Computational
Optimal Transport
http://optimaltransport.github.io

Introduction
Gabriel Peyré
www.numerical-tours.com
ÉCOLE NORMALES U P É R I E U R E

https://optimaltransport.github.io

Probability Distributions in Data Sciences
Sliced Wasserstein projection of X to style
image color statistics Y

Probability distributions and histograms<latexit sha1_base64="NHBrzuHY/DIiTVSeYz9st2EbBcI=">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</latexit>! images, vision, graphics and machine learning, . . .

Source image (X )Source image after color transfer
Style image (Y )J. Rabin

Wasserstein Regularization

Probability Distributions in Data Sciences
Sliced Wasserstein projection of X to style
image color statistics Y

Probability distributions and histograms<latexit sha1_base64="NHBrzuHY/DIiTVSeYz9st2EbBcI=">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</latexit>! images, vision, graphics and machine learning, . . .

Source image (X )Source image after color transfer
Style image (Y )J. Rabin

Wasserstein Regularization

Unsupervised learning
def. 1 Pn
Observations:= n i=1 xi<latexit sha1_base64="NVr4d97mahJgu8Mr3K/nifIrenM=">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</latexit><latexit
sha1_base64="NVr4d97mahJgu8Mr3K/nifIrenM=">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

Parametric model: ✓ 7! ↵✓<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit><latexit
sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">AAA/qHiczVtbdxu3EUbSW6xe4rSPfei2shs7dX1kNeekbU6OYl0sK6Yt2RJ9SRirvKyojZdcencpW2aZP9PH/oT+gr62T/0H9U/oW2cGwAJLYnewbnpOhCMJC+KbGQyAuQDL3iSOsnxt7V9vvf2d737v+z9458LKD3/045+8e/G9nz7MkmnaD9v9JE7Sx71uFsbROGznUR6Hjydp2B314vBR79kWfv7oLEyzKBkf5eeT8MtRdziOTqJ+N4em44ufdPLwJeBmB920OwrzNOoHo2QQxn+cB5c6+WnQGXUnWZ4Esw4xm6XhYN7pxsfw2fzS8cXVtetr9BMsV26oyqpQPwfJe7/4QHTEQCSiL6ZiJEIxFjnUY9EVGZQvxA2xJibQ9qWYQVsKtYg+D8VcrAB2Cr1C6NGF1mfwdwhPX6jWMTwjzYzQfeASw28KyEBcBkwC/VKoI7eAPp8SZWytoj0jmijbOfzvKVojaM3FKbRyON3THzeA1hOQDTVTP+Yc+v2exhpB7wm1oBb6ituUtIcjDKzR50BhAm1YH8DnKdT7hNTzERAmIx3hHHTp839TT2zF577qOxWvaTTfFv19+/RyuUEJxJ64J7ZES7TFttgRhyRvfVkh3BaMYwI6TkHyIWgIpb0C0lyF/+uwp9bER1DbBRl71CeEsQTiAP6fk/wrStZA3BU3xZG4TbyvQP0u1K42mCeup17bGc2G/7pZYVZZSjNyXitBTuOfkgVIxcvavrh2Bmo2T2g9ZGpH1s+p1OCWeCD2a+cPOY+B/guyRCPSIHKcQfsQZMIVGgM/tIm4lv8qvharUP9a/E2t4wDkStXKPiV68smf5p+gIM33oczFBaLZBqlXilEG8CRnIClWf1ec0d4IyJJP4TO5l2Jae2PqVTe6e1DmVNOrpgdlRq3zWuQWFBdyi0XuQnEhd1lkC4oL2WKRB1BcyAOFrMPeh+LC3me5PoDiQj5gkUdQXMgjFvkQigv5kEV+DsWF/JxF3oLiQt5ikXeguJB3WGQbigvZZpE7UFzIHYWs3mto1xKiEzH76ibUyzzQT8XQcpOVb5NsrAu76bEr+xVYfl9uw383dttDp2EFdsdj9ZxUYPn1swtWzo3lrclt8pEu7G0WuwcrwI3dY7Gfia8qsJ957JdnFVh+x7TI37iwvP28C09u7F0Wew9qbizvZfahxY3d97D5kwrsAYu9L55XYH2sflqB5e3+IcViLuyhh8/IK7C812hD7ODG8vb0IcQgbizvcx5Bqxv7iMU+pqjRhX3MYp9QdOrCPvHwk68qsNpTyuh9SDFgCDu2jlq32JVYmwC1LsM/LnwL1tBHDVjMsMAMCTNiEbsFYtcT0SoQLW+5ssKOZhSx8lwOC8ShJ6JX+Cas5Wz/QdF/QLkZj9guENsLiLosAOdaj+WMogvdwiHzwnNhzWdMSWG/sRaq9VBveTViv4SQa/uUVv41ytUxkxlQ3lpN7bTw8RIZ0HMd4gXlenqUmgePywurYKNesqieA9VjUecO1DmLmjpQUxZ15kCdsSiz821cx2MFGP3jXMzoSef+3FlK+exiHzzuDng/bNmH/z5nKfVZOZ4IoKfU2bOxxSnUZpRJm8xum3JkedIQgmSy5746Y8InPJucqV0n7fC88OWB0Cee/nQikmdY0MF4MaAd1YzOHWqZU3wna83wt4udr2vN8Duk8TnF8bLWDJ8r6fM3kP1IYY/eAHsI+2mitG/qTWnIMxRJQ9c527xX2E08r3xJO0e2NeW/RTWpA1NvRiOzxpCVxtCEhtFlZumyGRWMkWRsq2tB45GMVXZr6k1lSMhXjpUc5qnpzGCfgZoZXW9G4wDiqi3KrGdWvekKnRSjMfVmNB4KebY+p3hd15vRGNKz1IepN6OBZypdlc2belPrjRqQGbKp65glpShGnzhH5L3rz1LsiHzZx+CJytMigq+nZCLPajq9ws/US7RoXZrIgd5/akVIZRozsc5mP1KGvOR7l+kY/4uab4EWA9i18lyfO5OOQUJ9YhDSKflTolaPKY9M49ZZHFqSkwVUR7XmbCxn+MoznXLbMbVyWZMZrdFjh+xsRmtvQhFbizTL6aFVOcNVFDkNtUoa4uk10d0r2sHJgvbXWNxkATEpVlqfbofkLXV9FunS+qGl48vqFiWHIu9UzPrFs+ATwqWUkSRkbVCWOp52P33KY7ehP7wmzAm0/CygGUV7dUZWI6Ibn4yNQ/RZroyUZ/RsaLfpnq18axUoKhMhb1TxjBtPuwO6z7RtLccb9aXPz2Q9I6tr7tHrb8cMeuhAN80+tsBX3IPaEUTzbXg68sg/zJ1bQvpOxW+L+8+E5q8+27Zv9joFDWltwpJ9rMuAT0tUXgAa14LMoP1pLNLR+M4SJT4jd8lj8sqy3b9Md/r6zZEurfDqtVx9SjIgruvENaA9I+/75dMiBynBzPnJOkWd9aNEfk04ogXluD61OEu9jOldmpByywnFszHtNW5vlHvbZ0eLn2hOB0K/VYF3xQnZx4CsXwDeKaE1GdCv/VaOfrdC2oOYLKSP1YmK2MYV6UTsGotol8s1It8XMustJEs2Jf6arr27MlqLMs6XXmC+sLa1TloUCYbENVW23eztet+DSPMmhL1KJEWzVq4Q/6v0V//qdbK6tCJQwzgDmbJ0rvlIKNNAHXXJx9fbIN3XlvJSIcNTJbXxfkamSyXJtilPQnnQVw+Ac5+eJS9cJSnJnS31kV607qQVKU8W9IijRe/aUVZ/qPwvyn2NfOQq7bmOMG/Y6Nhf9+VOeBf51vMqU/ejnf1fqBtdl7WGFANhTlelhriz95ByLFvKmN5Jkm/fhIrSstbThV71fMa0FkfWXv4ztP4S/mq59bMfnV7JKmzSGpAUzJPRiGwJlnr48dos8dIrU9Myz4afWZO6l93yJlmxtG4mMz5rTOWAVs1Lddag6z7jP6I7PKMB3a6tyPFCBH2k7vx8qTej7U95ylLmYyeNijyktHMeP6oDliqfi2vUK5bWGksL38azT9Tt3Wn7zlsUO/QpwpHZwYB2QUQxjW6tz4MkBeT1obJf9u7qUAvupR5ZKKRcfqsQz+76Qr5DKmX8tfIcCdlRs+P0OzsvVB9twzpU/90SckTZbEZ7RyM+pB6hkt+WI1jY8dctnx7QmXeXYhbp1+szUru3mYWg5K9NNif3guEVi/dVND6mOcjF61pue0v54Z6VIQZC3vfFRQSPs/y60Sxj3iZXhs7V3dFbRrOEniSlfF5GjT2yUcunDPIWawza1+vuOkk9E594WAeZV5qdYq8v/GyqshIc2wfQEzVvZt7Vg+cXe3Pk+L3JfVaXPMdIxYGzhec3o9VVnqT8XKeH6QJfo4+pGKucUEfvJpMqYzriY28uUqJmXCTGh0uzUTSRv5nkTWSW9za+lHVvTbmczUsbc0o5CfceJCJcEdQVZ8R0lRlHb4lej7A2NdnCUcLzrkTl4La1xZOfC0u+SLZeqPVIseWNqryFpm57DGPDpYUMyfrFgjsXkb37Qr6PWpVB2bnBx1DwbyBc+ZP2Jj6na4cQReJ79X7fiqi/yX+u6vLEL6AW3MmDhey0q8ZZ7lE/y88t6jZ9Hw7+PCLQNSd9RH6wqeySMi+5Td2f/gvaw6kIWelNz+ZjsLnwI1nm1GQ8EdklfjSR0N9jajoWzcFnJGUu/nzkuT83ihOhv0/XbAyaOj+CMocmPPT9vN+cm97Nedmc6vW1zMWXh7Th+kZC4/BmrDrbMP18LFRqzcg3zwGtw0kNde0t/tdxaD6GU3Nevtwy+q7TVx6zLvuF6swSo9nme8Zw81nN1Rz9eSbF6Eys4+Yno7ag0Uwl1mi+efoYTZo1oHnNhDwp5KWTeHsVGXl9qeDJuUuGRPzHSwaJr5LBl4o+wa+mpHvw1PQ3/Vyj0p/5yGToVMlUpob0ZKbdoT0c0mmBudkZ0HPxtuXxxdUbi986X66016//4fqN++urn26qL6S/I34ufgWB/A3xkfhU3IYAsw0i/EX8XfxD/HPjNxv3Nx5tPJFd335LYX4mSj8bvf8C/bg34w==</latexit>↵✓Probability Distributions in Data Sciences
Sliced Wasserstein projection of X to style
image color statistics Y

Probability distributions and histograms<latexit sha1_base64="NHBrzuHY/DIiTVSeYz9st2EbBcI=">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</latexit>! images, vision, graphics and machine learning, . . .

