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Course on Optimal Transport

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Course on Optimal Tranport, Symposium on Geometry Processing, Paris 2018

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Course on Optimal Transport

  1. 1. Mathématiques - Informatique Bruno Lévy ALICE Géométrie & Lumière CENTRE INRIA Nancy Grand-Est Symposium on Geometry Processing – Paris – 2018 Course on Numerical Optimal Transport – Bruno Lévy
  2. 2. Part. 1. Goals and Motivations Part. 2. Introduction to Optimal Transport Part. 3. Semi-Discrete Optimal Transport Part. 4. Applications in Computational Physics OVERVIEW
  3. 3. Goals and Motivations 1
  4. 4. Part. 1 Optimal Transport Goal #1: “Understanding”
  5. 5. Part. 1 Optimal Transport Goal #1: “Understanding” What I can’t create I don’t understand Richard Feynman
  6. 6. Part. 1 Optimal Transport Goal #1: “Understanding” Jos Stam, Stable Fluids, 1999 The art of fluid sim. Understand fluids Explain Program
  7. 7. Part. 1 Optimal Transport Goal #1: “Understanding” I wrote code
  8. 8. Part. 1 Optimal Transport Goal #1: “Understanding” Your mission statement: 1. Understand the stuff 2. Explain it in simple terms Be a good teacher, to others and to yourself Know what you know and what you don’t know Try to know what you don’t know 3. Program it
  9. 9. Part. 1 Optimal Transport Measuring distances between functions
  10. 10. Part. 1 Optimal Transport Measuring distances between functions
  11. 11. Part. 1 Optimal Transport Measuring distances between functions
  12. 12. Part. 1 Optimal Transport Measuring distances between function
  13. 13. Part. 1 Optimal Transport Interpolating functions
  14. 14. Part. 1 Optimal Transport Interpolating functions
  15. 15. Part. 1 Optimal Transport Interpolating functions
  16. 16. Part. 1 Optimal Transport Interpolating functions
  17. 17. Part. 1 Optimal Transport Interpolating functions
  18. 18. Part. 1 Optimal Transport Interpolating functions
  19. 19. Part. 1 Optimal Transport Interpolating functions
  20. 20. Part. 1 Optimal Transport Interpolating functions
  21. 21. Part. 1 Optimal Transport Interpolating functions
  22. 22. Part. 1 Optimal Transport Interpolating functions
  23. 23. Part. 1 Optimal Transport Interpolating functions
  24. 24. Part. 1 Optimal Transport Interpolating functions
  25. 25. Part. 1 Optimal Transport Gaspard Monge - 1784
  26. 26. Part. 1 Optimal Transport Gaspard Monge – geometry and light ANR TOMMI Workshop
  27. 27. Cédric Villani Optimal Transport Old & New Topics on Optimal Transport Yann Brenier The polar factorization theorem (Brenier Transport) Part. 1 Optimal Transport Monge-Brenier-Villani, the french connection
  28. 28. Part. 1 Optimal Transport [Castro, Merigot, Thibert 2014] Optimal transport geometry and light [Caffarelli, Kochengin, and Oliker 1999]
  29. 29. Part. 1 Optimal Transport – Image Processing Barycenters / mixing textures Video-style transfer, A.I., “data sciences” [Nicolas Bonneel, Julien Rabin, Gabriel Peyŕe, Hanspeter Pfister] [Nicolas Bonneel, Kalyan Sunkavalli, Sylvain Paris, Hanspeter Pfister] [Marco Cuturi, Gabriel Peyré]
  30. 30. Part. 1 Optimal Transport Optimal transport - geometry and light [Chwartzburg, Testuz, Tagliasacchi, Pauly, SIGGRAPH 2014]
  31. 31. Part. 1. Motivations Discretization of functionals involving the Monge-Ampère operator, Benamou, Carlier, Mérigot, Oudet arXiv:1408.4536 The variational formulation of the Fokker-Planck equation Jordan, Kinderlehrer and Otto SIAM J. on Mathematical Analysis Geostrophic current
  32. 32. Part. 1 Optimal Transport How to “morph” a shape into another one of same mass while minimizing the “effort” ? ? T
  33. 33. Part. 1 Optimal Transport How to “morph” a shape into another one of same mass while minimizing the “effort” ? ? T The “effort” of the best T defines a distance between the shapes
  34. 34. Part. 1 Optimal Transport How to “morph” a shape into another one while preserving mass and minimizing the effort ? ? T
  35. 35. Part. 1 Optimal Transport
  36. 36. Part. 1 Optimal Transport How to “morph” a shape into another one while preserving mass and minimizing the effort ? ? T “minimum action principle”
  37. 37. Part. 1 Optimal Transport How to “morph” a shape into another one while preserving mass and minimizing the effort ? ? T “minimum action principle”“conservation law”
  38. 38. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator”
  39. 39. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator” New ways of simulating physics with a computer
  40. 40. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator” New ways of simulating physics with a computer Fast Fourier Transform
  41. 41. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator” New ways of simulating physics with a computer Fast Fourier Transform Fast OT algo. ???
