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Computational OT #1 - Monge and Kantorovitch

Computational OT #1 - Monge and Kantorovitch

A course on computational optimal transport

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Gabriel Peyré

April 09, 2025
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  1. Theoretical Foundations Computational Optimal Transport Gabriel Peyré É C O

    L E N O R M A L E S U P É R I E U R E www.numerical-tours.com http://optimaltransport.github.io
  2. " Semi-discrete Optimal Transport ¸ = 3 ¸ = 50

    ¸ = 100 Matching Iterations of the semi-discrete OT algorithm minimizing (5.8) (here dient descent is used). The support (yj ) j of the discrete measure — is the red points, while the continuous measure – is the uniform measure The blue cells display the Laguerre partition (Lg(¸) (yj )) j where g ( ¸ ) is dual potential computed at iteration ¸. wton solver which is applied to sampling in computer graph- sed in [De Goes et al., 2012], see also [Lévy, 2015] for appli- -D volume and surface processing. An important area of ap- semi-discrete method is for the resolution of incompressible mic (Euler’s equations) using Lagrangian methods de Goes ], Gallouët and Mérigot [2017]. The semi-discrete OT solver ompressibility at each iteration by imposing that the (possi- d) points cloud approximates a uniform inside the domain. rgence (with linear rate) of damped Newton iterations is f (a, b) and s igure 6.1: Example of computation of W1(a, b) on a planar graph with uniform weights wi,j = 1. eft: potential f solution of (6.5) (increasing value from red to blue). The green color of the edges is oportional to |(Òf) i,j |. Right: flow s solution of (6.6), where bold black edges display non-zero si,j , hich saturate to wi,j = 1. These saturating flow edge on the right match the light green edge on the ft where |(Òf) i,j | = 1. 1. Algorithmic Foundations 2. Entropic Regularization 3. (Gradient) Flows in ML (to be or not to be a gradient field)
  3. www.numerical-tours.com Unbalanced Optimal Transport Optimal Transport with Linear Programming Semi-discrete

    Optimal Transport Entropic Regularization of Optimal Transport Advanced Topics on Sinkhorn Algorithm
  4. Comparing Distributions for Learning <latexit sha1_base64="vT0xGll5ZYO8gmdwavYy0Ckm65g=">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</latexit> ATAC-seq

  5. Parametric model: ✓ 7! ↵✓ <latexit 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<latexit 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<latexit

    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<latexit 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<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Comparing Distributions for Learning <latexit 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<latexit 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<latexit 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↵✓ <latexit 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ATAC-seq
  6. min ✓ D(↵✓, ) <latexit 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<latexit 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<latexit 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Density fitting: <latexit 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<latexit 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<latexit 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<latexit 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Parametric model: ✓ 7! ↵✓ <latexit 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<latexit 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<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Comparing Distributions for Learning <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit 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ATAC-seq
  7. Monge optimal matching: <latexit 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<latexit 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<latexit 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<latexit

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[Monge 1784] <latexit 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Monge’s Problem Points (xi)i, (yj)j Permutation: <latexit 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<latexit 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<latexit 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: {1, . . . , n} ! {1, . . . , n} <latexit 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<latexit 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<latexit 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D(X, Y ) = min n X i=1 d(xi, y (i) )
  8. Monge optimal matching: <latexit 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<latexit 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<latexit 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<latexit

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[Monge 1784] <latexit 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Monge’s Problem Points (xi)i, (yj)j Permutation: <latexit 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<latexit 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<latexit 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: {1, . . . , n} ! {1, . . . , n} <latexit 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<latexit 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D(X, Y ) = min n X i=1 d(xi, y (i) ) ! Seems intractable: n! possibilities. <latexit 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<latexit 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<latexit 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<latexit sha1_base64="xAfoQVY7WeuWfvbTW5Z5P5YLXic=">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</latexit> ! Di↵erent number of points?
  9. 1-D Optimal Transport x1 x2 x3 x4 x5 y1 y2

    y3 y4 y5 min 2⌃n n X i=1 |xi y (i) |p, p > 1 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  10. 1-D Optimal Transport xxx x1 x2 x3 x4 x5 y1

    y2 y3 y4 y5 min 2⌃n n X i=1 |xi y (i) |p, p > 1 <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="728Z8fBn49NzO1ft9oZkwnWJbMw=">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</latexit>
  11. 1-D Optimal Transport xxx x1 x2 x3 x4 x5 y1

    y2 y3 y4 y5 n n ! n(n-1)/2 n log(n) 10 3628800 45 23 11 39916800 55 26 12 479001600 66 30 25 1,551x1025 300 80 70 1,198x10100 21415 297 QuickSort: O(n log(n)). <latexit 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<latexit 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Sorting algorithms: insertion n(n 1)/2 worst case. <latexit 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<latexit 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<latexit 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min 2⌃n n X i=1 |xi y (i) |p, p > 1 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  12. 1-D OT Interpolation comparison <latexit 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<latexit 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<latexit 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Grades 1 <latexit 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<latexit 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<latexit 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Grades 2 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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0 10 20 x1 x2 x3 x4 x5 y4 y1 y5 y3 y2 Grades 1 <latexit 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<latexit 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Grades 2 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Grades /20 <latexit 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  13. 1-D OT Interpolation comparison <latexit 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<latexit 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<latexit 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Grades 1 <latexit 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<latexit 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<latexit 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Grades 2 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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0 10 20 x1 x2 x3 x4 x5 y4 y1 y5 y3 y2 Grades 1 <latexit 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<latexit 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Grades 2 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Grades /20 <latexit 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  14. Grayscale Histogram Equalization f[argsort(f.flatten())] = np.sort(g.flatten()) <latexit sha1_base64="XvM/j+TRFdrUIRjd7RjpxM4w5ws=">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</latexit> f