Source image (X )Source image after color transfer
Style image (Y )J. Rabin

Wasserstein Regularization

Unsupervised learning
def. 1 Pn
Observations:= n i=1 xi

Parametric model: ✓ ! ↵✓Density fitting:<latexit sha1_base64="ulCUACOhTCX5a4s1cPqdMYjWUUA=">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min D(↵✓ , )<latexit sha1_base64="9IYkKnkiR+VYnIAjCg9CiCnhkKw=">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</latexit>

✓

↵✓
D

Overview• Monge Formulation• Continuous Optimal Transport• Kantorovitch Formulation• Applications

Fom bakeries to cafés

30

10

mi n

12 m i n

27 m

31
m

in

in

m

in
cij

y1

y2

y3

y4

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

y5

y6

34 15

Fom bakeries to cafés

Cout:

cij

y1

y2

y3

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

10+7+15+10+14+9 = 65 min

y4

9

y5

y6

34 15

From best to worst

Coût=64

Coût=65

Coût=66

Coût=152

Combinatorial Search
Cost (ci,j )ni,j=1 2 Rn⇥n<latexit sha1_base64="thcVLe8+VbtQ1j6L+GfVIruzbdg=">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
sha1_base64="thcVLe8+VbtQ1j6L+GfVIruzbdg=">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</latexit>

Permutations:def.
⌃n = { : {1, . . . , n} ! {1, . . . , n}}Monge’s problem:Pnmin
i=1 ci, (i)<latexit sha1_base64="moOTTFH1ZhSL0jeHC+A3a8JRR18=">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</latexit><latexit
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2⌃n

i

ci,j
j

Combinatorial Searchi

Cost (ci,j )ni,j=1 2 Rn⇥nci,jjPermutations:def.
⌃n = { : {1, . . . , n} ! {1, . . . , n}}Atoms in the universe: 1079

Monge’s problem:Pnmin
i=1 ci, (i)<latexit sha1_base64="vdJqxM7ekgKL9j16SoToRrNKVf4=">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</latexit><latexit
sha1_base64="vdJqxM7ekgKL9j16SoToRrNKVf4=">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</latexit>

2⌃n
n
0
1
2
3
4
5
6
7
8

n!
1
1
2
6
24
120
720
5 040
40 320

n
9
10
11

n!
3628803628800
39916800

12…
25479001600…
1,551x1025

…

…

70

1,198x10100

Neurons in the brain: 1011 .
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Gaspard Monge (1746-1818)(1784)

1-D Optimal Transportmin<latexit sha1_base64="728Z8fBn49NzO1ft9oZkwnWJbMw=">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</latexit><latexit
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x4 x3

y1

1-D Optimal Transportmin<latexit sha1_base64="728Z8fBn49NzO1ft9oZkwnWJbMw=">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2⌃n

n
X
i=1

y2

y (i) |p ,

p>1

x1

x5

y4

|xi

xxx

x 2 y3y5x4 x3

Sorting algorithms: insertion n(ny11)/2 worst case.

<latexit sha1_base64="aiEeh1bwVYal7uZhpfXFWGXXd7M=">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</latexit><latexit
sha1_base64="aiEeh1bwVYal7uZhpfXFWGXXd7M=">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</latexit>nn!

n(n-1)/2

n log(n)103628800

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)).
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1-D OT Interpolation

Grades /20

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y3 y4 x 2
x3

x1

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5

y2

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y5 Grades 1

y1

20

Grades 2

Grades 2<latexit sha1_base64="LlyYO5VvZANbuESiDr4wsnjZUj4=">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1-D OT Interpolation

Grades /20

0Grades 1<latexit sha1_base64="iE5wS1qBLRXsWTbsZCoShktKZbg=">AAA9j3ictVttcxu3EYbTl8Tum5P2W2c61yruJB3HI6mZaToZdWxLsqyYsWWTsp2EtocvJ/rsI4++IyXbDP9Lv7Z/or+j/6D91L/Q3QVwwJG4W0B1dSMJB+J5drEHLHaBY3+aJsVsc/OfF977wQ9/9OP3P7h46Sc//dnPf3H5w48eFtk8H8THgyzN8sf9XhGnySQ+niWzNH48zePeuJ/Gj/ovd/HzR6dxXiTZpDN7M42fjHujSXKSDHozqHp2+Vdd4ljk8XAZHeS9YVxEW88ub2xe26SfaL2wpQobQv0cZR/+Zlt0xVBkYiDmYixiMREzKKeiJwq4vhNbYlNMoe6JWEBdDqWEPo/FUlwC7BxaxdCiB7Uv4e8I7r5TtRO4R86C0AOQksJvDshIXAFMBu1yKKO0iD6fEzPW1nEviBN1ewP/+4prDLUz8RxqOZxu6YvDvszEifiC+pBAn6ZUg70bKJY5WQU1j6xezYBhCnVYHsLnOZQHhNR2jghTUN/Rtj36/F/UEmvxfqDazsW/ScsrcEWirXqflQw9cUr8ET3NOXwm9UlB8ggYYtVHLJ2RrcfU+wm0X0D9XbiWVNI26cO1oNplI3IXLhdyl0UewOVCHrDIFlwuZItFHsHlQh4pJGJzsrkb34bLhW+zku/D5ULeZ5EP4HIhH7DIh3C5kA9Z5LdwuZDfsshbcLmQt1jkHbhcyDsssgOXC9lhkcdwuZDHLHIfLhdyXyHrZ2oOV0Y8CTMrb0C5KgM9RQo1N1j9bpJ3dGFveszpQQ2Wn9V78N+N3fOwaVyD3fcYdyc1WH7kHYCPdGN5X3SbVhMX9jaLPYQR4MYestivxIsa7FceM+1lDZafay1o58by3vdruHNjv2axd6HkxvJr1D2ocWPveawY0xrsEYu9L17VYH28fl6D5f1+G/yKG8uvUx1o78b6eNN5DZb3pw8hgnFj+dXqEdS6sY9Y7GPxugb7mMV+A97djf3GY4V9W4PVa+wlWkFGFI/EMGOb2HrlrMTSFNh6jPy0XFtSio37UM9hRiVmRJgxizgoEQeeiFaJaHnrVZR+tKB4l5fSLhFtT0S/XJuwNGPbD8v2WEo9EHslYm8F0RSR4rPWfTml6ELXcMhZuXJhyadPWem/sRSr8dDseTXiXgUhx/ZzGvlXKVvCDAot1cT2vFzjJTKi+ybEGWVvupdaBo+blV7BRr1mUX0Hqs+i3jhQb1jU3IGas6hTB+qURZmZb+O6HiPA2B+fxYLu5AiQMXL9FUFUcANWndswRyMYP0cQBT6gmnvwv025N3c1aYbZPK6TuMvxpOKJcygtxAbUm6xwj/LrlGZYDJrJlvdUjo93uLexUHNOeuFluZJH5Y6JP09C+oxKHowWI5pPYTx3qGZJ0Z0sheFvl/Nel8Lw+2TxJUXxshSGnyntZ+fQvaOwnXNg2zCbpsr6phzKIfdfJIcuX6JVFz0uPtWxGjPI9zqQ/1A9mcNzPJddKkn7mHIYR2H1r6j0L4TD2Lmw7BzGgtGTjHp1KQruyUTlvaYcqkNGq+hE6WHuQp8MthmqJ6PLYRxHEHHtUs69sMqho3da9saUwzgeCrnvuaRIXpfDOEZ0L+1hymEcuNvSU3m+KYd6drSAzJ1NOdSrT2gXGPeA5JiXNSYqyilOmiu2hOKD5t0aO+ZfX8dwz+ZpmSM0M5nYtp6nX65lzRrpeCEGrzYL1APji7kVg1U5FmKbza+kDrPK+r7OY9Z4tHwLrBjB7JdnANyeeQoa6j0J9N4pMG6xWVe1Zxq3zeJwlJysoLqqdsZGi0au3DWq1j2jWi4vM701duySvy5o7E0pJmyRZTk7tGqfcB0jZ6FWxUI8X4jt3qr5WrX+JoubriCm5Ugb0ImQPElrzlNdVm9bNr6iTnlmcMkzHzN+cbf5RHkbzHky8kWoS5NMu53eR7LrcF29Kswet/wsoieK/uqUvEZCJ1IFm4Xq3WIZjS/o3nAf05kcypAcA3iOkWKZCnlqhrvouJ8ekUe1/S0nG+2ld+hkuSCvq/1xM3pkoUcOdHiOswsrxl0odSBnOIa7jkeWc6m0VUYWz8Vn5eloRk+wOaNPKx5Sc0h/E1c8ZFOW/bzCcgZoHA0yS/fnWOXR+O4aE5/1u/QxuWvV81+hk1t9vt2jMV4/mut3YoYkdZukRjRr5KmuvFuVIDVYOD/Zpvi1uZcoL0Qi+lBO6lNLsrTLhE78Y8pgpxQZpzTbuNlRbW3vT61+oiUdCX12jqfZGXnIiPxfBOtTRmMyol/73QF9gi49Qko+0sfvJGV044p1EnaMmTguEfKtBjPeYvJlc5Kvee3ZVdBYlBmDXAeWK2Nb26RFsWBMUnPl3c3cbl59EGnek7BHiWQ0Y+UTkv8p/dW/epxsrI0ItDA+gUL5OtfzyChnQRv1aJVv9kG6ra3lx6UOT5XWZv0zOn1c0WyPMi7UB1frIUge0L2UhaMkJ72LtTZyHW3azUXm6YodsbcnlMVLvz9SKzDqfZVWyQ2ac10aJSMYBbMyi9BtuV3kVbnNsqrsftzF/4Xd2LpqNWSMhNnBlRbi9vdjytZsLVMY1XL8vqTZ5LZ6vtKqWc6ExuLYmsvfQ+1v4a/WW9/78fQrXuEmjQHJYO6MRWRNtNbCT9bNiiw9MjWXuTfyzJjUreya8+TX0ruZHPs0mOWIRs1rtWuhy+fheGFxvPC0YYfOGo0Vdb32RM/Y3KKjTit95YVI6wQwz1lmPiLTqMRDSzuX8mMdsqx8jq9Rb1muTZarB7PVPg2w57wP0j3XV2f39+XqHolbFNsMKAKT+cuQZmlCMZeubc7UJANK/lz5V3v2d6kGpffJgyKzfI8TZ4w8dRrQtSw1/b1a2TLy88Yj6PeWzlQb7WO7VP7jGnJMc6KgeakRn1OLWOlv6xGteKRrVswR0c5/j2IqGXc058x2a/NMoko8YfJNOauMLJkpTMj+3M7b4Vr2emjlrxHlhHMVXfeBK/wJI4PE6J0Ed2RZ0BPCVU6eJMiItk/+c91PyVO8iaXRNdJ6IXY8fIzMes1Yt8eW7rHu2x+gJVrdPHVXC15e6i2Rk3eeE70erWpjFaMuVu7Px9VTq1z1vskO8xW5xh5zamNnFibLq2K64ktvKVKjMCkS4yMlrBch+odpHqKzPJ3yZdatNXN1p0H6mOeUL3HvgSLCFd194ozmPmX60V/j6xPWZpM1HBPuxmVqf8D2tLgrdXFtHZK1FxtXo9RaiepWCs1urxbGf0sPGZP3SwW3ZyNb27p3K1kKvwsjGQZCvtFblx/anF/ChX8j4coOtUSfvcM2xLc3xK7YfwdvQ7xSZbmjGVEN+oLhSu7dU/2stmi20SuL3eb3keAvIwFbc9ontJKG6i6Zec1tdn/+M/ICuYhZ7U3L8D7YUvierEsK6U9Cno3vTSL0d3FC+6Il+PSkKsVfjjzX4HpxIvR3msL6oNn5HlQlhMjQ7zH4PXPTOlyWLanZXutSfGXIVUCfuGgcnvzV5yqmnY+Hyq0n8u4loHc4aWDXq8X/2g8tx0gKl+UrraDvmr3weOqyXax2ZDEeDp8zRprPaK6X6C8zK3tnoiW3PBn3RUFPKrN68+75MR41Y0DLWgi5D8prJ/H2KDL6+rLguYBLh0z8R/zjAv9thFclR50eIUz6nKKeTbfg2fQ3Ll2905/56GR46nSqspk8ok1vxO6KQ3ELfnfLCDD07VD5XUr5H7Hu788OofaEvIfeRZc7B12qi2n3w5yiDele7TE+u7yxtfot5PXC8fa1P1/bur+9cf2m+oLyB+LX4neQlmyJP4nr4jZ09xhUeiv+Kv4m/r7z0c4XO3/ZuS6bvndBYX4pKj87h/8FIzbbxg==</latexit><latexit
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y3 y4 x 2
x3