  42. 42. Optimal Transport an elementary introduction 2
  43. 43. Part. 2 Optimal Transport – Monge’s problem (X;μ) (Y;ν) Two measures μ, ν such that ∫dμ(x) = ∫dν(x) X Y
  44. 44. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y (X;μ) (Y;ν) (Borel subset = subset that can be measured)
  45. 45. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y B (X;μ) (Y;ν)
  46. 46. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y B T-1(B) (X;μ) (Y;ν)
  47. 47. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y Notation: if T is a transport map between μ and ν then one writes ν = T#μ (ν is the pushforward of μ) (X;μ) (Y;ν)
  48. 48. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) (X;μ) (Y;ν)
  49. 49. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) • Difficult to study • If μ has an atom (isolated Dirac), it can only be mapped to another Dirac (T needs to be a map)
  50. 50. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Transport from a measure concentrated on L1 onto another one concentrated on L2 and L3
  51. 51. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Transport from a measure concentrated on L1 onto another one concentrated on L2 and L3
  52. 52. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Transport from a measure concentrated on L1 onto another one concentrated on L2 and L3
  53. 53. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Transport from a measure concentrated on L1 onto another one concentrated on L2 and L3 The infimum is never realized by a map, need for a relaxation
  54. 54. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem (1942): Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y)
  55. 55. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) “γ(x,y)” : How much sand goes from x to y
  56. 56. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Everything that is transported from x sums to “μ(x)”
  57. 57. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Everything that is transported to y sums to “ν(y)”
  58. 58. Part. 2 Optimal Transport – Kantorovich Transport plan – example 1/4 : translation of a segment
  59. 59. Part. 2 Optimal Transport – Kantorovich Transport plan – example 1/4 : translation of a segment
  60. 60. Part. 2 Optimal Transport – Kantorovich Transport plan – example 2/4 : spitting a segment
  61. 61. Part. 2 Optimal Transport – Kantorovich
  62. 62. Part. 2 Optimal Transport – Kantorovich
  63. 63. Part. 2 Optimal Transport – Kantorovich
  64. 64. Part. 2 Optimal Transport – Kantorovich Transport plan – example 3/4 : splitting a Dirac into two Diracs
  65. 65. Part. 2 Optimal Transport – Kantorovich Transport plan – example 3/4 : splitting a Dirac into two Diracs (No transport map)
  66. 66. Part. 2 Optimal Transport – Kantorovich Transport plan – example 4/4 : splitting a Dirac into two segments
  67. 67. Part. 2 Optimal Transport – Kantorovich Transport plan – example 4/4 : splitting a Dirac into two segments (No transport map)
  68. 68. Part. 2 Optimal Transport – Duality
  69. 69. Part. 2 Optimal Transport – Duality Duality is easier to understand with a discrete version Then we’ll go back to the continuous setting.
