ( p, q ) } Proposition: KL( ⇡|⇠ ) def. = P i,j ⇡i,j log ⇣ ⇡i,j ⇠i,j ⌘ + ⇠i,j ⇡i,j KL divergence: where ⇠ = e c One has: h⇡, ci + E(⇡) = KL(⇡|⇠) + C
28. Sinkhorn / IPFP Algorithm Iterative Bregman projections: ⇡(0) = ⇠ ⇠ ⇡(1) ⇡(2) ⇡(3) ⇡(4) ⇡(5) ⇡ ⇡(`+1) = ProjC`%K ( ⇡(`) ) [Bregman 1957]
29. Sinkhorn / IPFP Algorithm Iterative Bregman projections: ⇡(0) = ⇠ ⇠ ⇡(1) ⇡(2) ⇡(3) ⇡(4) ⇡(5) ⇡ ⇡(`+1) = ProjC`%K ( ⇡(`) ) Theorem: ⇡(`) ! ProjC1 \...\CK ( ⇠ ) [Bregman 1957] If {Ci }i are a ne sets,
30. Sinkhorn / IPFP Algorithm Iterative Bregman projections: ⇡(0) = ⇠ ⇠ ⇡(1) ⇡(2) ⇡(3) ⇡(4) ⇡(5) ⇡ ⇡(`+1) = ProjC`%K ( ⇡(`) ) Theorem: ⇡(`) ! ProjC1 \...\CK ( ⇠ ) Fixed marginals: Proposition: ProjC1 ( ⇡ ) = diag ⇣ p ⇡1 ⌘ ⇡ ProjC2 ( ⇡ ) = ⇡ diag ⇣ q ⇡T 1 ⌘ ( C1 def. = {⇡ ; ⇡1 = p} , C2 def. = ⇡ ; ⇡T 1 = q . [Bregman 1957] If {Ci }i are a ne sets,
31. Diagonal Scaling, Fast Implementation Sinkhorn algorithm: ⇡(0) = ⇠ [Sinkhorn 1967] [Deming,Stephan 1940] ⇡(2`+1) = diag(p/⇡(2`)1)⇡(2`) ⇡(2`+2) = ⇡(2`+1) diag(q/⇡(2`+1),T 1)
32. Diagonal Scaling, Fast Implementation Sinkhorn algorithm: ⇡(0) = ⇠ [Sinkhorn 1967] [Deming,Stephan 1940] Proposition: ⇡ = diag(u )⇠ diag(v ) where ⇠ = e c . ⇡(2`+1) = diag(p/⇡(2`)1)⇡(2`) ⇡(2`+2) = ⇡(2`+1) diag(q/⇡(2`+1),T 1)
33. Diagonal Scaling, Fast Implementation Sinkhorn algorithm: ⇡(0) = ⇠ [Sinkhorn 1967] [Deming,Stephan 1940] Proposition: ⇡ = diag(u )⇠ diag(v ) where ⇠ = e c . ⇡(`) = diag(u(`))⇠ diag(v(`)) ⇡(2`+1) = diag(p/⇡(2`)1)⇡(2`) ⇡(2`+2) = ⇡(2`+1) diag(q/⇡(2`+1),T 1)
34. Diagonal Scaling, Fast Implementation Sinkhorn algorithm: ⇡(0) = ⇠ [Sinkhorn 1967] [Deming,Stephan 1940] v(0) = 1 Sinkhorn, revisited: u(`) = p ⇠v(`) v(`+1) = q ⇠T u(`) Proposition: ⇡ = diag(u )⇠ diag(v ) where ⇠ = e c . ⇡(`) = diag(u(`))⇠ diag(v(`)) ⇡(2`+1) = diag(p/⇡(2`)1)⇡(2`) ⇡(2`+2) = ⇡(2`+1) diag(q/⇡(2`+1),T 1)
35. Diagonal Scaling, Fast Implementation Sinkhorn algorithm: ! Only matrix-vector multiplications. ⇡(0) = ⇠ [Sinkhorn 1967] [Deming,Stephan 1940] v(0) = 1 Sinkhorn, revisited: u(`) = p ⇠v(`) v(`+1) = q ⇠T u(`) Proposition: ⇡ = diag(u )⇠ diag(v ) where ⇠ = e c . ⇡(`) = diag(u(`))⇠ diag(v(`)) ⇡(2`+1) = diag(p/⇡(2`)1)⇡(2`) ⇡(2`+2) = ⇡(2`+1) diag(q/⇡(2`+1),T 1)
36. Diagonal Scaling, Fast Implementation Sinkhorn algorithm: ! Only matrix-vector multiplications. ! Highly parallelizable. ⇡(0) = ⇠ [Sinkhorn 1967] [Deming,Stephan 1940] v(0) = 1 Sinkhorn, revisited: u(`) = p ⇠v(`) v(`+1) = q ⇠T u(`) Proposition: ⇡ = diag(u )⇠ diag(v ) where ⇠ = e c . ⇡(`) = diag(u(`))⇠ diag(v(`)) ⇡(2`+1) = diag(p/⇡(2`)1)⇡(2`) ⇡(2`+2) = ⇡(2`+1) diag(q/⇡(2`+1),T 1)