x1

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5

y2

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y5 Grades 1

y1

20

Grades 2

Grades 2<latexit sha1_base64="LlyYO5VvZANbuESiDr4wsnjZUj4=">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Grayscale Histogram Equalization

Grayscale Histogram Equalization

Application: Grayscale Histogram Equalization[~,I] = sort(f(:));
f(I) = linspace(0,1,length(f(:)));

f0↵0

f[argsort(f.flatten())] = np.linspace(0,1,n*n)

f 14

f 12

f 34

↵ 14

↵ 12

↵ 34

f1
↵1

In 2-Dxi , yj 2 R2qyj || = (x1ici,j = ||xin = 10xjyj1 )2 + (x2in = 70yj2 )2xi

n = 300

Proposition: two segments never cross.
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xi

xj

xi

xj

x i0

xj 0

x i0

xj 0

Sliced Wasserstein projection of
In 3-D: Color Image image
PaletteEqualizationcolor
statistics Y

Optimal transport framework Sliced Wasserstein projection Applications

Application to Color Transfer

optimal
Source image (XApplication)
to Color Transfer
transport
Sliced Wasserstein projection of X to style
Optimal transport framework Sliced Wasserstein projection Applications

image color statistics Y

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Sliced Wasserstein projection
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Style image (Y )Source image after color tra
J. RabinStyle image (Y )Wasserstein Regularization

Color Image Palette Equalization
OptimaltransportColor Image Palette Equalization
Optimaltransport

Overview• Monge Formulation• Continuous Optimal Transport• Kantorovitch Formulation• Applications

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X = RdProbability Measures

Positive
Radon measure
on a metric space X .
2.2. Assignment
and Monge↵ProblemP↵ = i ai xid↵(x) = ⇢↵ (x)dx

Discrete d = 1 Discrete d = 2 Density d = 1

Figure 2.1: Schematic display of discrete distributions – =a ” (red cori=1 i xi
responds to empirical uniform distribution ai = 1/n, and blue to arbitrary distributions) 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 ).

qn

is the dimension), can have a density d–(x) = ﬂ– (x)dx w.r.t. theLebesgue measure, often denoted ﬂ– = d–dx , which means that⁄⁄9

Density d = 2

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X = RdProbability Measures

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2.2. Assignment
and Monge↵ProblemP↵ = i ai xid↵(x) = ⇢↵ (x)dx

Discrete d = 1 Discrete d = 2 Density d = 1

9Density d = 2

qn
RFigure 2.1: Schematic display of discrete distributions – =a ” (red cori=1 i x
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↵X
Change of variables:.
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T ]↵
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y = T (x)def.

= T ]↵()

Densities d↵dx = ⇢↵ :Y

g(y)d (y) =XEg(T (x))d↵(x)⇢↵ (x) = | det(@T (x))|⇢ (T (x))

YContinuous Monge’s Problem

GaspardMonge

inf=T ] ↵Z↵c(x, T (x))d↵(x)
X

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Discrete case:Pn↵ = i=1 xiPn= j=1 yj

min

2⌃n

n
X
i=1

Ci, (i)T : xi ! y (i)Ci,j = c(xi , yj )Continuous Monge’s Problem
infGaspardMonge=T ] ↵Z↵c(x, T (x))d↵(x)
X

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Discrete case:Pn↵ = i=1 xiPn= j=1 yj

min

2⌃n

n
X
i=1

Ci, (i)T : xi ! y (i)Ci,j = c(xi , yj )Non-symmetry, non-existence, non-uniqueness:
T

↵

T

T

↵

↵

T0

?

↵

↵

Brenier’s Theorem
Hypotheses:c(x, y) = ||x<latexit sha1_base64="dLOWuHxfzRpj9kO3xo3BkczKa6k=">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] ↵↵<latexit 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min

y||2

d↵
dx = ⇢↵ density.

Zc(x, T (x))d↵(x)
X

TBrenier’s Theorem
Hypotheses:c(x, y) = ||x<latexit sha1_base64="dLOWuHxfzRpj9kO3xo3BkczKa6k=">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] ↵↵<latexit 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min

y||2

d↵
dx = ⇢↵ density.

ZTc(x, T (x))d↵(x)
X

Theorem: [Brenier, 1991]
There exists a unique Monge map T .
It is the unique T = r' such that' is convex and (r')] ↵ = .

YannBrenierBrenier’s Theorem
Hypotheses:c(x, y) = ||x<latexit sha1_base64="dLOWuHxfzRpj9kO3xo3BkczKa6k=">AABFZHictVzvctu4EUeu/67ptc31pp8602HPSeeu47q2e5125uZmLrEdxxcncSLZyV2UeEiJkpXQokJKihOdXqJP02+d9i36Ap32U1+hi12AACWQC7qpObZACL/dxRJY7C5AR+NkmE82N/9x5b3vfPd73//B+z+8+qMPfvyTn1778GcneTrNuvFxN03S7EkU5nEyHMXHk+EkiZ+Mszg8j5L4cfRyR37/eBZn+TAdtSdvxvGz83AwGvaH3XACVafX1q93+lnYnXd6nTDZWcBncLEIvgg62Vl6Ose660EvHuXDyZuN02trmxub+BOsFrZUYU2on6P0w4+viY7oiVR0xVSci1iMxATKiQhFDtdTsSU2xRjqnok51GVQGuL3sViIq4CdQqsYWoRQ+xL+DuDuqaodwb2kmSO6C1wS+M0AGYgbgEmhXQZlyS3A76dIWdZW0Z4jTSnbG/iMFK1zqJ2IM6jlcLqlL072ZSL64k/YhyH0aYw1snddRWWKWpGSB1avJkBhDHWy3IPvMyh3Ean1HCAmx75L3Yb4/b+wpayV913Vdir+jVLegCsQLdX7tKAQihnSD/BpTuE7kicBzgOgEKs+ytJr1PU59n4E7edQfx+uBZa0TiK45li7qEXuwOVC7rDIfbhcyH0WeQiXC3nIIo/gciGPFFJiM9S5G9+Cy4VvsZwfwuVCPmSRj+ByIR+xyBO4XMgTFvkNXC7kNyzyNlwu5G0WeRcuF/Iui2zD5UK2WeQxXC7kMYvcg8uF3FPI6pmawZUinSEzK29CucxDWooEam6y8t1C6+jC3vKY090KLD+rd+HTjd310Glcgd3zGHf9Ciw/8vbBRrqxvC26g6uJC3uHxR7ACHBjD1jsV+JFBfYrj5n2sgLLz7VDaOfG8tb3Hty5sfdY7H0oubH8GvUAatzYBx4rxrgCe8RiH4pXFVgfq59VYHm73wK74sby61Qb2ruxPtZ0WoHl7ekJeDBuLL9aPYZaN/Yxi30iLiqwT1js12Dd3divPVbYtxVYvcZexRVkgP5IDDO2jlpYzEpZGgO1kOGfFGtLgr5xBPUcZlBgBog5ZxH7BWLfE3FYIA695coLO5qjv8tzaRWIliciKtYmWZqw7XtFe1lKPBC7BWJ3CVHnkcpnrfsyQ+9C13DISbFyyZJPn9LCfstSrMZDveXViAclBI3tMxz56xgtyQhKaqqO2lmxxhMywPs6xGuM3nQvNQ8eNymsgo26YFGRAxWxqDcO1BsWNXWgpixq5kDNWJSZ+Tau4zECjP7ls5jjHY0A8pGrrwC8gpuw6tyBORrA+DkCL/
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] ↵↵<latexit 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min

y||2

d↵
dx = ⇢↵ density.

ZTc(x, T (x))d↵(x)
X

Theorem: [Brenier, 1991]
There exists a unique Monge map T .
It is the unique T = r' such that' is convex and (r')] ↵ = .

YannBrenier! Monge-Ampère equation (non-linear, degenerate elliptic).

⇢↵ (x) = | det(@ 2 '(x))|⇢ (T (x))

s.t. ' convex.

Regularity Theory! Regularity of T requires convex target.
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Alessio
Figalli

Regularity Theory! Regularity of T requires convex target.
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Alessio
Figalli1-D Optimal Transportdef.

<latexit 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Cumulative function:

Quantile function:

Optimal transport ↵ ! :↵C↵ (x) =

Rx

1

d↵

C↵] : ↵ ! U[0,1]C 1 ] : U[0,1] !

T = C 1 C↵
T

C↵ 1

C 1

T

1

1-D Optimal Transportdef.

<latexit 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Cumulative function:

Quantile function:

Optimal transport ↵ ! :↵C↵ (x) =

Rx

1

d↵

C↵] : ↵ ! U[0,1]C 1 ] : U[0,1] !

T = C 1 C↵
T

C↵ 1

C 1

T

1

def.

C↵ (x) =<latexit sha1_base64="lkYmEXSwz6mt4AGHn22Q31ywqqo=">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</latexit>

↵

1

d↵

C 1

C↵

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C

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Wp (↵, ) =<latexit sha1_base64="V2c4oi6Ui+8aAQQRSUjfv4krA88=">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</latexit>

W1 (↵, ) = ||↵<latexit 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=R

R

R1|C↵ (x)||2K =

||↵
||↵

1
|C
(t)
↵
0

R1

C 1 (t)|p dt

C (x)|dx|C↵ (t)0

C (t)|2 dt||KS = supx |C↵ (x)AUC(↵, ) = 1

R1

C
C
0 ↵

C (x)|
1

(x)

def.