  70. 70. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  71. 71. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn Є IRmn
  72. 72. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn Є IRmnЄ IRmn
  73. 73. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn cij = || xi – yj ||2 Є IRmnЄ IRmn
  74. 74. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn cij = || xi – yj ||2 mn X m Є IRmnЄ IRmn
  75. 75. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn cij = || xi – yj ||2 mn X n mn X m Є IRmnЄ IRmn
  76. 76. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn Є IRmnЄ IRmn mn X n mn X m
  77. 77. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 < u, v > denotes the dot product between u and v
  78. 78. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  79. 79. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  80. 80. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise Consider now: Inf [ Sup[ L(φ,ψ) ] ] φ Є IRm ψ Є IRn γ ≥ 0 (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  81. 81. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise Consider now: Inf [ Sup[ L(φ,ψ) ] ] = Inf [ < c, γ> ] φ Є IRm ψ Є IRn γ ≥ 0 γ ≥ 0 P1 γ = u P2 γ = v (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  82. 82. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise Consider now: Inf [ Sup[ L(φ,ψ) ] ] = Inf [ < c, γ> ] (DMK) φ Є IRm ψ Є IRn γ ≥ 0 γ ≥ 0 P1 γ = u P2 γ = v (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  83. 83. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  84. 84. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] Exchange Inf Sup (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  85. 85. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Exchange Inf Sup Expand/Reorder/Collect (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  86. 86. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Exchange Inf Sup Expand/Reorder/Collect Interpret (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0
  87. 87. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Interpret Sup[ <φ,u> + <ψ, v> ] φ Є IRm ψ Є IRn P1 t φ + P2 t ψ ≤ c (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 (DDMK)
  88. 88. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Interpret Sup[ <φ,u> + <ψ, v> ] φ Є IRm ψ Є IRn P1 t φ + P2 t ψ ≤ c φi + ψj ≤ cij (i,j) A (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 (DDMK)
  89. 89. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem (Continuous): Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φdμ + ∫Y ψdν
  90. 90. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φdμ + ∫Y ψdν Your point of view: Try to minimize transport cost
  91. 91. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φdμ + ∫Y ψdν Point of view of a “transport company”: Try to maximize transport price Your point of view: Try to minimize transport cost
  92. 92. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν What they charge for loading at x Your point of view: Try to minimize transport cost
  93. 93. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν What they charge for loading at x What they charge for unloading at y Your point of view: Try to minimize transport cost
  94. 94. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν What they charge for loading at x What they charge for unloading at y Price (loading + unloading) cannot be greater than transport cost (else you do the job yourself) Your point of view: Try to minimize transport cost
  95. 95. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν
  96. 96. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν If we got two functions φ and ψ that satisfy the constraint Then it is possible to obtain a better solution by replacing ψ with φc defined by: For all y, φc(y) = inf x in X ½|| x – y ||2 - φ(y)
  97. 97. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν If we got two functions φ and ψ that satisfy the constraint Then it is possible to obtain a better solution by replacing ψ with φc defined by: For all y, φc(y) = inf x in X ½|| x – y ||2 - φ(y) • φc is called the c-conjugate function of φ • If there is a function φ such that ψ = φc then ψ is said to be c-concave • If ψ is c-concave, then ψ cc = ψ
  98. 98. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
  99. 99. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν ψ is called a “Kantorovich potential”
  100. 100. Part. 2 Optimal Transport – c-subdifferential What about our initial problem ? ψ is called a “Kantorovich potential” Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
  101. 101. Part. 2 Optimal Transport – c-subdifferential What about our initial problem ? (i.e., this is T() that we want to find …) ψ is called a “Kantorovich potential” Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
  102. 102. Part. 2 Optimal Transport – c-subdifferential
  103. 103. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10.
  104. 104. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter):
  105. 105. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter):
  106. 106. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter):
  107. 107. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter):
  108. 108. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter):
  109. 109. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter):
  110. 110. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter): w
  111. 111. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter): -w
  112. 112. Part. 2 Optimal Transport – c-subdifferential Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
  113. 113. Part. 2 Optimal Transport – c-subdifferential Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν { grad ψ(x) with ψ(x) := (½ x2- ψ(x))
  114. 114. Part. 2 Optimal Transport – convexity Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν { grad ψ(x) with ψ(x) := (½ x2- ψ(x)) ψ is convex
  115. 115. Part. 2 Optimal Transport – convexity Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν { grad ψ(x) with ψ(x) := (½ x2- ψ(x)) ψ is convex
  116. 116. Part. 2 Optimal Transport – convexity Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν { grad ψ(x) with ψ(x) := (½ x2- ψ(x)) ψ is convex