37. Translation-invariant Ground Metrics Assuming ci,j = 'i j on a discrete grid (e.g. periodic b.c.). ⇠v =  ? v where  def. = e '/
38. Translation-invariant Ground Metrics Assuming ci,j = 'i j on a discrete grid (e.g. periodic b.c.). Example: ci,j = || xi xj ||2,  = Gaussian filter. ⇠v =  ? v where  def. = e '/
39. Translation-invariant Ground Metrics Assuming ci,j = 'i j on a discrete grid (e.g. periodic b.c.). Example: ci,j = || xi xj ||2,  = Gaussian filter. v(`+1) = q ⇣  ? ⇣ p  ? v(`) 1 ⌘⌘ 1 Convolutive Sinkhorn: ⇠v =  ? v where  def. = e '/ a b def. = ( aibi)i, ? def. = convolution ! ⇠v computed in O ( N log( N )) operations (FFT, IIR approximation)
40. Translation-invariant Ground Metrics Assuming ci,j = 'i j on a discrete grid (e.g. periodic b.c.). Example: ci,j = || xi xj ||2,  = Gaussian filter. v(`+1) = q ⇣  ? ⇣ p  ? v(`) 1 ⌘⌘ 1 Convolutive Sinkhorn: ⇠v =  ? v where  def. = e '/ a b def. = ( aibi)i, ? def. = convolution p q ` ⇡(`) ! ⇠v computed in O ( N log( N )) operations (FFT, IIR approximation)
41. Overview • Optimal Transport • Regularized Transport • Wasserstein Barycenters • Heat Kernel Approximation
42. Wasserstein Barycenters For µ = P i p i xi , ⌫ = P j q j yj , W2(µ, ⌫) = W(p, q) for ci,j = || xi yj ||2 W2 def. = Wasserstein distance for measures.
43. Wasserstein Barycenters µ µ1 µ3 W2 (µ1 , µ ) W 2 (µ 2 ,µ ) W2 (µ3 ,µ ) µ2 µ? 2 argmin µ P k k W2(µk, µ) Barycenters of measures ( µk)k: P k k = 1 For µ = P i p i xi , ⌫ = P j q j yj , W2(µ, ⌫) = W(p, q) for ci,j = || xi yj ||2 W2 def. = Wasserstein distance for measures.
44. Wasserstein Barycenters µ µ1 µ3 W2 (µ1 , µ ) W 2 (µ 2 ,µ ) W2 (µ3 ,µ ) µ2 µ? 2 argmin µ P k k W2(µk, µ) Barycenters of measures ( µk)k: P k k = 1 If µ k = xk then µ? = P k kxk Generalizes Euclidean barycenter: For µ = P i p i xi , ⌫ = P j q j yj , W2(µ, ⌫) = W(p, q) for ci,j = || xi yj ||2 W2 def. = Wasserstein distance for measures.
45. µ exists and is unique. Theorem: if µ1 does not vanish on small sets, Wasserstein Barycenters [Agueh, Carlier, 2010] µ µ1 µ3 W2 (µ1 , µ ) W 2 (µ 2 ,µ ) W2 (µ3 ,µ ) µ2 µ? 2 argmin µ P k k W2(µk, µ) Barycenters of measures ( µk)k: P k k = 1 If µ k = xk then µ? = P k kxk Generalizes Euclidean barycenter: For µ = P i p i xi , ⌫ = P j q j yj , W2(µ, ⌫) = W(p, q) for ci,j = || xi yj ||2 W2 def. = Wasserstein distance for measures.