C↵ (x) =<latexit sha1_base64="lkYmEXSwz6mt4AGHn22Q31ywqqo=">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</latexit>

↵

1

d↵

C 1

C↵

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C

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Wp (↵, ) =<latexit sha1_base64="V2c4oi6Ui+8aAQQRSUjfv4krA88=">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</latexit>

W1 (↵, ) = ||↵<latexit 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=R

R

R1|C↵ (x)||2K =

||↵
||↵

1
|C
(t)
↵
0

R1

C 1 (t)|p dt

C (x)|dx|C↵ (t)0

C (t)|2 dt||KS = supx |C↵ (x)AUC(↵, ) = 1

R1

C
C
0 ↵

C (x)|
1(x)

1-D Optimal Transport Interpolationdef.

<latexit 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Rx

1

d↵

Optimal transport interpolation ↵0 $ ↵1
8 t 2 [0, 1], C↵t1 = (1

↵0

↵ 12

↵1

t)C↵01 + tC↵11

C↵0

C↵1

1-D Optimal Transport Interpolationdef.

<latexit 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sha1_base64="NagChymWaASb8bvGA6rN+HpPF8Q=">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</latexit>C↵ (x) =

Rx

1

d↵

Optimal transport interpolation ↵0 $ ↵1
8 t 2 [0, 1], C↵t1 = (1

↵0

↵ 12

↵1

t)C↵01 + tC↵11

C↵0

C↵1

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T (x) =(xm↵ ) + m(xm↵ )2 + m x↵'(x) =

2 ↵T = r'

Brenier=)
W22 (↵, ) = (m↵mm' is convex.
T ⌘ OTm )2 + ( ↵↵)2

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d↵
1
=
e
d/2
dx
(2⇡) |⌃↵ |⌃↵ = U↵ diag(||xm↵ ||2 1
⌃↵
2

>
)U
↵
↵

T (x) = A(x m↵ ) + m
q
1
1
1
1
2
2
2
A = ⌃↵⌃↵ ⌃ ⌃↵ ⌃↵ 2

T

2 u2

m

m↵
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1 u1

Proposition:n
A 2 S+A⌃↵ A = ⌃Brenier=)T ⌘ OT

OT Between Gaussians<latexit sha1_base64="o+Os9I3V1/gBGlzVJJ9cwsUwA9Q=">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</latexit><latexit
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d↵
1
=
e
d/2
dx
(2⇡) |⌃↵ |⌃↵ = U↵ diag(||xm↵ ||2 1
⌃↵
2

T

2 u2

m

m↵

>
)U
↵
↵

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1 u1

T (x) = A(x m↵ ) + m
q
1
1
1
1
2
2
2
A = ⌃↵⌃↵ ⌃ ⌃↵ ⌃↵ 2

Proposition:n
A 2 S+A⌃↵ A = ⌃Brenier) = ||m↵ m || + B(⌃↵ , ⌃ )=)T ⌘ OT
q
⇣
⌘
1
1
def.
2 ⌃↵2 ⌃ ⌃↵2
Bures distance: B(⌃↵ , ⌃ )2 = tr ⌃↵ + ⌃W22 (↵,

2

2

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1 u1

T (x) = A(x m↵ ) + m
q
1
1
1
1
2
2
2
A = ⌃↵⌃↵ ⌃ ⌃↵ ⌃↵ 2

Proposition:n
A 2 S+A⌃↵ A = ⌃Brenier) = ||m↵ m || + B(⌃↵ , ⌃ )=)T ⌘ OT
q
⇣
⌘
1
1
def.
2 ⌃↵2 ⌃ ⌃↵2
Bures distance: B(⌃↵ , ⌃ )2 = tr ⌃↵ + ⌃W22 (↵,

2

If ⌃↵ ⌃ = ⌃ ⌃↵ :pp
2B(⌃↵ , ⌃ ) = || ⌃↵
⌃ ||2

2

↵

Interpolation Between GaussiansT

2 u2

Optimal transport map T ] ↵ = .
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T (x) = A(x m↵ ) + m
⇣ 1
⌘ 12
1
1
1
2
2
2
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1 u1
def.

Displacement interpolation: ↵t = ((1mt = (1mm↵⌃t = [(1t)Id + tT )] ↵ = N (mt , ⌃t )t)Id + tA]⌃↵ [(1t)Id + tA]

Interpolation Between GaussiansT

2 u2

Optimal transport map T ] ↵ = .
<latexit sha1_base64="oYYQEL56qLdM5TG7uikRSS0wyGc=">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</latexit><latexit
sha1_base64="oYYQEL56qLdM5TG7uikRSS0wyGc=">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</latexit>

T (x) = A(x m↵ ) + m
⇣ 1
⌘ 12
1
1
1
2
2
2
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1 u1
def.

Displacement interpolation: ↵t = ((1mt = (1mm↵⌃t = [(1t)Id + tT )] ↵ = N (mt , ⌃t )t)Id + tA]⌃↵ [(1t)Id + tA]

Overview• Monge Formulation• Continuous Optimal Transport• Kantorovitch Formulation• Applications

Leonid Kantorovich (1912-1986)Леонид Витальевич Канторович

[Kantorovich 1942]

Kantorovitch’s Formulation
Input distributions↵=Pn

i=1 ai xi

Points (xi )i , (yj )j=Pm

Weights ai > 0, bj > 0.
PnPm
i=1 ai =j=1 bj = 1

j=1 bj yj

Kantorovitch’s Formulation
Input distributions↵=Pn

i=1 ai xi

Points (xi )i , (yj )j=Pm

j=1 bj yj

Weights ai > 0, bj > 0.
PnPm
i=1 ai =j=1 bj = 1

Couplings:<latexit sha1_base64="MCNqjwUgnSKqr3h3uG6OPk9oPyc=">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</latexit><latexit
sha1_base64="MCNqjwUgnSKqr3h3uG6OPk9oPyc=">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</latexit>

def.

U(a, b) =<latexit sha1_base64="kzj0wPD/kexOzeTHYdqyAzdpsr8=">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</latexit><latexit
sha1_base64="kzj0wPD/kexOzeTHYdqyAzdpsr8=">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</latexit>>
P 2 Rn⇥m
;
P1
=
a,
P
1n = bm+ab<latexit sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">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<latexit
sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">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</latexit>

PKantorovitch’s Formulation
Input distributions↵=Pn

i=1 ai xi

Points (xi )i , (yj )j

=

Pm

,j
i
C

j=1 bj yj

xi

Weights ai > 0, bj > 0.
PnPm
i=1 ai =j=1 bj = 1

aCouplings:<latexit sha1_base64="MCNqjwUgnSKqr3h3uG6OPk9oPyc=">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</latexit><latexit
sha1_base64="MCNqjwUgnSKqr3h3uG6OPk9oPyc=">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</latexit>

def.

U(a, b) =<latexit sha1_base64="kzj0wPD/kexOzeTHYdqyAzdpsr8=">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</latexit><latexit
sha1_base64="kzj0wPD/kexOzeTHYdqyAzdpsr8=">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</latexit>>
P 2 Rn⇥m
;
P1
=
a,
P
1n = bm+[Kantorovich 1942]

min

nP

i,j Ci,j Pi,j ; P 2 U(a, b)

yj

o

b
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PKantorovitch’s Formulation
Input distributions↵=Pn

i=1 ai xi

Points (xi )i , (yj )j

=

Pm

,j
i
C

j=1 bj yjyjxi

Weights ai > 0, bj > 0.
PnPm
i=1 ai =j=1 bj = 1ab<latexit sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">AABB0nictVxfcxPJER8u/w7yD5LHvGxiSHEpQoxzVUnqKlVnbGN8CDBINtydgNJKayFYa4VWsgGdH1J5zVfIa/Ix8jnyDZKnfIX0n5mdWWl2e9YhTNmeHc2vu6d3pqe7Z0Q8SUf5bH39nxc++ta3v/Pd73188dL3f/DDH/348pWfHObZfNpPDvpZmk2fxr08SUfj5GA2mqXJ08k06R3HafIkfr2Fnz85Sab5KBt3Zu8mybPj3nA8Ohr1ezNoenH5ytUuEVnE6Tw5i7rx1ReX19ZvrtO/aLVyS1fWlP63n12JDlRXDVSm+mqujlWixmoG9VT1VA7la3VLrasJtD1TC2ibQm1EnyfqTF0C7Bx6JdCjB62v4fcQnr7WrWN4Rpo5ofvAJYWfKSAjdQ0wGfSbQh25RfT5nChjaxXtBdFE2d7B31jTOobWmXoJrRLO9AzF4Vhm6kj9nsYwgjFNqAVH19dU5qQVlDxyRjUDChNow/oAPp9CvU9Io+eIMDmNHXXbo8//RT2xFZ/7uu9c/ZukvAYlUm09+qyg0FMnRD+itzmHz1ieFDgPgUKix4i1U9L1MY1+DP0X0P4AyhnVjE5iKAtqPatFbkHxIbdE5C4UH3JXRLag+JAtEbkPxYfc10jETknnfnwbig/fFjk/guJDPhKRj6H4kI9F5CEUH/JQRH4FxYf8SkTegeJD3hGR96D4kPdEZAeKD9kRkQdQfMgDEbkDxYfc0cjqlTqFkhGdkbAqN6Fe5oGWIoWWTVG+22QdfdjbAWu6X4GVV/U2/PVjtwN0mlRgdwLm3VEFVp55u2Aj/VjZFt2l3cSHvSti92AG+LF7IvYL9aoC+0XASntdgZXXWgv6+bGy9b0PT37sfRH7AGp+rLxHPYQWP/ZhwI4xqcDui9hH6k0FNsTqTyuwst1vg13xY+V9qgP9/dgQazqvwMr29BA8GD9W3q2eQKsf+0TEPlVvK7BPReyXYN392C8Ddtj3FVizx16iHWRI/kgCK7aOWq9YlVibALWewD8t9paUfOMY2iXMsMAMCXMsInYLxG4golUgWsFy5YUdzcnflbm0C0Q7EBEXexPWZmL/QdEfa2kAYrtAbC8h6jxSfNdmLCfkXZgWCTkrdi6shYwpK+w31hI9H+otr0E8LCF4br+kmX+DoiWMoFBTddReFns8IyN6rkOcUvRmRml4yLhZYRVc1FsRFXtQsYh650G9E1FzD2ouok48qBMRZVe+i+sGzACrf3wXC3riGcA+cnWJwCvYhF3nLqzRCObPPniBj6nlIfxtU+wtlTrJMJrHfRKzHM9KlngKtYVag3YbFW5TfJ3SCktAMu75UMf4+IS5jYVec2yFz4qdPCoyJuF0RiTPsKCD3mJE66kZnXvUckbeHdea4e8W697UmuF3SONn5MVzrRl+pqWfnUP2jsZ2zoFtw2qaaO3belManH9hGqZ+iXZdtLj4Vo/1nEF6bxvS39NvZu8c72WLaqwfW29GI3fGl5fG14SG1XPu6LkZFfSe2Os1tajxSMY67rX1pjJktIuOtRz2qembwT4D/WZMvRmNffC4tijmXjj1prN3UozG1pvROFSc9zwjT97Um9EY0jPrw9ab0cBsS0/H+bbe1LKjBjh2tvWmVn1MWWDMAfGc5xbrFU3JT5praiPyD+qzNa7Pv7qPYc7meREj1FOyvm01nbjYy+olMv5CAlZt1lAO9C/mjg9WprFQG2J8xTLMSvv7Kh27x6PmW6DFCFY/nwFIOfMUJDQ5CbTeKVC8JUZd5ZEZ3IaIw1lytITq6taZ6C1avpw1Kre9oFYpLrOjtXrskr3Oae5NyCdskWYlPbQq33AVRUlDrZKGZHpNdPder9ey9tdF3GQJMSlmWp9OhPgkrT5O9Wm97ej4mj7lmUHhMx87fzHbfKStDcY8GdkilKWOp9vP5JHcNtxXbyib4+bPInqjaK9OyGqM6EQqF6NQky1mb3xBz5b2AZ3JIQ+m0Yf3GGkqE8WnZphFx3x6RBbVtbcSb9SXydBxPSera+xxPXrooIcedPMYZwt2jAdQ60DMcABPnYAo51Khq4w0PlW/Lk5HM3qD9RF9WrKQhgbbm6RkIeui7JclKqeAxtnAUXo4jWU6Bt9doSRH/T55bOxatvzX6O<latexit
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P

Couplings:<latexit sha1_base64="MCNqjwUgnSKqr3h3uG6OPk9oPyc=">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</latexit><latexit
sha1_base64="MCNqjwUgnSKqr3h3uG6OPk9oPyc=">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</latexit>

def.

