Le transport optimal: de Gaspard Monge à la science des données

Le transport optimal: de Gaspard Monge à la science des données

Exposé grand public.

E34ded36efe4b7abb12510d4e525fee8?s=128

Gabriel Peyré

May 29, 2017
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  1. Le transport optimal: de Gaspard Monge à la science des

    données Gabriel Peyré https://optimaltransport.github.io É C O L E N O R M A L E S U P É R I E U R E
  2. • Le transport de Monge • Le transport de Kantorovitch

    • Applications
  3. Des boulangeries aux cafés cij y1 y2 y3 y4 y5

    y6 x1 12 10 31 27 10 30 x2 22 7 25 15 11 14 x3 19 7 19 10 15 15 x4 10 6 21 19 14 24 x5 15 23 14 24 31 34 x6 35 26 16 9 34 15 12 min 10 min 31 min 30 m in 27 min ? <latexit 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? <latexit 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<latexit 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  4. Des boulangeries aux cafés Cout: 10+7+15+10+14+9 = 65 min cij

    y1 y2 y3 y4 y5 y6 x1 12 10 31 27 10 30 x2 22 7 25 15 11 14 x3 19 7 19 10 15 15 x4 10 6 21 19 14 24 x5 15 23 14 24 31 34 x6 35 26 16 9 34 15 cij y1 y2 y3 y4 y5 y6 x1 12 10 31 27 10 30 x2 22 7 25 15 11 14 x3 19 7 19 10 15 15 x4 10 6 21 19 14 24 x5 15 23 14 24 31 34 x6 35 26 16 9 34 15 ? <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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  5. Du meilleur … au moins bon Coˆ ut=64 Coˆ ut=65

    Coˆ ut=66 Coˆ ut=152
  6. Gaspard Monge (1746-1818) Il joue un grand rôle dans la

    Révolution française, tant du point de vue politique que du point de vue de l'instauration d'un nouveau système éducatif : il participe à la création de l'École normale de l'an III et de l'École polytechnique (en 1794), deux écoles où il enseigne la géométrie. Il concourt également avec Berthollet, Chaptal et Laplace à la création de l'École d'arts et métiers. Il est également membre de la commission des sciences et des arts lors de la campagne d'Italie (1796–1797), et chargé de mission dans l'expédition d'Égypte (1798–1799). (1784)
  7. Nombre de Possibilités 1 vers 1 2 {1, 2, 3,

    4, 5, 6}. 3 vers 3 2 {1, . . . , 6} { 1, 2 }. 2 vers 2 2 {1, . . . , 6} { 1 }. 4 vers 4 2 {1, . . . , 6} { 1, . . . , 3 }. 5 vers 5 2 {1, . . . , 6} { 1, . . . , 4 }. Total : 6 ⇥ 5 ⇥ 4 ⇥ 3 ⇥ 2 ⇥ 1 = 720 possibilit´ es. 6 vers 6 2 {1, . . . , 6} { 1, . . . , 5 }.
  8. Nombre de Possibilités n n ! 0 1 1 1 2

    2 3 6 4 24 5 120 6 720 7 5 040 8 40 320 n n ! 9 362880 10 3628800 11 39916800 12 479001600 … … 25 1,551x1025 … … 70 1,198x10100 Atomes dans l’univers: 1079 Neurones dans le cerveau: 1011. 1 vers 1 2 {1, 2, 3, 4, 5, 6}. 3 vers 3 2 {1, . . . , 6} { 1, 2 }. 2 vers 2 2 {1, . . . , 6} { 1 }. 4 vers 4 2 {1, . . . , 6} { 1, . . . , 3 }. 5 vers 5 2 {1, . . . , 6} { 1, . . . , 4 }. Total : 6 ⇥ 5 ⇥ 4 ⇥ 3 ⇥ 2 ⇥ 1 = 720 possibilit´ es. 6 vers 6 2 {1, . . . , 6} { 1, . . . , 5 }.
  9. Exemples n=70 Cout=200 Cout=1493

  10. En tram x1 x2 x3 x4 x5 y1 y2 y3

    y4 y5
  11. En tram ! Solution: classer les (xi)i, (yj)j par ordre

    croissant. xxx x1 x2 x3 x4 x5 y1 y2 y3 y4 y5
  12. En tram Algorithmes de tri: par selection, pire cas n(n

    1)/2 op´ erations. 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 Tri rapide: n log(n). ! Solution: classer les (xi)i, (yj)j par ordre croissant. xxx x1 x2 x3 x4 x5 y1 y2 y3 y4 y5
  13. Comparer des classes Notes /20: Classe 1 Classe 2 0

    10 20 x1 x2 x3 x4 x5 y4 y1 y5 y3 y2 Classe 1 Classe 2 Comparaison
  14. Comparer des classes Notes /20: Classe 1 Classe 2 0

    10 20 x1 x2 x3 x4 x5 y4 y1 y5 y3 y2 Classe 1 Classe 2
  15. Images en niveaux de gris

  16. Images en niveaux de gris

  17. En dimension 2 Le cas 2D: xi xj n =

    10 n = 70 n = 300 ci,j = ||xi yj || = q (x1 i y1 j )2 + (x2 i y2 j )2 xi, yj 2 R2
  18. En dimension 2 Le cas 2D: xi xj n =