  15. Grayscale Histogram Equalization f[argsort(f.flatten())] = np.sort(g.flatten()) <latexit sha1_base64="XvM/j+TRFdrUIRjd7RjpxM4w5ws=">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</latexit> f

  16. In 2-D xi xj n = 10 n = 70

    n = 300 xi xj xj0 xi0 xi xj xj0 xi0 ci,j = ||xi yj || = q (x1 i y1 j )2 + (x2 i y2 j )2 xi, yj 2 R2 Proposition: two segments never cross. <latexit 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<latexit 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<latexit 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  17. Source image (X) Style image (Y) Sliced Wasserstein projection of

    image color statistics Y Source image after color tra Optimal transport framework Sliced Wasserstein projection Applications Application to Color Transfer Source image (X) Sliced Wasserstein projection 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 optimal In 3-D: Color Image Palette Equalization Reference <latexit 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sha1_base64="3GKcotffcbQ7ZK6RRi5K3JhAuCs=">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</latexit> Output <latexit 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<latexit 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<latexit 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<latexit 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Input <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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transport <latexit 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  18. Probability Measures d↵(x) = ⇢↵(x)dx 2.2. Assignment and Monge Problem

    9 Discrete d = 1 Discrete d = 2 Density d = 1 Density d = 2 Figure 2.1: Schematic display of discrete distributions – = q n i =1 ai”xi (red cor- responds to empirical uniform distribution ai = 1/n, and blue to arbitrary distri- butions) and densities d–(x) = fl– (x)dx (in violet), in both 1-D and 2-D. Discrete distributions in 1-D are displayed using vertical segments (with length equal to ai ) and in 2-D using point clouds (radius equal to ai ). is the dimension), can have a density d–(x) = fl – (x)dx w.r.t. the Lebesgue measure, often denoted fl – = d – d x , which means that ⁄ ⁄ ↵ = P i ai xi Positive Radon measure ↵ on a metric space X. X = Rd <latexit 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  19. Probability Measures d↵(x) = ⇢↵(x)dx 2.2. Assignment and Monge Problem

    9 Discrete d = 1 Discrete d = 2 Density d = 1 Density d = 2 Figure 2.1: Schematic display of discrete distributions – = q n i =1 ai”xi (red cor- responds to empirical uniform distribution ai = 1/n, and blue to arbitrary distri- butions) and densities d–(x) = fl– (x)dx (in violet), in both 1-D and 2-D. Discrete distributions in 1-D are displayed using vertical segments (with length equal to ai ) and in 2-D using point clouds (radius equal to ai ). is the dimension), can have a density d–(x) = fl – (x)dx w.r.t. the Lebesgue measure, often denoted fl – = d – d x , which means that ⁄ ⁄ ↵ = P i ai xi Positive Radon measure ↵ on a metric space X. X = Rd <latexit 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<latexit 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Probability (normalized) measure: ↵(X) = R X d↵(x) = 1 Integration against continuous functions: Measure of sets A ⇢ X: ↵(A) = R A d↵(x) > 0 R X g(x)d↵(x) > 0 d↵(x) = ⇢↵(x)dx ↵ = P i ai xi R X gd↵ = P i aig(xi)
  20. Push Forward ↵ T <latexit 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<latexit 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<latexit 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<latexit 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Push-forward: <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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T]↵ <latexit 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Map: T : X ! Y <latexit 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<latexit 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<latexit 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P i xi 7 ! P i T (xi) <latexit 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<latexit 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P i ai xi 7 ! P i ai T (xi) <latexit 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<latexit 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x 7 ! T (x) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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8 < : <latexit 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<latexit 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<latexit 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<latexit 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  21. Push Forward ↵ T <latexit 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<latexit 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<latexit 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<latexit 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Push-forward: <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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T]↵ <latexit 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Map: T : X ! Y <latexit 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<latexit 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<latexit 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P i xi 7 ! P i T (xi) <latexit 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<latexit 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P i ai xi 7 ! P i ai T (xi) <latexit 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<latexit 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x 7 ! T (x) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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8 < : <latexit 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<latexit 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<latexit 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seful for applica- mplex instead of rmance increase. on further speeds of the final trans- recision, but that polation scheme thod to generate the performance D image retrieval rs [Rubner et al. . 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<latexit 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y = T(x) <latexit 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<latexit 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<latexit sha1_base64="SLqhf2ynx7RkQrk5I2Me8kQffP0=">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</latexit> (T]↵)(E) def. = ↵(T 1(E)) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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General case: <latexit 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<latexit 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<latexit 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  22. Push Forward ↵ T <latexit 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<latexit 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<latexit 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<latexit 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Push-forward: <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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T]↵ <latexit 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Map: T : X ! Y <latexit 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<latexit 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P i xi 7 ! P i T (xi) <latexit 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<latexit 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P i ai xi 7 ! P i ai T (xi) <latexit 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<latexit 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x 7 ! T (x) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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8 < : <latexit 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<latexit 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<latexit 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seful for applica- mplex instead of rmance increase. on further speeds of the final trans- recision, but that polation scheme thod to generate the performance D image retrieval rs [Rubner et al. . 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<latexit 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y = T(x) <latexit 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<latexit 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<latexit sha1_base64="SLqhf2ynx7RkQrk5I2Me8kQffP0=">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</latexit> (T]↵)(E) def. = ↵(T 1(E)) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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General case: <latexit 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<latexit 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Change of variables: <latexit 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= T]↵ <latexit 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</latexit> <latexit 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<latexit 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Densities d↵ dx = ⇢↵: <latexit 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<latexit 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sha1_base64="wKnxbOt7+IXe41dpkCS+2UP/V+A=">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</latexit> ⇢↵(x) = | det(@T(x))|⇢ (T(x)) <latexit 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<latexit 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R Y g(y)d (y) = R X g(T(x))d↵(x) <latexit 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</latexit>
  23. Random vectors Radon measures P(X 2 A) = P(Xn 2