46. Entropic Wasserstein Barycenters Barycenter: min p2⌃N P k kW (pk, p) p p1 p2 p3
47. Entropic Wasserstein Barycenters In term of couplings: 8 k, p = ⇡k 1 where min { P k kKL(⇡k |⇠) ; (⇡k)k 2 C1 \ C2 } ⇠ = e c Barycenter: min p2⌃N P k kW (pk, p) p p1 p2 p3 C1 def. = (⇡k)k ; 8 k, ⇡T k 1 = pk C2 def. = {(⇡k)k ; 9p, 8 k, ⇡k 1 = p}
48. Entropic Wasserstein Barycenters In term of couplings: Proposition: p = Q k (⇡k 1) k 8 k, p = ⇡k 1 where min { P k kKL(⇡k |⇠) ; (⇡k)k 2 C1 \ C2 } ⇠ = e c Barycenter: min p2⌃N P k kW (pk, p) p p1 p2 p3 C1 def. = (⇡k)k ; 8 k, ⇡T k 1 = pk C2 def. = {(⇡k)k ; 9p, 8 k, ⇡k 1 = p} ProjC1 ( ⇡k)k = ✓ ⇡k diag ✓ pk ⇡T k 1 ◆◆ k ProjC2 ( ⇡k)k = ✓ diag ✓ p ⇡k 1 ◆ ⇡k ◆ k
49. Entropic Wasserstein Barycenters In term of couplings: Proposition: p = Q k (⇡k 1) k 8 k, p = ⇡k 1 where min { P k kKL(⇡k |⇠) ; (⇡k)k 2 C1 \ C2 } Sinkhorn-like algorithm: ⇠ = e c ( ⇡(2`+1) k )k = ProjC1 ( ⇡(2`) k )k ( ⇡(2`+2) k )k = ProjC2 ( ⇡(2`+1) k )k 8 k, ⇡(0) k = ⇠ Barycenter: min p2⌃N P k kW (pk, p) p p1 p2 p3 C1 def. = (⇡k)k ; 8 k, ⇡T k 1 = pk C2 def. = {(⇡k)k ; 9p, 8 k, ⇡k 1 = p} ProjC1 ( ⇡k)k = ✓ ⇡k diag ✓ pk ⇡T k 1 ◆◆ k ProjC2 ( ⇡k)k = ✓ diag ✓ p ⇡k 1 ◆ ⇡k ◆ k
50. Barycenter of 3 Shapes p1 p2 p3
51. Barycenter of 3-D Volumes
52. Barycenter of 3-D Volumes
53. Color Transfer µ ⌫ Input images: ( f, g ) (chrominance components) Input measures: f g µ(A) = U(f 1(A)), ⌫(A) = U(g 1(A))
54. Color Transfer µ ⌫ Input images: ( f, g ) (chrominance components) Input measures: f g µ(A) = U(f 1(A)), ⌫(A) = U(g 1(A))
55. Color Transfer µ ⌫ Input images: ( f, g ) (chrominance components) Input measures: f T T f g ˜ T g µ(A) = U(f 1(A)), ⌫(A) = U(g 1(A))
56. Raw image sequence Color Harmonization . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing
57. Raw image sequence Compute Wasserstein barycenter Project on the barycenter Color Harmonization . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; Raw image sequence J. Rabin – GREYC, University of Caen Approximate Wasserstein Metric for Texture Synthesis and Mixing erstein Barycenter Sliced Wasserstein Barycenter Experimental results Applications Conclusion olor transfer Color harmonization of several images . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; stein Barycenter Sliced Wasserstein Barycenter Experimental results Applications Conclusion lor transfer Color harmonization of several images . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter; in Barycenter Sliced Wasserstein Barycenter Experimental results Applications Conclusion or transfer Color harmonization of several images . Step 1: compute Sliced-Wasserstein Barycenter of color statistics; . Step 2: compute Sliced-Wasserstein projection of each image onto the Barycenter;
58. Overview • Optimal Transport • Regularized Transport • Wasserstein Barycenters • Heat Kernel Approximation • Wasserstein Gradient Flows
59. Optimal Transport on Surfaces Triangulated mesh: M. Geodesic distance: dM.
60. Optimal Transport on Surfaces Ground cost: ci,j = dM(xi, xj) 2 . Triangulated mesh: M. Geodesic distance: dM. Level sets xi d ( xi, ·)
61. Optimal Transport on Surfaces Ground cost: ci,j = dM(xi, xj) 2 . Triangulated mesh: M. Geodesic distance: dM. Level sets xi d ( xi, ·) Computing c (Fast-Marching): N2 log( N ) ! too costly.
62. Entropic Transport on Surfaces Heat equation on M: @ u ( x, ·) = Mu ( x, ·) , u0( x, ·) = x
63. Entropic Transport on Surfaces Heat equation on M: Sinkhorn kernel: Theorem: [Varadhan] log( u ) !0 ! d2 M @ u ( x, ·) = Mu ( x, ·) , u0( x, ·) = x ⇠ = e d2 M ⇡ ut ⇡ Id L 1 M L
64. Barycenter on a Surface 1 p0 p1
65. Barycenter on a Surface 1 p0 p1 p0 p2 p3 p4 p6 p1 = (1, . . . , 1)/6
66. MRI Data Procesing [with A. Gramfort] ariational Wasserstein Problems Labels L2 barycenter W barycenter Ground cost ci,j = dM(xi, xj): geodesic on cortical surface.
67. Conclusion Source image (X) Style image (Y) Source ima J. Rabin Wasserstein Regu Histogram features in imaging and machine learning. ! histograms are now trendy!
68. Conclusion EMD Entropy Discrete analog: Cuturi, NIPS 2013 Entropic regularization for optimal transport. Source image (X) Style image (Y) Source ima J. Rabin Wasserstein Regu Histogram features in imaging and machine learning. ! histograms are now trendy!
69. Conclusion EMD Entropy Discrete analog: Cuturi, NIPS 2013 Entropic regularization for optimal transport. Source image (X) Style image (Y) Source ima J. Rabin Wasserstein Regu Histogram features in imaging and machine learning. ! histograms are now trendy! Barycenters in Wasserstein space: Figure 4: Simulation results with focal random signals generated in are