  117. 117. Part. 2 Optimal Transport – no collision If T(.) exists, then T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) {grad ψ (x) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν ψ is convex Two transported particles cannot “collide”
  118. 118. Part. 2 Optimal Transport – no collision If T(.) exists, then T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) {grad ψ (x) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν ψ is convex Two transported particles cannot “collide”
  119. 119. Part. 2 Optimal Transport – no collision If T(.) exists, then T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) {grad ψ (x) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν ψ is convex Two transported particles cannot “collide”
  120. 120. Part. 2 Optimal Transport – Monge-Ampere What about our initial problem ? If T(.) exists, then one can show that: T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) { for all borel set A, ∫A dμ = ∫T(A) (|JT|) dν (change of variable) Jacobian of T (1st order derivatives) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν grad ψ(x) with ψ(x) := (½ x2- ψ(x))
  121. 121. Part. 2 Optimal Transport – Monge-Ampere What about our initial problem ? If T(.) exists, then one can show that: T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) { for all borel set A, ∫A dμ = ∫T(A) (|JT|) dν = ∫T(A) (H ψ ) dν Det. of the Hessian of ψ (2nd order derivatives) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν grad ψ(x) with ψ(x) := (½ x2- ψ(x))
  122. 122. Part. 2 Optimal Transport – Monge-Ampere What about our initial problem ? T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) { When μ and ν have a density u and v, (H ψ(x)). v(grad ψ(x)) = u(x) Monge-Ampère equation for all borel set A, ∫A dμ = ∫T(A) (|JT|) dν = ∫T(A) (H ψ ) dν Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν {grad ψ(x) with ψ(x) := (½ x2- ψ(x))
  123. 123. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x)
  124. 124. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) After several rewrites and under some conditions…. (Kantorovich formulation, dual, c-convex functions)
  125. 125. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) After several rewrites and under some conditions…. (Kantorovich formulation, dual, c-convex functions) Monge-Ampère equation (When μ and ν have a density u and v resp.) Solve (H ψ(x)). v(grad ψ(x)) = u(x)
  126. 126. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) After several rewrites and under some conditions…. (Kantorovich formulation, dual, c-convex functions) Brenier, Mc Cann, Trudinger: The optimal transport map is then given by: T(x) = grad ψ(x) Solve (H ψ(x)). v(grad ψ(x)) = u(x) Monge-Ampère equation (When μ and ν have a density u and v resp.)
  127. 127. Semi-Discrete Optimal Transport 3
  128. 128. Part. 3 Optimal Transport – how to program ? Continuous (X;μ) (Y;ν)
  129. 129. Part. 3 Optimal Transport – how to program ? Continuous Semi-discrete (X;μ) (Y;ν)
  130. 130. Part. 3 Optimal Transport – how to program ? Continuous Semi-discrete Discrete (X;μ) (Y;ν)
  131. 131. Part. 3 Optimal Transport – how to program ? Continuous Semi-discrete Discrete (X;μ) (Y;ν)
  132. 132. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) (X;μ) (Y;ν)
  133. 133. Part. 3 Optimal Transport – semi-discrete (X;μ) (Y;ν) ∑j ψ(yj) vj ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK)
  134. 134. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) ∑j ψ(yj) vj
  135. 135. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) ∫X inf yj ЄY [ || x – yj ||2 - ψ(yj) ] dμ ∑j ψ(yj) vj
  136. 136. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) ∑j ∫Lagψ(yj) || x – yj ||2 - ψ(yj) dμ ∫X inf yj ЄY [ || x – yj ||2 - ψ(yj) ] dμ ∑j ψ(yj) vj
  137. 137. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j
  138. 138. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j Laguerre diagram of the yj’s (with the L2 cost || x – y ||2 used here, Power diagram)
  139. 139. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j Laguerre diagram of the yj’s (with the L2 cost || x – y ||2 used here, Power diagram) Weight of yj in the power diagram
  140. 140. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j Laguerre diagram of the yj’s (with the L2 cost || x – y ||2 used here, Power diagram) Weight of yj in the power diagram ψ is determined by the weight vector [ψ(y1) ψ(y2) … ψ(ym)]
  141. 141. Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) } Part. 3 Power Diagrams
  142. 142. Power diagram: Pow(xi) = { x | d2(x,xi) – ψi < d2(x,xj) – ψj } Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) } Part. 3 Power Diagrams
  143. 143. Part. 3 Power Diagrams
  144. 144. Part. 3 Optimal Transport Theorem: (direct consequence of MK duality alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ): Given a measure μ with density, a set of points (yj), a set of positive coefficients vj such that ∑ vj = ∫ dμ(x), it is possible to find the weights W = [ψ(y1) ψ(y2) … ψ(ym)] such that the map TS W is the unique optimal transport map between μ and ν = ∑ vj δ(yj)
  145. 145. Part. 3 Optimal Transport Theorem: (direct consequence of MK duality alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ): Given a measure μ with density, a set of points (yj), a set of positive coefficients vj such that ∑ vj = ∫ dμ(x), it is possible to find the weights W = [ψ(y1) ψ(y2) … ψ(ym)] such that the map TS W is the unique optimal transport map between μ and ν = ∑ vj δ(yj) Proof: G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Is a concave function of the weight vector [ψ(y1) ψ(y2) … ψ(ym)]
  146. 146. Part. 3 Optimal Transport Theorem: (direct consequence of MK duality alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ): Given a measure μ with density, a set of points (yj), a set of positive coefficients vj such that ∑ vj = ∫ dμ(x), it is possible to find the weights W = [ψ(y1) ψ(y2) … ψ(ym)] such that the map TS W is the unique optimal transport map between μ and ν = ∑ vj δ(yj) Proof: G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Is a concave function of the weight vector [ψ(y1) ψ(y2) … ψ(ym)]
  147. 147. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) The (unknown) weights W = [ψ(y1) ψ(y2) … ψ(ym)]
  148. 148. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) T : an arbitrary but fixed assignment.