    A) n!+1 ! P(X 2 A) 8 set A 8 continuous function f Weak⇤ convergence: Convergence in law: Measures and Random Variables R A d↵(x) R fd↵n n!+1 ! R fd↵ Weak convergence: . . . ↵1 ↵2 ↵3 ↵ . . . Push-forward: X ∼ α T(X) ∼ T♯ α T]↵ <latexit 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↵ T <latexit 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  24. Continuous Monge’s Problem ↵ Y <latexit 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<latexit sha1_base64="V/ijx18qSlgI3jCwJ6JJh1g2VwY=">AAA9inictVttcxu3EYbTl8Rq0zjpx850rlXcSTquRlI802Y8mon1YkmxYskmJdsJbQ1fTtTZJx7NIyXLjH5Jv7a/o7+j/6D91L/Q3QVwwJG4W0B1dSMJB+J5drEHLHaBY2eYJvl4efmfNz74yU9/9vMPP7q58ItffvyrT259+tlRnk1G3fiwm6XZ6FmnncdpMogPx8k4jZ8NR3H7rJPGTzuvN/Dzp+fxKE+yQXN8OYxfnLX7g+Qk6bbHUHV865MWcUw76SS+aj2/PL61uLy0TD/RfGFFFRaF+jnIPv3tqmiJnshEV0zEmYjFQIyhnIq2yOH6QayIZTGEuhdiCnUjKCX0eSyuxAJgJ9AqhhZtqH0Nf/tw94OqHcA9cuaE7oKUFH5HgIzEbcBk0G4EZZQW0ecTYsbaKu4pcaJul/C/o7jOoHYsTqGWw+mWvjjsy1iciL9QHxLo05BqsHddxTIhq6DmkdWrMTAMoQ7LPfh8BOUuIbWdI8Lk1He0bZs+/xe1xFq876q2E/Fv0vI2XJFoqN5nBUNbnBN/RE9zAp9JfVKQ3AeGWPURSxdk6zPq/QDaT6H+EVxXVNI26cA1pdqrWuQGXC7kBovchsuF3GaRe3C5kHss8gAuF/JAIRE7Ipu78Q24XPgGK/kxXC7kYxb5BC4X8gmLPILLhTxikd/D5UJ+zyIfwOVCPmCRD+FyIR+yyCZcLmSTRR7C5UIessgtuFzILYWsnqkjuDLiSZhZeR/KZRnoKVKouc/qt07e0YVd95jT3QosP6s34b8bu+lh07gCu+Ux7k4qsPzI2wYf6cbyvmiHVhMXdofF7sIIcGN3Wey34lUF9luPmfa6AsvPtT1o58by3vc7uHNjv2Oxj6DkxvJr1D7UuLH7HivGsAJ7wGIfizcVWB+vP6rA8n6/AX7FjeXXqSa0d2N9vOmkAsv70yOIYNxYfrV6CrVu7FMW+0y8rcA+Y7HPwbu7sc89Vth3FVi9xi7QCtKneCSGGVvH1i5mJZaGwNZm5KfF2pJSbNyBeg7TLzB9wpyxiO0Cse2J2CsQe9565YUfzSne5aU0CkTDE9Ep1iYsjdn2vaI9llIPxGaB2JxB1EWk+Kx1X84putA1HHJcrFxY8ulTVvhvLMVqPNR7Xo3YLyHk2D6lkX+HsiXMoNBSdWynxRovkRHd1yEuKHvTvdQyeNy48Ao26i2L6jhQHRZ16UBdsqiJAzVhUecO1DmLMjPfxrU8RoCxPz6LKd3JESBj5OorgqjgPqw6OzBHIxg/BxAFPqGaffjfoNybu+o0w2we10nc5XhR8sQjKE3FItSbrHCT8uuUZlgMmsmW+yrHxzvc25iqOSe98FWxkkfFjok/T0L69AsejBYjmk9hPA+p5oqiO1kKw+8U816XwvBbZPEriuJlKQw/VtqPr6F7U2Gb18A2YDYNlfVNOZRD7r9IDl1eoFUXPS4+1TM1ZpDvbSD/rnoyu9d4LhtUkvYx5TCO3OpfXupfCIexc27ZOYwFoycZ9epSFNyTgcp7TTlUh4xW0YHSw9yFPhls01NPRpfDOA4g4tqgnHtqlUNH77DojSmHcRwJue95RZG8Lodx9Ole2sOUwzhwt6Wt8nxTDvXsaAGZO5tyqFcf0C4w7gHJMS9rTFQ0ojhpotgSig/qd2vsmH9+HcM9m5dFjlDPZGLbap5OsZbVa6TjhRi82jhQD4wvJlYMVuaYilU2v5I6jEvr+zyPWePR8ntgxQhmvzwD4PbMU9BQ70mg906BcYXNuso907hVFoej5GQG1VK1YzZaNHLlrlG57phqubzM9NbYsUX+OqexN6SYcI8sy9lhr/IJVzFyFtorWYjnC7HdOzVfy9ZfZnHDGcSwGGldOhGSJ2n1earL6g3LxrfVKc8YLnnmY8Yv7jafKG+DOU9Gvgh1qZNpt9P7SHYdrqt3hNnjlp9F9ETRX52T10joRCpns1C9Wyyj8SndG+5DOpNDGZKjC88xUixDIU/NcBcd99Mj8qi2v+Vko730Dp0s5+R1tT+uR/ctdN+BDs9xNmDFeASlJuQMh3DX9MhyFgpbZWTxkfhTcTqa0ROsz+jTkofUHNLfxCUPWZdln5ZYLgCNo0Fm6f4cszwa35pj4rN+lz4mdy17/tt0cqvPt9s0xqtHc/VOTI+krpLUiGaNPNWVd7MSpAZT5yerFL/W9xLlhUhEH8pJfWlJlnYZ0Il/TBnskCLjlGYbNzvKre39qdlPtKQDoc/O8TQ7Iw8Zkf+LYH3KaExG9Gu/O6BP0KVHSMlH+vidpIhuXLFOwo4xE8clQr7VYMZbTL5sQvI1rz27chqLMmOQ68DVzNjWNtmjWDAmqSPl3c3crl99EGnek7BHiWQ0Y+ULkv8l/dW/epwszo0ItDA+gVz5OtfzyChnQRu1aZWv90G6ra3l54UOL5XWZv0zOn1e0myTMi7UB1frHkju0r2UhaNkRHrnc23kOlq3m4vMwxk7Ym9PKIuXfr+vVmDU+w6tkos051o0SvowCsZFFqHbcrvIs3LrZZXZ/bjz/wu7sXXZasgYCbODKy3E7e/HlK3ZWqYwquX4fU2zyW310UyrejkDGotn1lz+EWp/B3+13vrej6dT8grrNAYkg7kzFpE10VwLP1nrJVl6ZGouc2/kmTGpW9k118mvpXczOfZ5MMsBjZq3atdCl6/D8crieOVpwyadNRor6nrtiY7Z3KKpTit95YVIawYwT1hmPiLTqMRDSzuX8mPtsax8jq9R71iuZZarDbPVPg2w57wP0j3XZ2f3j8XqHokHFNt0KQKT+UuPZmlCMZeurc/UJANKvqv8qz37W1SD0jvkQZFZvseJM0aeOnXpuio0/YNa2TLy88Yj6PeWLlQb7WNbVP5qDnlGcyKneakRd6lFrPS39YhmPNKSFXNEtPPfpphKxh31ObPd2jyTqBRPmHxTziojS2YKA7I/t/O2O5e97lr5a0Q54URF1x3gCn/CyCAxeifBHVnm9IRwlZMnCTKi7ZD/nPdT8hRvYGm0RFpPxZqHj5FZrxnr9tjSPdZ9+yO0RKubp+5qwctLvSVy8q5zotemVe1MxajTmfvrcbXVKle+r7PDZEausceE2tiZhcnyypiWuOctRWoUJkVifKSE9SJE/zDNQ3SWp1O+zLq1Zi7vNEgfc0r5EvceKCJc0d0XzmjuS6YfnTm+DmFtNlnDMeFuXKb2B2xPi7tSN+fWIVl7s3Y1Sq2VqGql0Oz2amH8t/SQMXm/VHB7NrK1rXurlKXwuzCSoSvkG71V+aHNeQ8u/BsJV3aoJfrsHTYgvr0vNsTWe3gb4o0qyx3NiGrQF/Rmcu+26me5Rb2N3ljsNr+PBH8ZCdia0z6hlTRUd8nMa26z+/NfkBcYiZjV3rQM74Mthe/JvKSQ/iTk2fjeJEJ/Fye0L1qCT0/KUvzlyHMNrhcnQn+nKawPmp3vQVlCiAz9HoPfMzetw2XZkurtNS/FV4ZcBfSJi8bhyV91rmLa+XiokfVE3r8E9A4nNex6tfhf+6HlGEnhsnyl5fRds1ceT122i9WOLMbD4XPGSPMZzdUS/WVmRe9MtOSWJ+O+KOhJZVZv3j8/xqNmDGhZUyH3QXntJN4eRUZfXxY8F3DpkIn/iH/c4L+N8KbgqNIjhEmfU1Sz6RY8m/7Gpat3+jMfnQxPlU5lNpNHNOiN2A2xKx7A70YRAYa+HSq/Syn/I9b9/dke1J6Q99C76HLnoEV1Me1+mFO0Ht2rPcbjW4srs99Cni8cri59vbTy+O7iN+vqC8ofid+I30NasiL+LL4RO9DdQ4FfVP6r+Jv4+9rHa1+tfb12Tzb94IbC/FqUftY2/wuFudpq</latexit> <latexit