U(a, b) =<latexit sha1_base64="kzj0wPD/kexOzeTHYdqyAzdpsr8=">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</latexit><latexit
sha1_base64="kzj0wPD/kexOzeTHYdqyAzdpsr8=">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</latexit>>
P 2 Rn⇥m
;
P1
=
a,
P
1n = bm+[Kantorovich 1942]

min

nP

i,j Ci,j Pi,j ; P 2 U(a, b)
3o! Linear program, simplex O(n log(n)).

GeorgesDantzig

Kantorovitch’s Exact Relaxation↵=

xi<latexit sha1_base64="q6ZsACa7ksLxQ6laNJc690EL4kA=">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2PermnnX
i=1

Ci, (i)=Pn

j=1 yjKantorovitch’s Exact Relaxation
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sha1_base64="q6ZsACa7ksLxQ6laNJc690EL4kA=">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</latexit>yj

Bistn =>
P 2 Rn⇥n
;
P1
=
1,
P
1=1
+

Kantorovitch (1942):<latexit sha1_base64="4em3xVJ/KwPVYxTU4rHFOirPd8E=">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</latexit><latexit
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2PermnnX
i=1

j=1 yjPermutations “⇢” Bi-stochastic matrices:

Monge (1784):min=

i=1 xiPn(relaxation)↵=

Pn

Ci, (i)

def.

>

min

P2Bistnn XnX
i=1 j=1

Pi,j Ci,jKantorovitch’s Exact Relaxation
xi

Bistn =>
P 2 Rn⇥n
;
P1
=
1,
P
1=1
+

Kantorovitch (1942):

def.

<latexit sha1_base64="4em3xVJ/KwPVYxTU4rHFOirPd8E=">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</latexit><latexit
sha1_base64="4em3xVJ/KwPVYxTU4rHFOirPd8E=">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</latexit>

2PermnnX

j=1 yjPermutations “⇢” Bi-stochastic matrices:yj

Monge (1784):min=

i=1 xiPn(relaxation)↵=

Pn

Ci, (i)>i=1

n! permutations<latexit sha1_base64="FpF/yIIF10mrIAzpJoxtJO3DXQM=">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</latexit><latexit
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min

P2Bistnn X
nX

Pi,j Ci,j

i=1 j=1

O(n3 ) algorithm<latexit sha1_base64="E5UrblKdTajBJeA+caTcOqhquMw=">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
sha1_base64="E5UrblKdTajBJeA+caTcOqhquMw=">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</latexit>

Kantorovitch’s Exact Relaxation
xi

Bistn =>
P 2 Rn⇥n
;
P1
=
1,
P
1=1
+

Kantorovitch (1942):

def.

<latexit sha1_base64="4em3xVJ/KwPVYxTU4rHFOirPd8E=">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</latexit><latexit
sha1_base64="4em3xVJ/KwPVYxTU4rHFOirPd8E=">AABEIXictVxLcxu5EYY3r7Xz2k2OucxG65SdchxJcdVuZStVa1OyrDVtyyYle3dlu/gY0bSHHJpD0g+uKj8j16TyQ3LOLZVbKn8gOeUvpB/AAENipjGKY5QkDIivu9EDNLoboLuTZJjNNjf/ce69b337O9/93vvnL3z/Bz/80Y8/+PAnR1k6n/biw16apNNH3U4WJ8NxfDgbzpL40WQad0bdJH7YfdHAzx8u4mk2TMft2ZtJ/HjUGYyHJ8NeZwZNj++k40EcXdr65NNrl3/79IONzaub9C9ar2zpyobS/w7SDz/6vTpWfZWqnpqrkYrVWM2gnqiOyqB8rbbUpppA22O1hLYp1Ib0eaxO1QXAzqFXDD060PoCfg/g6WvdOoZnpJkRugdcEviZAjJSFwGTQr8p1JFbRJ/PiTK2ltFeEk2U7Q387WpaI2idqWfQKuFMz1AcjmWmTtSnNIYhjGlCLTi6nqYyJ62g5JEzqhlQmEAb1vvw+RTqPUIaPUeEyWjsqNsOff4v6omt+NzTfefq3yTlRSiRaunRpzmFjloQ/Yje5hw+Y3kS4DwACrEeI9Zeka5HNPox9F9C+10op1QzOulCWVLraSWyAcWHbIjIPSg+5J6IbELxIZsi8gCKD3mgkYidks79+BYUH74lcr4PxYe8LyIfQPEhH4jIIyg+5JGI/AqKD/mViLwJxYe8KSJvQ/Ehb4vINhQfsi0iD6H4kIcicheKD7mrkeUrdQolJTpDYVVeh3qRB1qKBFqui/LdIOvow94IWNO9Eqy8qnfgrx+7E6DTuAS7GzDvTkqw8szbAxvpx8q26BbtJj7sLRG7DzPAj90XsV+o5yXYLwJW2osSrLzWmtDPj5Wt7x148mPviNi7UPNj5T3qHrT4sfcCdoxJCfZAxN5XL0uwIVZ/WoKV7X4L7IofK+9Tbejvx4ZY03kJVranR+DB+LHybvUQWv3YhyL2kXpdgn0kYr8E6+7Hfhmww74twZo99gLtIAPyR2JYsVXUOvmqxNoEqHUE/km+tyTkG3ehXcIMcsyAMCMRsZcj9gIRzRzRDJYry+1oRv6uzKWVI1qBiG6+N2FtJvbv5/2xlgQgdnLEzgqiyiPFd23GsiDvwrRIyFm+c2EtZExpbr+xFuv5UG15DeJeAcFz+xnN/CsULWEEhZqqovYs3+MZGdFzFeIVRW9mlIaHjJvlVsFFvRZRXQ+qK6LeeFBvRNTcg5qLqIUHtRBRduW7uOOAGWD1j+9iSU88A9hHLi8ReAXXYde5BWs0gvlzAF7gA2q5B39bFHtLpUoyjOZxn8Qsx+OCJZ5Cbak2oN1GhTsUXye0wmKQjHve0zE+PmFuY6nXHFvh03wnj/KMSTidIckzyOmgtxjReqpH5za1nJJ3x7V6+Fv5uje1evhd0vgpefFcq4efaelnZ5C9rbHtM2BbsJomWvu2XpcG51+YhqlfoF0XLS6+1ZGeM0jvdU36+/rN7J/hvTSoxvqx9Xo0Mmd8WWF8dWhYPWeOnutRQe+JvV5Ti2qPZKzjXluvK0NKu+hYy2Gf6r4Z7NPXb8bU69E4AI+rQTH30qnXnb2TfDS2Xo/GkeK85yl58qZej8aAnlkftl6PBmZbOjrOt/W6lh01wLGzrde16mPKAmMOiOc8t1ivaEp+0lxTG5J/UJ2tcX3+9X0MczZP8hihmpL1bcvpdPO9rFoi4y/EYNVmNeVA/2Lu+GBFGku1LcZXLMOssL+v07F7PGq+CVqMYPXzGYCUM09AQpOTQOudAMUtMeoqjszgtkUczpKTFdSxbp2J3qLly1mjYttTapXiMjtaq8djstcZzb0J+YRN0qykh2bpGy6jKGmoWdCQTK+O7t7q9VrU/qaIm6wgJvlM69GJEJ+kVcepPq23HB1f1Kc8Myh85mPnL2abT7S1wZgnJVuEslTxdPuZPJLbhvvqFWVz3PxZRG8U7dWCrMaQTqQyMQo12WL2xpf0bGkf0pkc8mAaPXiPkaYyUXxqhll0zKdHZFFdeyvxRn2ZDB3XM7K6xh5XowcOeuBB149xGrBj3IVaG2KGQ3hqB0Q5F3JdpaTxqfpVfjqa0husjuiTgoU0NNjexAULWRVlPytQeQVonA0cpYfTWKVj8MdrlOSo3yePjV2Llv8indya8+0OzfHy2VyeiekT123iGtGq4VNdflrlwBIsvZ9sk/9aPUrkV4cj2lCJ6xOHM+tlTCf+MUWwE/KME1pt0uoo9nbzU6ufGE4Hypyd42l2ShYyIvsXwf6U0pyM6Me9O2BO0NkiJGQjQ+zOMPdufL7OUJxj1o8bKr7VYOdbTLZsTvwNXXd1ZTQXOWLgfeB0ZW4bnTTJF4yJ61Rbd7u2q3cfRNp7Eu4sYYp2rlwi/pfpt/kx82RjbUaghvENZNrW+d5HSjEL6qhDu3y1DTJ9XSk/zmV4oqW2+5+V6eOCZDsUcaE8uFv3gXOPnpkXzpIpyZ2t9eF9tCqbi5QnK3rE0Z5QFM92f6B3YJT7Cu2SG7TmjmmWDGAWzPIowvSVssirfKt5FamH0c7+L9StrotaQ4qRshlc1pCU348pWnOlTGBW8/x9QavJr/XpSq9qPmOaiyNnLX8DrR/BbyO3eQ6j0y1YhRs0B5iCfbIa4ZZorUcYrxsFXmZmGlr22fKzc9L0clvOEl+zdbMx9qI2lQOaNa911sLUz0LjuUPjeaAO23TWaLVo2o0leirGFm19WhnKrw63dg3Kc5Gy7JEZ1DBASjeWCqPaF6nKMb5BvRVpbYq0OrBa3dMAd82HIP1rfXV1f5Pv7pG6Sb5Njzwwjl/6tEqH5HOZ1upIjSkg52vavrqr/5hakHuXLChS5nucuGL41KlH5TSX9Bd6Z0vJzluLYO4tvdJ9jI09pvpv1pAjWhMZrUuDuEY9Yi2/K0e0YpGuOj5HRJn/DvlU7HdUx8xub/tOooI/YeNNXlWWF0cKY9K/lHnbX4te9534NaKYcK696y7Qqv+GkQJjTCbB71lm9IZwl+OTBPZou2Q/1+0Un+KNHYmuktRL9bsAG8NRr53r7twyIzZj+yX0RK3bt+7rIfNLgjlK/M5yotehXW2kfdTlyvPZaHX0Lld8rtLDfIWv1cec+riRhY3yiphj9VkwF5aoHhfGhHCpN4o68teTvI7MfDoVStn0NpSLmQa2Mc8oXpLugSLC591d8npzl4VxdNfodQnrUuMWiRJm41KdH3AtLWalzq/tQ9x6vnI3SpydqGynMNTd3cLab7aQMVm/REk5G+7tyn5ciFLkLAxT6Cm+0VsWH7o0P4OCvyPliw4Nx5DcYQv82+uqoXbfwW2Il7rOGc2IWtAW9Fdi744eZ7FHtY5eOtRd+iEcwnkMQdeS9EPaSevKzpRlyV3q4fRfkRWYqliU3vasPwaXizySdU51xjMkyyaPZqjMd3HqjsVwCBlJkUs4Hz7XkEZxosx3muqNwVCXR1DkUIeHuccQ9s5t7/q8XE7V+lrnEsqDdwFz4mJwePJXHqvYfiEWauq8kXfPAa3DSQV1s1v8r+MwfCyn+rxCuWX0XbPnAW+d+8U6I4v+cP01Y7mFzOZyjuE803x01lvy82O/L6r1plJnNO+ePvqjdg4YXkvFeVBZOsa7s8jKG0oFzwV8MqTqP+ov5+RvI7zMaZTJUYeSOacop2Z6yNTMNy59ozOfhchk6ZTJVKRm44gW3YhtqH11E34auQdY93Yof5eS/yLW//3ZPrSekPUwWXTOHBxTW0zZD3uK1qdne3+2TGK8y8t3e9vQgmfhTWrFe753qT/e9W0Xxlb+DRJe63dUqvqFiGT1dM+uqy6MoHjyxjkg8z3fiO7ScxaLb56NAs4Wzf2pVYmW9Il8s6Bbiu86UvZork70WT2eHOAN+06eH4rUr6mto+087rkS54NSzgcrnDPSTpHDa+ez6rtZZVwaDpd+njtb6H4pxdn2PK86N7pTyoXvoFfjBxX4gSNli7T/giLhqarO5s0raM61TO4J61iZTCTrAePMTv6+qyPbRQWvRcD4b5eibzuS7oEsXcp/R3TCNiV6idbNLknPNx2rM6m3KqQ136Nkmvauo50H5tZidZY+0fOOMxfmfyNY0nrA3dTcTJSyJ3EJnS7dCIyJEt+RlM4NZHlkaUJkkajMRUnmAd/eXQTIsgigcyJIcyJSGIiS6BX89IONrdX/H2O9crR9dWvz6tb97Y3Pb+j/O+N99TP1c3UJ9o1P1OcwQw/UIe3hf1B/VH9q/Lnx18bfGn/nru+d05ifqsK/xj//C8l5Hgc=</latexit>