  149. 149. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) T : an arbitrary but fixed assignment.
  150. 150. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) T : an arbitrary but fixed assignment.
  151. 151. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) T : an arbitrary but fixed assignment.
  152. 152. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) W fT(W) fT(W)
  153. 153. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) W fT(W) fT(W)
  154. 154. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W fT(W) W fT(W) Fixed T
  155. 155. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f TW (W) : defined by Laguerre diagram fT(W) fixed W W
  156. 156. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f TW (W) = minT fT(W) fT(W) fixed W W
  157. 157. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f: W → f TW (W) is concave !! (because its graph is the lower enveloppe of linear functions) fT(W) W fT(W) f TW (W) = minT fT(W)
  158. 158. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f: W → f TW (W) is concave (because its graph is the lower enveloppe of linear functions) fT(W) W fT(W) f TW (W) = minT fT(W) Consider g(W) = f TW (W) + ∑vj ψj
  159. 159. Part. 3 Optimal Transport – the AHA paper fT(W) W fT(W) f TW (W) = minT fT(W) Consider g(W) = f TW (W) + ∑vj ψj vj -∫Lag ψ (yj) dμ(x) and g is concave.∂ g / ∂ ψ j = Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f: W → f TW (W) is concave (because its graph is the lower enveloppe of linear functions)
  160. 160. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) =
  161. 161. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: G(ψ) = g(W) =∑j ∫Lag ψ (yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj is concave ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) =
  162. 162. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: G(ψ) = g(W) =∑j ∫Lag ψ (yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj is concave vj -∫Lag(yj)dμ(x) (= 0 at the maximum)∂ G / ∂ ψ j = ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) =
  163. 163. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: G(ψ) = g(W) =∑j ∫Lag ψ (yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj is concave vj -∫Lag(yj)dμ(x) (= 0 at the maximum)∂ G / ∂ ψ j = ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) = Desired mass at yj Mass transported to yj
  164. 164. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j =
  165. 165. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x)
  166. 166. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x) yi yj
  167. 167. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x) yi yj ψ j ψ j + δψ j
  168. 168. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x) yi yj ψ j ψ j + δψ j Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x)
  169. 169. Part. 3 Optimal Transport – the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0
  170. 170. Part. 3 Optimal Transport – the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0
  171. 171. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n
  172. 172. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)||
  173. 173. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)|| δh
  174. 174. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)|| δh ∂ h / ∂ ψ j = -1/ || gradx c(x,yi) – gradx c(x,yj) ||
  175. 175. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)|| δh ∂ h / ∂ ψ j = -1/ || gradx c(x,yi) – gradx c(x,yj) || ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫Lag(yi) ∩Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x)
  176. 176. Part. 3 the Hessian ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x) ∂ 2 / ∂ ψ i 2 F = - ∑ ∂ 2 / ∂ ψ i ∂ ψ j
  177. 177. Part. 3 the Hessian ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x) ∂ 2 / ∂ ψ i 2 F = - ∑ ∂ 2 / ∂ ψ i ∂ ψ j c(x,y) = || x – y ||2 ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) 1 / || xj – xi || dμ(x)