    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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Gaspard Monge T <latexit 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<latexit 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<latexit 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inf =T ]↵ Z X c(x, T(x))d↵(x) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  25. Continuous Monge’s Problem ↵ Y <latexit 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<latexit sha1_base64="V/ijx18qSlgI3jCwJ6JJh1g2VwY=">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</latexit> <latexit

    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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Gaspard Monge T <latexit 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<latexit 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<latexit 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Discrete case: <latexit 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<latexit 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<latexit sha1_base64="q7OQYOXIvuTuAKQi9e1ICZoeO5Y=">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</latexit> ↵ = Pn i=1 xi <latexit 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<latexit 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<latexit 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= Pn j=1 yj <latexit 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<latexit 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<latexit 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<latexit sha1_base64="B0/7Q0xaRpkid90QuR/UJOBxWDQ=">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</latexit> Ci,j = c(xi, yj) <latexit 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<latexit 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min 2⌃n n X i=1 Ci, (i) <latexit 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<latexit sha1_base64="9c50g6gbnvG3Zcd4CgGqg0hmEvc=">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</latexit> inf =T ]↵ Z X c(x, T(x))d↵(x) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="H4SzjqeKxRpGkCl+FqrHBNnEXiQ=">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</latexit>
  26. Continuous Monge’s Problem ↵ Y <latexit 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<latexit sha1_base64="V/ijx18qSlgI3jCwJ6JJh1g2VwY=">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</latexit> <latexit