2PermnnX

j=1 yjPermutations “⇢” Bi-stochastic matrices:yj

Monge (1784):min=

i=1 xiPn(relaxation)↵=

Pn

Ci, (i)>i=1

n! permutations<latexit sha1_base64="FpF/yIIF10mrIAzpJoxtJO3DXQM=">AABEJ3ictVzdcxPJER8uH3eQL0ge87LEkOJSHLF9V5VUXaXqQDbGhwGDZMPdGSh9rIVgpRVaSQZ0rvwleU0qf0X+hLylkse8JU/5F9IfMzuz0uz2rEOYsj07ml93T+9MT3fPiM44GWTT9fV/nPvgO9/93vc//Oj8hR/88Ec//snFSz89zNLZpBsfdNMknTzptLM4GYzig+lgmsRPxpO4Pewk8ePOqwZ+/ngeT7JBOmpN347jp8N2fzQ4HnTbU2h6fvHSldHlK9E4ngxnU2rKnl9cW7+xTv+i1cqGrqwp/W8/vXT59+pI9VSqumqmhipWIzWFeqLaKoPyjdpQ62oMbU/VAtomUBvQ57E6VRcAO4NeMfRoQ+sr+N2Hp2906wiekWZG6C5wSeBnAshIXQVMCv0mUEduEX0+I8rYWkZ7QTRRtrfwt6NpDaF1ql5Aq4QzPUNxOJapOla/pTEMYExjasHRdTWVGWkFJY+cUU2BwhjasN6DzydQ7xLS6DkiTEZjR9226fN/UU9sxeeu7jtT/yYpr0KJVFOPPs0ptNWc6Ef0NmfwGcuTAOc+UIj1GLF2Qroe0uhH0H8B7fehnFLN6KQDZUGtp5XIBhQfsiEid6D4kDsicg+KD7knIveh+JD7GonYCencj29C8eGbIueHUHzIhyLyERQf8pGIPITiQx6KyK+h+JBfi8jbUHzI2yLyLhQf8q6IbEHxIVsi8gCKD3kgIreh+JDbGlm+UidQUqIzEFblTagXeaClSKDlpijfLbKOPuytgDXdLcHKq3oL/vqxWwE6jUuw2wHz7rgEK8+8HbCRfqxsi+7QbuLD3hGxuzAD/NhdEfulelmC/TJgpb0qwcprbQ/6+bGy9b0HT37sPRF7H2p+rLxHPYAWP/ZBwI4xLsHui9iH6nUJNsTqT0qwst1vgl3xY+V9qgX9/dgQazorwcr29BA8GD9W3q0eQ6sf+1jEPlFvSrBPROxXYN392K8Cdth3JVizx16gHaRP/kgMK7aKWjtflVgbA7W2wD/J95aEfOMOtEuYfo7pE2YoInZyxE4gYi9H7AXLleV2NCN/V+bSzBHNQEQn35uwNhX79/L+WEsCEFs5YmsJUeWR4rs2Y5mTd2FaJOQ037mwFjKmNLffWIv1fKi2vAbxoIDguf2CZv51ipYwgkJNVVF7ke/xjIzouQpxQtGbGaXhIeOmuVVwUW9EVMeD6oiotx7UWxE186BmImruQc1FlF35Lu4oYAZY/eO7WNATzwD2kctLBF7BTdh17sAajWD+7IMX+IhaHsDfJsXeUqmSDKN53Ccxy/G0YIknUFuoNWi3UeEWxdcJrbAYJOOeD3SMj0+Y21joNcdW+DTfyaM8YxJOZ0Dy9HM66C1GtJ7q0blLLafk3XGtHv5Ovu5NrR5+mzR+Sl481+rhp1r66Rlkb2ls6wzYJqymsda+rdelwfkXpmHqF2jXRYuLb3Wo5wzSe1OT/q5+M7tneC8NqrF+bL0ejcwZX1YYXx0aVs+Zo+d6VNB7Yq/X1KLaIxnpuNfW68qQ0i460nLYp7pvBvv09Jsx9Xo09sHjalDMvXDqdWfvOB+Nrdejcag473lKnryp16PRp2fWh63Xo4HZlraO8229rmVHDXDsbOt1rfqIssCYA+I5zy3WK5qQnzTT1AbkH1Rna1yff3Ufw5zNszxGqKZkfdtyOp18L6uWyPgLMVi1aU050L+YOT5YkcZCbYrxFcswLezvq3TsHo+a3wMtRrD6+QxAypknIKHJSaD1ToDihhh1FUdmcJsiDmfJ8RLqSLdORW/R8uWsUbHtObVKcZkdrdXjEdnrjObemHzCPdKspIe90jdcRlHS0F5BQzK9Orp7p9drUfvrIm68hBjnM61LJ0J8klYdp/q03nR0fFWf8kyh8JmPnb+YbT7W1gZjnpRsEcpSxdPtZ/JIbhvuq9eVzXHzZxG9UbRXc7IaAzqRysQo1GSL2Rtf0LOlfUBncsiDaXThPUaayljxqRlm0TGfHpFFde2txBv1ZTJ0XM/I6hp7XI3uO+i+B10/xmnAjnEfai2IGQ7gqRUQ5VzIdZWSxifqk/x0NKU3WB3RJwULaWiwvYkLFrIqyn5RoHICaJwNHKWH01imY/BHK5TkqN8nj41di5b/Kp3cmvPtNs3x8tlcnonpEddN4hrRquFTXX5a5sASLLyfbJL/Wj1K5FeHI9pQieszhzPrZUQn/jFFsGPyjBNabdLqKPZ281PLnxhO+8qcneNpdkoWMiL7F8H+lNKcjOjHvTtgTtDZIiRkI0PsziD3bny+zkCcY9aPGyi+1WDnW0y2bEb8DV13dWU0Fzli4H3gdGluG53skS8YE9eJtu52bVfvPoi09yTcWcIU7Vy5Rvw/pt/mx8yTtZUZgRrGN5BpW+d7HynFLKijNu3y1TbI9HWlvJLL8ExLbfc/K9OVgmRbFHGhPLhb94Bzl56ZF86SCcmdrfThfbQqm4uUx0t6xNEeUxTPdr+vd2CU+zrtkmu05o5olvRhFkzzKML0lbLIy3yreRWph9HO/i/Ura6LWkOKkbIZXNaQlN+PKVpzpUxgVvP8fUWrya/1yVKvaj4jmotDZy1/C62X4beR2zyH0ekUrMItmgNMwT5ZjXBLtNIjjNetAi8zMw0t+2z52TlperktZ4mv2brZGHtem8o+zZo3Omth6meh8dKh8TJQhy06a7RaNO3GEj0XY4uWPq0M5VeHW6sG5ZlIWfbIDGoQIKUbS4VR7YlU5RjfoN6JtNZFWm1Yre5pgLvmQ5D+tb68ur/Nd/dI3SbfpkseGMcvPVqlA/K5TGt1pMYUkPNn2r66q/+IWpB7hywoUuZ7nLhi+NSpS+U0l/SXemdLyc5bi2DuLZ3oPsbGHlH90xXkkNZERuvSID6jHrGW35UjWrJINxyfI6LMf5t8KvY7qmNmt7d9J1HBn7DxJq8qy4sjhRHpX8q87a5Er7tO/BpRTDjT3nUHaNV/w0iBMSaT4PcsM3pDuMvxSQJ7tB2yn6t2ik/xRo5EN0jqhfpdgI3hqNfOdXdumRGbsf0KeqLW7Vv39ZD5JcEcJX5nOdFr06421D7qYun5bLTaepcrPlfpYbbE1+pjRn3cyMJGeUXMkfo8mAtLVI8LY0K41BtFHfnrSV5HZj6dCqVsehvKxUwD25gXFC9J90AR4fPurnm9uY+FcXRW6HUI61LjFokSZuNSnR9wLS1mpc6v7EPcer5yN0qcnahspzDU3d3C2m+2kDFZv0RJORvu7cp+VIhS5CwMU+gqvtFbFh+6ND+Hgr8j5YsODceQ3GET/NubqqG238NtiNe6zhnNiFrQFvSWYu+2HmexR7WOXjvUXfohHMJ5DEDXkvQD2knrys6UZcld6uH0T8gKTFQsSm971h+Dy0UeySqnOuMZkGWTRzNQ5rs4dcdiOISMpMglnA+fa0ijOFbmO031xmCoyyMocqjDw9xjCHvntnd9Xi6nan2tcgnlwbuAOXExODz5K49VbL8QCzVx3sj754DW4biCutkt/tdxGD6WU31eodwy+q7Zy4C3zv1inZFFf7j+mrHcQmZzOcdwnmk+Oust+fmx3xfVelOpM5r3Tx/9UTsHDK+F4jyoLB3j3Vlk5Q2lgucCPhlS9R/1l3PytxFe5zTK5KhDyZxTlFMzPWRq5huXvtGZz0JksnTKZCpSs3FEk27ENtSuug0/jdwDrHs7lL9LyX8R6//+bA9aj8l6mCw6Zw6OqC2m7Ic9RevRs70/WyYx3uXlu70taMGz8D1qxXu+96k/3vVtFcZW/g0SXuv3VKp6hYhk+XTPrqsOjKB48sY5IPM934ju0nMWi2+eDQPOFs39qWWJFvSJfLOgU4rvOFJ2aa6O9Vk9nhzgDft2nh+K1K+pra3tPO65Euf9Us77S5wz0k6Rwxvns+q7WWVcGg6XXp47m+t+KcXZ9jyvOje6VcqF76BX4/sV+L4jZZO0/4oi4YmqzubNKmjOtEzuCetImUwk6wHjzHb+vqsj23kFr3nA+O+Wou86ku6ALB3Kf0d0wjYheonWzTZJzzcdqzOpdyqkNd+jZJr2rqOdB+bWYnWWPtHzjjMX5n8jWNB6wN3U3EyUsidxCZ0O3QiMiRLfkZTODWR5ZGlCZJGozERJZgHf3p0HyDIPoHMsSHMsUuiLkugV/Pzi2sby/4+xWjncvLGxfmPj4ebaF7f0/53xkfq5+oW6BvvGb9QXMEP31QFwOlF/UH9Uf2r8ufHXxt8af+euH5zTmJ+pwr/GP/8LH0UhDQ==</latexit><latexit
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min