  178. 178. Part. 3 the Hessian ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x) ∂ 2 / ∂ ψ i 2 F = - ∑ ∂ 2 / ∂ ψ i ∂ ψ j c(x,y) = || x – y ||2 ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) 1 / || xj – xi || dμ(x) IP1 FEM Laplacian (not a big surprise)
  179. 179. Part. 3 Optimal Transport Let’s program it ! Hierarchical algorithm [Mérigot] Geometry, 3D [L], [L, Schwindt] Damped Newton algorithm, [Kitagawa, Mérigot, Thibert]
  180. 180. Optimal Transport applications in computational physics 4
  181. 181. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Short summary of the 1st chapter of Landau,Lifshitz Course of Theoretical Physics
  182. 182. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system position
  183. 183. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system position speed
  184. 184. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system position speed time
  185. 185. Euler Lagrange ∫t1 t2 L(x,x,t) dt The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Axiom 2: The movement (time evolution) of the physical system minimizes the following integral
  186. 186. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Axiom 2: The movement (time evolution) of the physical system minimizes the following integral
  187. 187. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Axiom 2: The movement (time evolution) of the physical system minimizes the following integral Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = t=0 t=1
  188. 188. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t
  189. 189. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte
  190. 190. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy
  191. 191. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum
  192. 192. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum
  193. 193. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Preserved quantities “Integrals of Motion” Noether’s theorem
  194. 194. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Free particle: Theorem 3: v = cte (Newton’s law I)
  195. 195. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2
  196. 196. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2
  197. 197. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Particle in a field: Expression of the Lagrangian: L = ½ m v2 – U(x) Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2
  198. 198. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Particle in a field: Expression of the Lagrangian: L = ½ m v2 – U(x) Expression of the Energy: E = ½ m v2 + U(x) Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2
  199. 199. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Particle in a field: Expression of the Lagrangian: L = ½ m v2 – U(x) Expression of the Energy: E = ½ m v2 + U(x) Theorem 4: mx = - U (Newton’s law II) Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2
  200. 200. Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. Lorentz change of frame x’ t’ = (x-vt) x γ (t – vx/c2) x γ γ = 1 / √ ( 1 – v2 / c2) The Least Action Principle (relativistic setting – just for fun…)
  201. 201. The Least Action Principle (relativistic setting – just for fun…) Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. Lorentz change of frame x’ t’ = (x-vt) x γ (t – vx/c2) x γ γ = 1 / √ ( 1 – v2 / c2) Theorem 5: E = ½ γ m v2 + mc2
  202. 202. The Least Action Principle (quantum physics setting – just for fun…)
  203. 203. ? T Fluids – Benamou Brenier ρ1 ρ2
  204. 204. ? T Fluids – Benamou Brenier Minimize A(ρ,v) = ∫t1 t2 (t2-t1) ∫Ω ρ(x,t) ||v(t,x)||2 dxdt s.t. ρ(t1,.) = ρ1 ; ρ(t2,.) = ρ2 ; d ρ dt = - div(ρv) ρ1 ρ2
  205. 205. ? T Fluids – Benamou Brenier ∫t1 t2 (t2-t1) ∫Ω ρ(x,t) ||v(t,x)||2 dxdt s.t. ρ(t1,.) = ρ1 ; ρ(t2,.) = ρ2 ; d ρ dt = - div(ρv) ρ1 ρ2 Minimize C(T) = ∫Ω || x – T(x) ||2 dx s.t. T is measure-preserving ρ1(x) Minimize A(ρ,v) =
  206. 206. Le schéma [Mérigot-Gallouet] Applications en graphisme: [De Goes et.al] (power particles) Part. 4 Optimal Transport – Fluids
  207. 207. Part. 4 Optimal Transport – Fluids Let’s code it !