    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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Gaspard Monge T <latexit 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<latexit 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<latexit 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Discrete case: <latexit 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<latexit 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<latexit sha1_base64="q7OQYOXIvuTuAKQi9e1ICZoeO5Y=">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</latexit> ↵ = Pn i=1 xi <latexit 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<latexit 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<latexit 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= Pn j=1 yj <latexit 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<latexit 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<latexit 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<latexit sha1_base64="B0/7Q0xaRpkid90QuR/UJOBxWDQ=">AABFgnictVxfc9u4EUeu1/aa/su1j31hz0knd5NLbd912um1M5fYjuPESZRIdnIXJR5SohUmlKiIkuJEp+/Ub9Bv0Zk+tY/tU79CF7sAAUogF3TTcGyDIH67iyWw2F2AicZpkk83N/9+4YPvffj9H/zwox9d/PFPfvqzn1/6+BfHeTab9OKjXpZmkydRmMdpMoqPpsk0jZ+MJ3E4jNL4cfRqRz5/PI8neZKNOtO34/jZMByMktOkF06h6uTSncvdzk7wx2DRRVqLSdxfnp0ky6CbZqPBMBzn06x4GKWzePl2ebLo5slVG5EsP11ePrm0sXl9E/8F64UtVdgQ6l8r+/iTS6Ir+iITPTETQxGLkZhCORWhyOF6KrbEphhD3TOxgLoJlBJ8HouluAjYGbSKoUUIta/g9wDunqraEdxLmjmie8AlhZ8JIANxBTAZtJtAWXIL8PkMKcvaKtoLpCllewt/I0VrCLVT8QJqOZxu6YuTfZmKU/EH7EMCfRpjjexdT1GZoVak5IHVqylQGEOdLPfh+QTKPURqPQeIybHvUrchPv8XtpS18r6n2s7Ev1HKK3AFoq16nxUUQjFH+gG+zRk8I3lS4DwACrHqoyy9QV0PsfcjaL+A+vtwLbGkdRLBtcDaZS1yBy4XcodF7sPlQu6zyEO4XMhDFtmCy4VsKaTETlDnbnwbLhe+zXJ+CJcL+ZBFPoLLhXzEIo/hciGPWeS3cLmQ37LIW3C5kLdY5F24XMi7LLIDlwvZYZFHcLmQRyxyDy4Xck8hq2fqBK4M6STMrLwB5TIPaSlSqLnByncTraMLe9NjTvcqsPys3oW/buyuh07jCuyex7g7rcDyI28fbKQby9ui27iauLC3WewBjAA39oDF3hEvK7B3PGbaqwosP9cOoZ0by1vfe3Dnxt5jsfeh5Mbya9QDqHFjH3isGOMKbIvFPhSvK7A+Vn9SgeXtfhvsihvLr1MdaO/G+ljTWQWWt6fH4MG4sfxq9Rhq3djHLPaJOKvAPmGx34B1d2O/8Vhh31Vg9Rp7EVeQAfojMczYOmphMStlaQzUQoZ/WqwtKfrGEdRzmEGBGSBmyCL2C8S+J+KwQBx6y5UXdjRHf5fn0i4QbU9EVKxNsjRl2/eL9rKUeiB2C8TuCqLOI5XvWvdljt6FruGQ02LlkiWfPmWF/ZalWI2HesurEQ9KCBrbL3DkX8NoSUZQUlN11F4UazwhA7yvQ7zB6E33UvPgcdPCKtioMxYVOVARi3rrQL1lUTMHasai5g7UnEWZmW/juh4jwOhfvosF3tEIIB+5+grAK7gBq85tmKMBjJ8WeIGPsOYB/G1j7M1ddZLJaF6ukzLL8axkiSdQWogNqDdR4S7G1ynOsBgko5YPVIwv72RuY6HmHFnhZbGSB0XGxJ9OgvIMCjrSWwxwPjWjcxdrlujdUakZ/nYx73WpGX4PNb5EL55KzfBTJf30HLJ3FLZzDmwbZtNYad+Um9Kg/AvR0OWLuOpKiyvf6lCNGUnvrCH9A/VmDs7xXnawRPox5WY0cqt/eal/TWgYPeeWnptRkd4Teb26FDTuyUjFvabcVIYMV9GRksPcNX0zsk1fvRldbkajBR7XDsbcC6vcdPSOi96YcjMax4Lynkv05HW5GY0B3pM+TLkZDZltCVWcb8pNLbvUAMXOptzUqo8wCyxzQDTmqcZ4RRP0k2aKWoL+QX22xvb519cxmbN5XsQI9ZSMb1tNJyrWsnqJtL8Qg1WbNpRD+hczywcr01iIbTa+IhmmpfV9nY5Z46XmD0GLAcx+2gPgcuYpSKhzEtJ6p0Bxi426yj3TuG0WJ0fJ6Qqqq2qnrLdo+FLWqFx3grVcXGZ6a/TYRXud49gbo094iJrl9HBY+YarKHIaOixpiKfXRHfv1Hwta3+TxY1XEONipPVwR4h20urjVJfW25aOr6hdnilctOdjxq/MNp8qayNjngxtkZSljqfdTueR7Dq5rl4TJsdNzwJ8o9JezdFqJLgjlbNRqM4Wkze+wHtD+wj35CQPotGD9xgoKmNBu2Yyiy7z6QFaVNvecrylvnSGjso5Wl1tj+vRAws9cKCbxzg7sGLch1IHYoYjuOt4RDkXC11lqPGJ+LzYHc3wDdZH9GnJQmoaZG/ikoWsi7JflKi8AbQcDRSl+9NYpaPx3TVKfNTvksfErmXLfwV3bvX+dohjvHo0V2di+sh1G7kGOGtoV5fuVjmQBAvnk230X+t7Kfk14ShtKMf1ucWZ9DLCHf8YI9gxesYpzjZudpRb2/mp1SeaU0vovXO5m52hhQzQ/gWwPmU4JgP8sc8O6B10sggp2kgfu5MU3o3L10nYMWb8uETQqQYz3mK0ZTPkr+nasyvHsUgRA60Dy5WxrXVyiL5gjFwnyrqbuV2/+kikOSdhjxKiaMbKVeT/Kf7WP3qcbKyNCKlh+QZyZetc7yPDmEXqKMRVvt4G6ba2lJcLGZ4rqc36Z2S6XJJsFyMuKY9crfvAuYf3xEuOkgnKna+1oXW0LpsrKY9X9Ch7e4pRPNn9gVqBpdzXcJXcwDnXxVEygFEwLaII3ZbLIq/yredVpu5HO/+/UDe6LmtNUgyEyeCShrj8fozRmi1lCqOaxu8rnE1urU9WWtXzGeFYHFpz+Tuo/TX81nLrez86Uckq3MQxQBTMndEI1QRrLfx43Szx0iNT0zL3hp8Zk7qVXXOe+Jqsm4mx542ptHDUnKmshS6fh8ZLi8ZLTx12cK/RaFHXa0t0wsYWHbVb6cuvCbdOA8ozljLvkWlU4iGlHUv5Ue2zVPkYX6PesbQ2WVohzFZ7N8Ce8z5I91xfnd3fFat7IG6hb9NDD4zilz7O0gR9Ll1bH6kRBcn5S2Vf7dnfxRrJPUILKinTOU45Y2jXqYfXspD0N2ply9DOG4ugzy29UW20je1i+Ys15BDnRI7zUiO+xBaxkt+WI1ixSNctnyPAzH+IPhX5HfUxs93avJOg5E+YeJNmleFFkcII9c9l3g7WotcDK34NMCacKe86AlrN37CkQBidSXB7ljm+IbnK0U4CebQR2s91O0W7eCNLouso9UL82cPGUNRrxro9tnSPdd8+g5ZS6+atu1rw/FJvjhy/8+zohbiqDZWPuli5Px+tUK1y5fs6PcxW+Bp9zLCNHVmYKK+M6YqvvLmQRM24EMaHS7NeNJG/meRNZKbdKV/KurWmXM40kI15gfESdw5UIlze3VWnN/cp049ojV6EWJsa1XCUZDYuU/kB29LKrFSwEiHZ9dyalFrrUdV6YXjYq4ax42QpY7SCqeByN9Ta7kO3FK3w2Rii0BN0srcqTrRpfgWX/B0IV5SoOfrkENvg594QO2LvPZyKeK3KlNkMsEbahP5KDB6qfpZb1OvotUXdpu/DwZ9HArrmpE9wRW0qO1HmJbep+9N/g9ZgImJWetOyeR9sLnxP1jk16U+CFo7vTSL0NzlN+6I5+PSkzMWfD+1vcL04FfrbpmZ90NT5HpQ5NOGhzzP4vXPTujkvm1O9vta5+PKgdUDvvGic3AGsjllMOx8LNbHeyPvnIK3DaQ11vVr8r/3QfAyn5rx8ueX4zdlLj7dO7WKVmZV+cfM5Y7j5jOZqjv48s6J3xmty8yP/L2j0pjKrN++fvvRLzRjQvBaC8qG8dIS3R5GR15eK3B9wyZCJ/4i/XuC/Snhd0KiSowklvV9RTU234KnpLy9dvdPPfGQydKpkKlMz8UQbT8buiANxC352Cg+w6SlR+qaS/kqs+zvaPtSeovXQ2XTKIHSxLsYsiNlN6+O9OUdbJbE800tnfDtQI/fED7FWnve9j+3lmd9OqW/VX5LQXL8nMtEvRSaru3xmXkXQg/IOHOWC9Pe+AZ6pp2wWnUAbeuwx0jkqipT0188LRPQxLlyVdIEIPVrqKEdOyhGeSYoraEelvvVwhI/VTr/cd5Dn88MiuxSI32JdqFYHuVLXab2lvgMuy/QU8wIRan8T4rPfiWvw95oqu+VsCT4H1TkXt85K/3N8s+V+nlnP6s+XLZ2jzXwjeQWzazr/N1ftMswVmD3J+vzubiUXOkdfjx/U4AeWlG0cA68wmp+I+ozkrIbmTMlk7xKPhM6mkh5kjBwWo64+Kp/X8Jp79P9uJfquJek+yBJhDj/AXcIJ0kuVbvZQejqtWZ8Nvl0jrf4WlGia85pmHOiTl/U7Dalz3Ns2hU5XchmguIKObUHonCd3BiU557wfe5yxCD16y/fVp6cclRkryczj++a5hyxzDzqnjDSnLIUBK4myDyeXNrZW/weR9cLx9vWtzetbD7c3vr6p/neRj8SvxCfiKqyovxdfw/hviSPg9BfxN/EP8c+9D/c+29va+4KafnBBYX4pSv/2/vRfQhxqeA==</latexit> Ci,j = c(xi, yj) <latexit 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<latexit 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min 2⌃n n X i=1 Ci, (i) <latexit 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<latexit sha1_base64="9c50g6gbnvG3Zcd4CgGqg0hmEvc=">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</latexit> inf =T ]↵ Z X c(x, T(x))d↵(x) <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="H4SzjqeKxRpGkCl+FqrHBNnEXiQ=">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</latexit> ? ↵ ↵ ↵ ↵ Non-symmetry, non-existence, non-uniqueness: <latexit 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<latexit 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<latexit 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↵
  27. Brenier’s Theorem Hypotheses: <latexit sha1_base64="Kz6oWTOyZxQfM22h7hulfw17JC4=">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</latexit> <latexit 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<latexit 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<latexit