P2Bistnn X
nX

Pi,j Ci,j

i=1 j=1

O(n3 ) algorithm<latexit sha1_base64="E5UrblKdTajBJeA+caTcOqhquMw=">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
sha1_base64="E5UrblKdTajBJeA+caTcOqhquMw=">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</latexit>

Theorem: [Birkho↵-von Neumann]
George David
BirkhoffJohn
von Neumann“Monge , Kantorovitch”

aOptimal Transport Distancesb<latexit sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">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</latexit><latexit<latexit
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sha1_base64="n2zdOYSE8oWxrpoEULZvL+5/PUM=">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</latexit>

p def.

Wp (↵, ) =

min
1

⇡=

⇡2M+ (X 2 )Pi,j Pi,j xi ,yj↵⇡c(x, y) = d(x, y)p⇢Zd(x, y)p d⇡(x, y) ; ⇡ 1 = ↵, ⇡ 2 =X2

Optimal Transport DistancesbaP

p def.

Wp (↵, ) =

⇡=

min
1

⇡2M+ (X 2 )Pc(x, y) = d(x, y)p⇢Zd(x, y)p d⇡(x, y) ; ⇡ 1 = ↵, ⇡ 2 =X2<latexit 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Weak (aka in law) convergence: ↵n *

|| xn

x2
x ||1 = 2↵

i,j Pi,j xi ,yj

Theorem: Wp is a distance and ↵n *

x1

⇡

x3

vs.

, Wp (↵n , ) ! 0
, R

...

8 f 2 C(X ),R
f d↵n ! X f d
Xx

Wp ( xn , x ) = d(xn , x)

A Glimpse at Algorithms
Linear programming:O(n3 log(n)2 )↵A Glimpse at Algorithms
Linear programming:O(n3 log(n)2 )O(n3 )
Hungarian/Auction:
Pn
Pn
1
1
↵ = n i=1 xi= n j=1 yj↵↵

A Glimpse at Algorithms
Linear programming:O(n3 log(n)2 )O(n3 )
Hungarian/Auction:
Pn
Pn
1
1
↵ = n i=1 xi= n j=1 yj

1-D case: sorting O(n log(n)).

↵

↵

↵

A Glimpse at Algorithms
Linear programming:O(n3 log(n)2 )O(n3 )
Hungarian/Auction:
Pn
Pn
1
1
↵ = n i=1 xi= n j=1 yj

1-D case: sorting O(n log(n)).

p=1
d = || · ||W1 (↵, ) = mindiv(u)=↵<latexit sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit><latexit
sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit>R||u(x)||dx! min-cost flow, on graphs O(n2 log(n)).

↵

↵

↵

A Glimpse at Algorithms
Linear programming:O(n3 log(n)2 )O(n3 )
Hungarian/Auction:
Pn
Pn
1
1
↵ = n i=1 xi= n j=1 yj

1-D case: sorting O(n log(n)).

p=1
d = || · ||W1 (↵, ) = mindiv(u)=↵<latexit sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit><latexit
sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit>R||u(x)||dx! min-cost flow, on graphs O(n2 log(n)).
Monge-Ampère/Benamou-Brenier, d = || · ||22 .

↵

↵

↵

A Glimpse at Algorithms
Linear programming:O(n3 log(n)2 )O(n3 )
Hungarian/Auction:
Pn
Pn
1
1
↵ = n i=1 xi= n j=1 yj

1-D case: sorting O(n log(n)).

p=1
d = || · ||W1 (↵, ) = mindiv(u)=↵<latexit sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit><latexit
sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit>R||u(x)||dx! min-cost flow, on graphs O(n2 log(n)).
Monge-Ampère/Benamou-Brenier, d = || · ||22 .

Semi-discrete: Laguerre cells, d = || · ||22 .

↵

↵

↵

A Glimpse at Algorithms
Linear programming:O(n3 log(n)2 )O(n3 )
Hungarian/Auction:
Pn
Pn
1
1
↵ = n i=1 xi= n j=1 yj

1-D case: sorting O(n log(n)).

p=1
d = || · ||W1 (↵, ) = mindiv(u)=↵<latexit sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit><latexit
sha1_base64="XsMmew57Vi3OsYPHSHUGZ9XmUqw=">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</latexit>R||u(x)||dx! min-cost flow, on graphs O(n2 log(n)).
Monge-Ampère/Benamou-Brenier, d = || · ||22 .

Semi-discrete: Laguerre cells, d = || · ||22 .

Entropic regularization: generic d.
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Overview• Monge Formulation• Continuous Optimal Transport• Kantorovitch Formulation• Applications

Wasserstein Barycenters
Barycenters of measures (↵s )Ss=1 :?

↵ 2 argmin
↵

P

P

s

s = 1

p
W
s
p (↵, ↵s )sGuillaume Carlier

Martial Agueh

↵1

↵3

↵2

2 ⌃3

min
<latexit 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s
p (↵, ↵s )WassersteinP<latexit 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s

s ↵s

EuclideanWasserstein Barycenters
Barycenters of measures (↵s )Ss=1 :?

↵ 2 argmin
↵

P

P

s

s = 1

p
W
s
p (↵, ↵s )sGuillaume Carlier

Martial Agueh

↵1

↵3

↵2

2 ⌃3

min
<latexit 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s
p (↵, ↵s )WassersteinP<latexit 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s

s ↵s

Euclidean

Examples[Solomon et al, SIGGRAPH 2015]

MRI Data Procesing [with A. Gramfort]
Ground cost c = dM : geodesic on cortical surface M .

L2 barycenter

W22 barycenter

Generalizations: Gradient Flows
Implicit stepping:<latexit sha1_base64="AZ5Bns6nVuPv1SK9pjo6pPxTkzk=">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</latexit>↵t+⌧ = argmin Wpp (↵t , ↵) + ⌧ f (↵)↵

Limit ⌧ ! 0:<latexit sha1_base64="fyvJjWdDXFW519cm5OzW7IKdgkQ=">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David Kinderlehrer

Felix Ottot<latexit sha1_base64="C9nT8d9sEcBu57RiNXNlIoGGDS0=">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</latexit><latexit

Gradient Flows Simulation

https://www.youtube.com/watch?v=tDQw21ntR64
Tim Whittaker (New Zealand)

Gradient Flows Simulation

https://www.youtube.com/watch?v=tDQw21ntR64
Tim Whittaker (New Zealand)Bag of Words↵

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Obama
speaksto themedia
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greetsObamapresident speaks
ChicagoIllinois

media
press

The
president
greetsthepressinChicagoword2vec embedding [Mikolov et al. 2013]