  208. 208. Part. 4 Optimal Transport – Fluids
  209. 209. Part. 4 Optimal Transport – Fluids
  210. 210. Part. 4 Optimal Transport – Fluids
  211. 211. Part. 4 To infinity and beyond…
  212. 212. René Descartes - 1663 Vortices in “ether” ? Part. 4 To infinity and beyond
  213. 213. The Data: #1: the Cosmic Microwave Background COBE 1992Part. 4 To infinity and beyond…
  214. 214. WMAP 2003 2006 2008 2010 Part. 4 To infinity and beyond… The Data: #1 the Cosmic Microwave Background
  215. 215. Part. 4 To infinity and beyond… The Data: #2 redshift acquisition surveys
  216. 216. Part. 4 To infinity and beyond… The Data: #2 redshift acquisition surveys
  217. 217. Part. 4 To infinity and beyond… The Data: #2 redshift acquisition surveys
  218. 218. The millenium simulation project, Max Planck Institute fur Astrophysik pc/h : parsec (= 3.2 années lumières)Part. 4 To infinity and beyond…
  219. 219. Part. 4 To infinity and beyond… The universal swimming pool
  220. 220. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Time = Now Early Universe Reconstruction
  221. 221. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris . . . . . Early Universe Reconstruction
  222. 222. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Time = BigBang (- 13.7 billion Y) Early Universe Reconstruction
  223. 223. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris “Time-warped” map of the universe Early Universe Reconstruction
  224. 224. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Cosmic Microware Background: “Fossil light” emitted 380 000 Y after BigBang and measured now “Time-warped” map of the universe Early Universe Reconstruction
  225. 225. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Cosmic Microware Background: “Fossil light” emitted 380 000 Y after BigBang and measured now Do they match ? “Time-warped” map of the universe Early Universe Reconstruction
  226. 226. Conclusions Open Questions References Online resources
  227. 227. Conclusions – Open questions * Connections with physics, Legendre transform and entropy ? [Cuturi & Peyré] – regularized discrete optimal transport – why does it work ? Hint 1: Minimum action principle subject to conservation laws Hint 2: Entropy = dual of temperature ; Legendre = Fourier[(+,*) → (Max,+)]… * More continuous numerical algorithms ? [Benamou & Brenier] fluid dynamics point of view – very elegant, but 4D problem !! FEM-type adaptive discretization of the subdifferential (graph of T) ? * Can we characterize OT in other semi-discrete settings ? measures supported on unions of spheres piecewise linear densities * Connections with computational geometry ? Singularity set [Figalli] = set of points where T is discontinuous Looks like a “mutual power diagram”, anisotropic Voronoi diagrams
  228. 228. Conclusions - References A Multiscale Approach to Optimal Transport, Quentin Mérigot, Computer Graphics Forum, 2011 Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations Xianfeng Gu, Feng Luo, Jian Sun, S.-T. Yau, ArXiv 2013 Minkowski-type theorems and least-squares clustering AHA! (Aurenhammer, Hoffmann, and Aronov), SIAM J. on math. ana. 1998 Topics on Optimal Transportation, 2003 Optimal Transport Old and New, 2008 Cédric Villani
  229. 229. Conclusions - References Polar factorization and monotone rearrangement of vector-valued functions Yann Brenier, Comm. On Pure and Applied Mathematics, June 1991 A computational fluid mechanics solution of the Monge-Kantorovich mass transfer problem, J.-D. Benamou, Y. Brenier, Numer. Math. 84 (2000), pp. 375-393 Pogorelov, Alexandrov – Gradient maps, Minkovsky problem (older than AHA paper, some overlap, in slightly different context, formalism used by Gu & Yau) Rockafeller – Convex optimization – Theorem to switch inf(sup()) – sup(inf()) with convex functions (used to justify Kantorovich duality) Filippo Santambrogio – Optimal Transport for Applied Mathematician, Calculus of Variations, PDEs and Modeling – Jan 15, 2015 Gabriel Peyré, Marco Cuturi, Computational Optimal Transport, 2018
  230. 230. Online resources All the sourcecode/documentation available from: http://alice.loria.fr/software/geogram Demo: www.loria.fr/~levy/GLUP/vorpaview * L., A numerical algorithm for semi-discrete L2 OT in 3D, ESAIM Math. Modeling and Analysis, 2015 * L. and E. Schwindt, Notions of OT and how to implement them on a computer, Computer and Graphics, 2018. J. Lévy – 1936-2018 - To you Dad, I miss you so much.