    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<latexit sha1_base64="Kz6oWTOyZxQfM22h7hulfw17JC4=">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</latexit> c(x, y) = ||x y||2 <latexit 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<latexit 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<latexit sha1_base64="8N+YrFgHwCNqJVdrD67sfxTeobw=">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</latexit> d↵ dx = ⇢↵ density. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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↵ <latexit 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<latexit 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W2 2 (↵, ) def. = inf =T ]↵ Z Rd ||x T(x)||2d↵(x) <latexit 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<latexit 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<latexit 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<latexit 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X = Rd <latexit 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<latexit 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<latexit sha1_base64="XN+iO57sWsbXjQR+wAEZLNcSmm8=">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</latexit>
  28. Brenier’s Theorem Hypotheses: <latexit sha1_base64="Kz6oWTOyZxQfM22h7hulfw17JC4=">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</latexit> <latexit 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<latexit 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<latexit

    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<latexit sha1_base64="Kz6oWTOyZxQfM22h7hulfw17JC4=">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</latexit> c(x, y) = ||x y||2 <latexit 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<latexit 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<latexit sha1_base64="8N+YrFgHwCNqJVdrD67sfxTeobw=">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</latexit> d↵ dx = ⇢↵ density. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Yann Brenier There exists a unique Monge map T. <latexit 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<latexit 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<latexit sha1_base64="nsglRAUi+tlHeI1cLZ6LHMM7xpo=">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</latexit> It is the unique T = r' such that <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="jRX8giyUu12rfn1251pXgPCW+54=">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</latexit> ' is convex and (r')]↵ = . <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tvjr0hwMzBZ6qILEwzAGLTMLlqQ=">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</latexit> Theorem: [Brenier, 1991] <latexit 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<latexit 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↵ <latexit 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<latexit 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W2 2 (↵, ) def. = inf =T ]↵ Z Rd ||x T(x)||2d↵(x) <latexit 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<latexit 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<latexit 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<latexit 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X = Rd <latexit 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<latexit 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<latexit sha1_base64="XN+iO57sWsbXjQR+wAEZLNcSmm8=">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</latexit>
  29. Brenier’s Theorem Hypotheses: <latexit sha1_base64="Kz6oWTOyZxQfM22h7hulfw17JC4=">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</latexit> <latexit 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<latexit 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<latexit