Word mover’s distance: [Kusner et al 2015]
Dist(D1 , D2 ) = W2 (↵, )<latexit sha1_base64="QCSRmVm3aqRzvp3CFa8R51ft4uA=">AABCIXictVxLcxu5EYY3r7Xz8ibHXCbROmWnvI6kuCqp2tqqlSVZ1lq2aZOSvLu0VTPkiKY95NAzpCyby1+TH5JTDrmlckvlDySVQy7JPf0ABhgSM8AojlGSMCC+7kYP0OhugI4myTCfrq//9dIH3/r2d777vQ8vX/n+D374ox9f/egnR3k6y3rxYS9N0uxpFOZxMhzHh9PhNImfTrI4HEVJfBy92sbPj8/iLB+m48707SR+NgoH4+HpsBdOoenk6kl3Gp9P5zvAaXF93iWC8yzuL3ZONhY3VUOUzGJo2Vzc+CxgwPHiZLPUvxsmS/27Uby4cXJ1bf3WOv0LVisbsrIm5L9W+lFwKLqiL1LREzMxErEYiynUExGKHMrXYkOsiwm0PRNzaMugNqTPY7EQVwA7g14x9Aih9RX8HsDT17J1DM9IMyd0D7gk8JMBMhDXAJNCvwzqyC2gz2dEGVuraM+JJsr2Fv5GktYIWqfiBbS6cKqnLw7HMhWn4nc0hiGMaUItOLqepDIjraDkgTGqKVCYQBvW+/B5BvUeIZWeA8LkNHbUbUif/516Yis+92TfmfgHSXkNSiDacvRpQSEUZ0Q/oLc5g89YngQ4D4BCLMeItTek6xGNfgz959D+EMqCakonEZQ5tS5qkdtQbMhtJ3IPig2550QeQLEhD5zIFhQbsiWRiM1I53Z8G4oN33ZyfgzFhnzsRD6BYkM+cSKPoNiQR07kV1BsyK+cyLtQbMi7TuR9KDbkfSeyA8WG7DiRh1BsyEMncheKDbkrkdUrNYOSEp2hY1VuQb3MAy1FAi1bTvnukHW0Ye94rOleBda9qnfgrx2746HTuAK76zHvTiuw7pm3BzbSjnXbonu0m9iw95zYfZgBduy+E/uFeFmB/cJjpb2qwLrX2gH0s2Pd1vcBPNmxD5zYh1CzY9171CNosWMfeewYkwpsy4l9LF5XYH2sflaBddv9NtgVO9a9T3Wgvx3rY01nFVi3PT0CD8aOde9Wx9Bqxx47sU/FeQX2qRP7JVh3O/ZLjx32XQVW7bFXaAcZkD8Sw4qtoxYWqxJrE6AWOvgnxd6SkG8cQbsLMygwA8KMnIi9ArHniTgoEAfecuWFHc3J33VzaReIticiKvYmrE2d/ftFf6wlHoidArGzhKjzSPFdq7GckXehWlzIabFzYc1nTGlhv7EWy/lQb3kV4lEJwXP7Bc38mxQtYQSFmqqj9qLY4xkZ0HMd4g1Fb2qUiocbNy2sgok6d6IiCypyot5aUG+dqJkFNXOiziyoMydKr3wT1/WYAVr/+C7m9MQzgH3k6hKAV7AFu849WKMBzJ8WeIFPqOUR/G1T7O0qdZJhNI/7JGY5npUscQa1uViDdh0V7lB8ndAKi0Ey7vlIxvj4hLmNuVxzbIUXxU4eFBkTfzpDkmdQ0EFvMaD11IzOfWpZkHfHtWb4e8W6V7Vm+F3S+IK8eK41w0+l9NMLyN6R2M4FsG1YTROpfV1vSoPzL0xD1a/QrosWF9/qSM4ZpHfekP6+fDP7F3gv21Rj/eh6Mxq5Mb68NL4mNLSec0PPzaig98Rer6oFjUcylnGvrjeVIaVddCzl0E9N3wz26cs3o+rNaLTA49qmmHtu1JvO3kkxGl1vRuNIcN5zQZ68qjejMaBn1oeuN6OB2ZZQxvm63tSyowY4dtb1plZ9TFlgzAHxnOcW7RVl5CfNJLUh+Qf12RrT51/dxzBn87yIEeopad+2mk5U7GX1Eil/IQarNm0oB/oXM8MHK9OYi01nfMUyTEv7+yodvcej5g9AiwGsfj4DcOXME5BQ5STQeidAccMZdZVHpnCbThzOktMlVFe2Tp3eoubLWaNy2wm1uuIyPVqtxy7Z65zm3oR8wgPSrEsPB5VvuIqiS0MHJQ256TXR3Tu5XsvaX3fiJkuISTHTenQixCdp9XGqTettQ8fX5CnPFAqf+ej5i9nmU2ltMOZJyRahLHU8zX4qj2S24b56U+gcN38W0BtFe3VGVmNIJ1K5MwpV2WL2xuf0rGkf0pkc8mAaPXiPgaQyEXxqhll0zKcHZFFNe+vijfpSGTqu52R1lT2uRw8M9MCCbh7jbMOO8RBqHYgZDuGp4xHlXCl0lZLGM/FJcTqa0husj+iTkoVUNNjexCULWRdlvyhReQNonA0cpfvTWKaj8N0VSu6o3yaPjl3Llv8andyq8+2Q5nj1bK7OxPSJ6yZxDWjV8KkuPy1zYAnm1k82yX+tHyXya8IRbaiL63ODM+tlTCf+MUWwE/KME1ptrtVR7m3mp5Y/UZxaQp2d42l2ShYyIPsXwP6U0pwM6Me8O6BO0NkiJGQjfezOsPBubL7O0DnHtB83FHyrQc+3mGzZjPgruubqymkucsTA+8BiaW4rnRyQLxgT10xad72263cfROp7EuYsYYp6rlwn/jfot/pR82RtZUaghvEN5NLW2d5HSjEL6iikXb7eBqm+ppQfFzI8l1Lr/U/L9HFJsh2KuFAe3K37wLlHz8wLZ0lGcucrfXgfrcvmIuXJkh5xtKcUxbPdH8gdGOW+SbvkGq25Ls2SAcyCaRFFqL6uLPIy33peZep+tPP/C3Wt67LWkGIgdAaXNeTK78cUrZlSJjCref6+otVk13q21Kuez5jm4shYy99A68/ht5JbPfvRiUpW4Q7NAaagn7RGuCVY6eHH606Jl5qZipZ+1vz0nFS9zJaLxNds3XSMfdaYSotmzbnMWqj6RWi8NGi89NRhh84atRZVu7JEJ87YoiNPK335NeHWaUB55qTs9sgUaughpRlL+VHtO6m6Y3yFeuekte6kFcJqNU8DzDXvg7Sv9eXV/U2xuwfiLvk2PfLAOH7p0yodks+lWusjNaaAnG9L+2qu/i61IPeILChS5nucuGL41KlHZVFI+ku5s6Vk57VFUPeW3sg+ysZ2qf6bFeSI1kRO61IhblOPWMpvyhEsWaRbhs8RUOY/JJ+K/Y76mNnsrd9JUPIndLzJq0rz4khhTPp3Zd72V6LXfSN+DSgmnEnvOgJazd8wUmCMyiTYPcuc3hDucnySwB5tRPZz1U7xKd7YkOgWST0Xn3nYGI569Vw355YasRrbr6Anal2/dVsPN7/Em6OL30VO9ELa1UbSR50vPV+MVih3ufJznR5mS3y1PmbUx4wsdJRXxnTFp95cWKJmXBjjw6XZKJrI30zyJjLz6ZQvZdVbUS5nGtjGvKB4yXUPFBE27+661Zu74RhHtEIvIqxJjVtclDAbl8r8gGlpMSt1eWUf4tbLtbtRYuxEVTuFom7uFtp+s4WMyfolwpWz4d6m7N1SlOLOwjCFnuAbvVXxoUnzUyj4OxC26FBx9MkdtsG/3RLbYvc93IZ4Leuc0QyoBW1Bfyn2DuU4yz3qdfTaoG7S9+Hgz2MIunZJP6SdtKnsTNktuUndn/4bsgKZiJ3S657Nx2BycY9klVOT8QzJsrlHMxTquzhNx6I4+IykzMWfD59ruEZxKtR3mpqNQVF3j6DMoQkPdY/B753r3s15mZzq9bXKxZcH7wLqxEXh8OSvOlbR/XwsVGa8kffPAa3DaQ11tVv8r+NQfDSn5rx8ueX0XbOXHm+d+8UyI4v+cPM1o7n5zOZqjv4802J02luy82O/L2j0plJjNO+fPvqjeg4oXnPBeVC3dIw3Z5GW15cKngvYZEjFv8QfLrm/jfC6oFElRxNK6pyimprq4aamvnFpG536zEcmTadKpjI1HUe06UbsttgXd+Fnu/AAm94O5e9S8l/E2r8/24fWU7IeKovOmYMutcWU/dCnaH161vdnqyTGu7x8t7cDLXgWfkCteM/3IfXHu76d0tiqv0HCa/2BSEW/FJEsn+7pdRXBCMonb5wDUt/zDeguPWex+ObZyONsUd2fWpZoTp+4bxZElfjIkLJHc3Uiz+rx5ABv2IdFfigQv6a2UNp53HNdnFuVnFtLnHPSTpnDufFZ/d2sKi7bBpd+kTs7k/1SirP1eV59bnSnkgvfQa/HD2rwA0PKNmn/FUXCmajP5s1qaM6kTOYJ61ioTCTrAePMsHjf9ZHtWQ2vM4/x369E3zck3QNZIsp/B3TClhG9ROpml6Tnm471mdR7NdLK71GeXF3bWP6/DFYrR5u3NtZvbTy+vfb5Hfn/HHwofiZ+Ia7DGv+t+ByotcQhcPij+Kf4t/jP1u+3/rT1562/cNcPLknMT0Xp39bf/gsBT8lt</latexit><latexit
sha1_base64="QCSRmVm3aqRzvp3CFa8R51ft4uA=">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</latexit>

Density Fitting and Generative Models
Observations:

def.

=

1
n

Pn

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i=1 xiParametric model: ✓ 7! ↵✓<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit><latexit
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Observations:

def.

=

1
n

Pn

↵✓
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Parametric model: ✓ 7! ↵✓<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit><latexit
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Density fitting: d↵✓ (x) = ⇢✓ (x)dxc ✓| ) =min KL(↵

def.

✓

X
i

log(⇢✓ (xi ))Maximum
likelihood (MLE)xi

Density Fitting and Generative Models
Observations:

def.

=

1
n

↵✓

Pn

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Parametric model: ✓ 7! ↵✓<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit><latexit
sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit>

Density fitting: d↵✓ (x) = ⇢✓ (x)dxc ✓| ) =min KL(↵

def.

✓

X

log(⇢✓ (xi ))i

Generative model fit: ↵✓ = g✓,] ⇣c ✓ | ) = +1
KL(↵

! MLE undefined.
! Need a weaker metric.
p
min W",p (↵✓ ,✓)Maximum
likelihood (MLE)

g✓

↵✓

⇣
Z

X

xi

Deep Discriminative vs Generative Models

z
x

d⇠

Generative

Discriminatived⇠ (x) = ⇢(⇠K (. . . ⇢(⇠2 (⇢(⇠1 (x) . . .)Deep networks:g✓ (z) = ⇢(✓K (. . . ⇢(✓2 (⇢(✓1 (z) . . .)
⇢
⇠1 ⇢
✓
⇠2 ⇢
✓1 ⇢ 2zg✓x

Deep Discriminative vs Generative Models

Generative

Discriminatived⇠ (x) = ⇢(⇠K (. . . ⇢(⇠2 (⇢(⇠1 (x) . . .)Deep networks:g✓ (z) = ⇢(✓K (. . . ⇢(✓2 (⇢(✓1 (z) . . .)
⇢
⇠1 ⇢
✓
⇠2 ⇢
✓1 ⇢ 2

z
x

z

g✓

d⇠

x

z2

g✓

x

z
d⇠
Z

X

z1

Examples of Images GenerationZ(a) WGAN MNIST(b) WGAN CelebA(a) WGAN MNIST⇣g✓(c) WGAN LSUN(b) WGAN CelebA(c) WGAN L

↵✓

h⇠

X
(d) MMD GAN MNIST
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Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and LSUN bedroom!
Need to learn the metricd(x,y) = ||hh⇠WGAN(y)|| (GANs)
datasets.
⇠ (x) from
Figure 2:
Generatedsamplesand MMD GAN on MNIST, CelebA, anddatasets.
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Ian Goodfellow

digits in Figure 1c are more natural with smooth outline and sharper strike. For CIFAR-10 dataset,Progressive Growing of GANs for Improved
Quality, Stability, and Variation
Tero Karras, Timo Aila, Samuli Laine,
Jaakko Lehtinen, ICLR 2018g✓

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XProgressive Growing of GANs for Improved
Quality, Stability, and Variation
Tero Karras, Timo Aila, Samuli Laine,
Jaakko Lehtinen, ICLR 2018g✓

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X

Conclusion: Toward High-dimensional OT
Monge

Kantorovich

Dantzig

Brenier

Otto

McCannVillaniFigalli

Optimal transport framework Sliced Wasserstein projection Applications

Application to Color TransferSi

Source image (X )Style image (Y )J. Rabin