  231. 231. Bonus Slides The Isoperimetric Inequality
  232. 232. The isoperimetric inequality For a given volume, ball is the shape that minimizes border area
  233. 233. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Explanation in [Dario Cordero Erauquin] course notes The isoperimetric inequality
  234. 234. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Explanation in [Dario Cordero Erauquin] course notes The isoperimetric inequality
  235. 235. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Explanation in [Dario Cordero Erauquin] course notes The isoperimetric inequality
  236. 236. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Consider a compact set Ω such that Vol(Ω) = Vol(B2 3) and f = the indicatrix function of Ω The isoperimetric inequality
  237. 237. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Consider a compact set Ω such that Vol(Ω) = Vol(B2 3) and f = the indicatrix function of Ω Vol(∂Ω) ≥ n Vol(B2 3)1/3 Vol(B2 3)2/3 The isoperimetric inequality
  238. 238. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Consider a compact set Ω such that Vol(Ω) = Vol(B2 3) and f = the indicatrix function of Ω Vol(∂Ω) ≥ n Vol(B2 3)1/3 Vol(B2 3)2/3 Vol(∂Ω) ≥ 4 π = Vol(∂B2 3) The isoperimetric inequality
  239. 239. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] The isoperimetric inequality
  240. 240. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 The isoperimetric inequality
  241. 241. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 The isoperimetric inequality
  242. 242. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality
  243. 243. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ Arithmetico-geometric ineq: det (H) 1/n ≤ trace(H)/n if H positive The isoperimetric inequality
  244. 244. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ Arithmetico-geometric ineq: det (H) 1/n ≤ trace(H)/n if H positive det (Hess Ψ) 1/n ≤ trace(Hess Ψ)/n The isoperimetric inequality
  245. 245. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ Arithmetico-geometric ineq: det (H) 1/n ≤ trace(H)/n if H positive det (Hess Ψ) 1/n ≤ trace(Hess Ψ)/n det (Hess Ψ) 1/n ≤ ∆ Ψ / n The isoperimetric inequality
  246. 246. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality
  247. 247. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality
  248. 248. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality
  249. 249. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ ∫f ∆ Ψ = - ∫ grad f . grad Ψ det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality
  250. 250. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ ∫f ∆ Ψ = - ∫ grad f . grad Ψ ≤ ∫ | grad f | (T = grad Ψ Є B2 n ) det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality
  251. 251. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ ∫f ∆ Ψ = - ∫ grad f . grad Ψ ≤ ∫ | grad f | (T = grad Ψ Є B2 n ) ∫ | grad f | ≥ n Vol(B2 n)1/n ■ det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality
  252. 252. Bonus Slides Plotting the potential & optics
  253. 253. The [AHA] paper summary: • The optimal weights minimize a convex function • The gradient and Hessian of this convex function is easy to compute Note: the weight w(s) correspond to the Kantorovich potential ψ(x) (solves a “discrete Monge-Ampere” equation) The algorithm: Summary: The algorithm computes the weights wi such that the power cells associated with the Diracs correspond to the preimages of the Diracs. μ ν Plotting the potential, “optics”
  254. 254. The [AHA] paper summary: • The optimal weights minimize a convex function • The gradient and Hessian of this convex function is easy to compute Note: the weight w(s) correspond to the Kantorovich potential ψ(x) (solves a “discrete Monge-Ampere” equation) The algorithm: Summary: The algorithm computes the weights wi such that the power cells associated with the Diracs correspond to the preimages of the Diracs. μ ν Plotting the potential, “optics”
  255. 255. The [AHA] paper summary: • The optimal weights minimize a convex function • The gradient and Hessian of this convex function is easy to compute Note: the weight w(s) correspond to the Kantorovich potential ψ(x) (solves a “discrete Monge-Ampere” equation) The algorithm: Summary: The algorithm computes the weights wi such that the power cells associated with the Diracs correspond to the preimages of the Diracs. μ ν Plotting the potential, “optics”
  256. 256. The [AHA] paper summary: • The optimal weights minimize a convex function • The gradient and Hessian of this convex function is easy to compute Note: the weight w(s) correspond to the Kantorovich potential ψ(x) (solves a “discrete Monge-Ampere” equation) The algorithm: Summary: The algorithm computes the weights wi such that the power cells associated with the Diracs correspond to the preimages of the Diracs. μ ν Plotting the potential, “optics”
  257. 257. The [AHA] paper summary: • The optimal weights minimize a convex function • The gradient and Hessian of this convex function is easy to compute Note: the weight w(s) correspond to the Kantorovich potential ψ(x) (solves a “discrete Monge-Ampere” equation) The algorithm: Summary: The algorithm computes the weights wi such that the power cells associated with the Diracs correspond to the preimages of the Diracs. μ ν Plotting the potential, “optics”
  258. 258. Plotting the potential, “optics”
  259. 259. Plotting the potential, “optics”
  260. 260. Plotting the potential, “optics”
  261. 261. Plotting the potential, “optics”
  262. 262. Plotting the potential, “optics”
  263. 263. Plotting the potential, “optics”
  264. 264. hi Plotting the potential, “optics”
  265. 265. Plotting the potential, “optics”
  266. 266. Plotting the potential, “optics”
  267. 267. Plotting the potential, “optics”
  268. 268. Plotting the potential, “optics”
  269. 269. Plotting the potential, “optics”
  270. 270. Numerical Experiment: A disk becomes two disks Plotting the potential, “optics”
  271. 271. End Of Line.

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