    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<latexit sha1_base64="Kz6oWTOyZxQfM22h7hulfw17JC4=">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</latexit> c(x, y) = ||x y||2 <latexit 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<latexit 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<latexit sha1_base64="8N+YrFgHwCNqJVdrD67sfxTeobw=">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</latexit> d↵ dx = ⇢↵ density. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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! Monge-Amp` ere equation (non-linear, degenerate elliptic). <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="vq9eQR/Pj8oaAYzIMN53xm+UT20=">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</latexit> ⇢↵(x) = | det(@2'(x))|⇢ (T(x)) <latexit 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<latexit 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<latexit sha1_base64="Sgg0ElPzJvjIt54QpbUNAPlp5pE=">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</latexit> s.t. ' convex. <latexit 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<latexit 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Yann Brenier There exists a unique Monge map T. <latexit 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<latexit 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<latexit sha1_base64="nsglRAUi+tlHeI1cLZ6LHMM7xpo=">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</latexit> It is the unique T = r' such that <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="jRX8giyUu12rfn1251pXgPCW+54=">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</latexit> ' is convex and (r')]↵ = . <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tvjr0hwMzBZ6qILEwzAGLTMLlqQ=">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</latexit> Theorem: [Brenier, 1991] <latexit 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<latexit 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↵ <latexit 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<latexit 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W2 2 (↵, ) def. = inf =T ]↵ Z Rd ||x T(x)||2d↵(x) <latexit 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<latexit 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<latexit 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<latexit 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X = Rd <latexit 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<latexit 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  30. Regularity Theory Alessio Figalli ! Regularity of T requires convex

    target. <latexit 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<latexit 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<latexit 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  31. Regularity Theory Alessio Figalli ! Regularity of T requires convex

    target. <latexit 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<latexit 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<latexit 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  32. <latexit sha1_base64="lkYmEXSwz6mt4AGHn22Q31ywqqo=">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</latexit> C↵(x) def. = R x 1 d↵ <latexit

    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<latexit 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<latexit 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<latexit 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↵ <latexit 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1-D Optimal Transport C↵] : ↵ 7 ! U[0,1] <latexit 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<latexit 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<latexit 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<latexit 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Cumulative function: <latexit 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<latexit 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<latexit 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<latexit 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  33. <latexit sha1_base64="lkYmEXSwz6mt4AGHn22Q31ywqqo=">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</latexit> C↵(x) def. = R x 1 d↵ <latexit

    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<latexit 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<latexit 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<latexit 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↵ <latexit 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1-D Optimal Transport C↵] : ↵ 7 ! U[0,1] <latexit 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<latexit 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<latexit 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<latexit 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Cumulative function: <latexit 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<latexit 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<latexit 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<latexit 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  34. <latexit sha1_base64="lkYmEXSwz6mt4AGHn22Q31ywqqo=">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</latexit> C↵(x) def. = R x 1 d↵ <latexit

    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<latexit 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<latexit 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<latexit 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↵ <latexit 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<latexit 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C 1 ↵ <latexit 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  35. <latexit sha1_base64="lkYmEXSwz6mt4AGHn22Q31ywqqo=">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</latexit> C↵(x) def. = R x 1 d↵ <latexit

    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<latexit 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<latexit 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<latexit 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↵ <latexit 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Optimal transport ↵ 7! : <latexit 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<latexit 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<latexit 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T = C 1 C↵ <latexit 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<latexit 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1-D Optimal Transport C 1 ] : U[0,1] 7 ! <latexit 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<latexit 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<latexit 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Quantile function: <latexit 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<latexit 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C↵] : ↵ 7 ! U[0,1] <latexit 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<latexit 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<latexit 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<latexit 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Cumulative function: <latexit 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<latexit 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<latexit 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<latexit 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C 1 ↵ <latexit 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  36. <latexit 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↵ <latexit 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<latexit

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<latexit 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<latexit sha1_base64="cOXoTQGOwUSDpobZ06nn8bk1NgM=">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</latexit> C↵(x) def. = R x 1 d↵ <latexit 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<latexit 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<latexit 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<latexit sha1_base64="NagChymWaASb8bvGA6rN+HpPF8Q=">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</latexit> W1(↵, ) = ||↵ ||W1 = R R |C↵(x) C (x)|dx <latexit 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<latexit 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<latexit 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<latexit sha1_base64="LTA4z6RcvQ5fpn9AKnIpN2fJHaE=">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</latexit> Wp(↵, )p def. = R ||T(x) x||pd↵(x) = R 1 0 |C 1 ↵ (t) C 1(t)|pdt <latexit 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<latexit 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<latexit 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<latexit 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  37. <latexit 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↵ <latexit 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<latexit

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<latexit 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<latexit sha1_base64="cOXoTQGOwUSDpobZ06nn8bk1NgM=">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</latexit> C↵(x) def. = R x 1 d↵ <latexit 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<latexit 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<latexit 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<latexit sha1_base64="NagChymWaASb8bvGA6rN+HpPF8Q=">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</latexit> W1(↵, ) = ||↵ ||W1 = R R |C↵(x) C (x)|dx <latexit 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<latexit 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<latexit 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<latexit sha1_base64="LTA4z6RcvQ5fpn9AKnIpN2fJHaE=">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</latexit> Wp(↵, )p def. = R ||T(x) x||pd↵(x) = R 1 0 |C 1 ↵ (t) C 1(t)|pdt <latexit 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<latexit 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<latexit 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<latexit 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  38. <latexit 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↵ <latexit 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<latexit

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<latexit 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<latexit sha1_base64="cOXoTQGOwUSDpobZ06nn8bk1NgM=">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</latexit> C↵(x) def. = R x 1 d↵ <latexit 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<latexit 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<latexit 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