    10 n = 70 n = 300 xi xj xj0 xi0 xi xj xj0 xi0 mauvais meilleur ci,j = ||xi yj || = q (x1 i y1 j )2 + (x2 i y2 j )2 xi, yj 2 R2 Propri´ et´ e: deux segments ne se croisent pas.
  19. En 2D : les formes Alessio Figalli Cedric Villani Quelle

    est la r´ egularit´ e <latexit 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<latexit 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<latexit 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du transport optimal? <latexit 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<latexit 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  20. En 2D : les formes Alessio Figalli Cedric Villani Quelle

    est la r´ egularit´ e <latexit 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<latexit 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<latexit 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du transport optimal? <latexit 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<latexit 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  21. En 3D : les couleurs 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 projectio 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 R´ ef´ erence Entr´ ee Sortie Transport optimal
  22. • Le transport de Kantorovitch • Le transport de Monge

    • Applications
  23. Leonid Kantorovich (1912-1986) Леонид Витальевич Канторович Au cours du siège

    de Léningrad, Kantorovitch est responsable de la sécurité de la Route de la vie. Il détermine la distance optimale à observer entre les voitures sur la surface gelée du lac Ladoga, en fonction de l'épaisseur de glace et de la température de l’air. En d é c e m b r e 1 9 4 1 – j a n v i e r 1 9 4 2 , Kantorovitch s'assure lui-même de la viabilité de la banquise en marchant entre les camions. Cependant bien des véhicules chargés d'approvisionnements sont détruits par les bombardements aériens nazis. En récompense de ses exploits et de son courage, les autorités attribuent à Kantorovitch l’ordre de la Guerre patriotique, et le décorent de la Médaille pour la Défense de Léningrad. Peu avant la Seconde Guerre mondiale, Leonid Kantorovitch découvre la programmation linéaire, l'optimisation linéaire et ses applications à l'optimisation de la production économique planifiée. Il est le seul chercheur soviétique à avoir reçu le « prix Nobel » d'économie (1975). Les théories de Kantorovitch ne sont publiées qu’après l'ère stalinienne. Un des apports de Kantorovitch est d'avoir incité à une meilleure prise en compte de la p r o d u c t i v i t é m a r g i n a l e d e l’investissement, afin de résoudre les difficultés liées à l’allocation des ressources au sein d’une économie socialiste.
  24. Poids et histogrammes A↵ectation entre ensembles de tailles di↵´ erentes:

    impossible. ? Points (xi)i, poids (pi)i. Points (yj)j, poids (qj)j. Contrainte: P i pi = P j qj Poids ⇠ masse ⇠ capacit´ e. 2 1 1 2 2
  25. 1 2 3 4 5 6 0 10 20 q1

    q2 q3 q4 q5 q6 Poids et histogrammes A↵ectation entre ensembles de tailles di↵´ erentes: impossible. ? Points (xi)i, poids (pi)i. Points (yj)j, poids (qj)j. Contrainte: P i pi = P j qj Poids ⇠ masse ⇠ capacit´ e. 2 1 1 2 2 1 2 3 4 5 6 0 10 20 p1 p2 p3 p4 p5 p6
  26. Couplage Optimal 1 2 3 4 5 6 0 10

    20 1 2 3 4 5 6 0 10 20 3 23 18 26 16 14 24 1 7 0 0 16 0 9 0 0 0 0 0 9 3 0 0 0 0 0 3 16 0 16 0 0 0 0 21 2 0 18 1 0 0 27 0 0 0 25 0 2 Transport de masse i $ j: Pi,j > 0 Conservation de la masse: P j Pi,j = pi P i Pi,j = qj Pi,j qj pi C L
  27. Couplage Optimal 1 2 3 4 5 6 0 10

    20 1 2 3 4 5 6 0 10 20 3 23 18 26 16 14 24 1 7 0 0 16 0 9 0 0 0 0 0 9 3 0 0 0 0 0 3 16 0 16 0 0 0 0 21 2 0 18 1 0 0 27 0 0 0 25 0 2 Transport de masse i $ j: Pi,j > 0 Conservation de la masse: P j Pi,j = pi P i Pi,j = qj Pi,j qj pi C L min P i,j >0 X i,j Pi,jci,j Probl` eme de Kantorovich: “Programmation lin´ eaire” L C & Georges Dantzig
  28. Exemples

  29. Monge (1784): <latexit 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Equivalence entre Monge et Kantorovitch min 2Permn n X i=1 Ci, (i) <latexit 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<latexit 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Kantorovitch (1942): <latexit 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(relaxation) <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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<latexit 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min P2Bistn n X i=1 n X j=1 Pi,jCi,j <latexit 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<latexit 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Permutations “⇢” <latexit 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<latexit 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matrices bi-stochastiques <latexit 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<latexit 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<latexit 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Bistn def. = n P 2 Rn⇥n + ; P i Pi,j = 1, P j Pi,j = 1 o <latexit 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<latexit 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<latexit 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Yann Brenier
  30. Monge (1784): <latexit 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Equivalence entre Monge et Kantorovitch min 2Permn n X i=1 Ci, (i) <latexit 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<latexit 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Kantorovitch (1942): <latexit 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(relaxation) <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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<latexit 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min P2Bistn n X i=1 n X j=1 Pi,jCi,j <latexit 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<latexit 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Permutations “⇢” <latexit 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<latexit 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matrices bi-stochastiques <latexit 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<latexit 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<latexit 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Bistn def. = n P 2 Rn⇥n + ; P i Pi,j = 1, P j Pi,j = 1 o <latexit 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<latexit 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<latexit 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John von Neumann George David Birkhoff “Monge , Kantorovitch” <latexit 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<latexit 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Theor` eme : [Birkho↵-von Neumann] <latexit 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<latexit 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<latexit 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<latexit 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Yann Brenier
  31. Monge (1784): <latexit 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Equivalence entre Monge et Kantorovitch min 2Permn n X i=1 Ci, (i) <latexit 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<latexit 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Kantorovitch (1942): <latexit 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(relaxation) <latexit 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<latexit 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