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

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

Overview• Linear Programming• PDE-based• Semi-discrete• Entropic Regularization

Linear Programming
Transportation
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def.

U(a, b) =<latexit sha1_base64="kgzb7F/dqp5hDBRkgfKQpy5m0Rc=">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
sha1_base64="kgzb7F/dqp5hDBRkgfKQpy5m0Rc=">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</latexit>>
P 2 Rn⇥m
;
P1
=
a,
P
1m = bn+Linear program:nPomini,j Pi,j Ci,j ; P 2 U(a, b)

o

P?
C

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Dantzig

Linear Programming
Transportation
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def.

U(a, b) =<latexit sha1_base64="kgzb7F/dqp5hDBRkgfKQpy5m0Rc=">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
sha1_base64="kgzb7F/dqp5hDBRkgfKQpy5m0Rc=">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</latexit>>
P 2 Rn⇥m
;
P1
=
a,
P
1m = bn+Linear program:nPomini,j Pi,j Ci,j ; P 2 U(a, b)

o

P?
C

U(a, b)<latexit sha1_base64="6K81TKRbGsC5Lqg6kSy+CEpMFfw=">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</latexit><latexit
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Theorem: 9P? solution extremal point of U(a, b)| (i, j) ; P?i,j = 0 | 6 n + m 1↵
Extremal points , no cycle
Non-extremal

ExtremalDantzig

Linear Programming
Transportation
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def.

U(a, b) =<latexit sha1_base64="kgzb7F/dqp5hDBRkgfKQpy5m0Rc=">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
sha1_base64="kgzb7F/dqp5hDBRkgfKQpy5m0Rc=">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</latexit>>
P 2 Rn⇥m
;
P1
=
a,
P
1m = bn+Linear program:nPomini,j Pi,j Ci,j ; P 2 U(a, b)

o

P?
C

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Theorem: 9P? solution extremal point of U(a, b)| (i, j) ; P?i,j = 0 | 6 n + m 1↵
Extremal points , no cycle
Non-extremal

Dantzig↵

ExtremalExample: if n = n, a = b = 1/n, P? permutation matrix.
! ⇠ n! extremal points.

Interior Point Methods
Linear programming:
x0 2 argmin {hx, ci ; i = 1, . . . , m, hai , xi 6 bi }<latexit 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= b2

xi

=

b1

, xi
ha 2

c
x0

ha3 , xi=b3

Interior Point Methods
Linear programming:
x0 2 argmin {hx, ci ; i = 1, . . . , m, hai , xi 6 bi }"!0<latexit 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Log-barrier approximation:P
def.
x" = argmin hx, ci " i log(bi
x

=
xi
1,

ha
<latexit

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sha1_base64="cWXr3qqkbrKWFFwsDjIpDSdFnLc=">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

= b2

x0

b1

, xi
ha 2

c

x"
ha3 , xi=b3

hai , xi)

Interior Point Methods
Linear programming:
x0 2 argmin {hx, ci ; i = 1, . . . , m, hai , xi 6 bi }"!0<latexit 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Log-barrier approximation:P
def.
x" = argmin hx, ci " i log(bixhai , xi)

Yurii
Nesterov

Interior
point method:p<latexit sha1_base64="lsfB2j+kwlsTsMdJrYGb1OZHqoc=">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
sha1_base64="lsfB2j+kwlsTsMdJrYGb1OZHqoc=">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</latexit>O( m log( m⌧ )) Newton iterations computes

x0

=
xi
1,

ha

= b2

c

b1

, xi
ha 2

feasible x̂" with hx̂"x"
ha3 , xi=b3x0 , ci 6 ⌧Arkadi
Nemirovski

Network Flowi w i, j<latexit sha1_base64="C3HSIygLm7jMq8Lhb9ojN1NPja8=">AABBzHictVzbchu5EYU3t7Vz8yaPeZlE65Q35TiS46oktZWqlSVZ1pq2aJOyvbu0XbyMaNpDDs0h5QuXr/mFvCZfku/IHyRP+YX0BRhgSMw0RnGMkoQBcbobPUCjuwG6N01G2Xx7+58XPvrOd7/3/R98fPHSD3/045/89PInP3uUpYtZPz7pp0k6e9LrZnEymsQn89E8iZ9MZ3F33Evix71Xe/j547N4lo3SSXv+bho/HXeHk9HpqN+dQ9PTDlFY9pJFvBo9v7y1fX2b/kWblR1d2VL6XzP9JDpRHTVQqeqrhRqrWE3UHOqJ6qoMyjdqR22rKbQ9VUtom0FtRJ/HaqUuAXYBvWLo0YXWV/B7CE/f6NYJPCPNjNB94JLAzwyQkboCmBT6zaCO3CL6fEGUsbWM9pJoomzv4G9P0xpD61y9gFYJZ3qG4nAsc3Wq/khjGMGYptSCo+trKgvSCkoeOaOaA4UptGF9AJ/PoN4npNFzRJiMxo667dLn/6Ke2IrPfd13of5NUl6BEqmWHn2aU+iqM6If0dtcwGcsTwKch0Ah1mPE2hvS9ZhGP4H+S2i/D2VFNaOTHpQlta4qkXtQfMg9EXkIxYc8FJENKD5kQ0Q2ofiQTY1E7Ix07se3oPjwLZHzAyg+5AMR+RCKD/lQRD6C4kM+EpFfQ/EhvxaRt6H4kLdF5F0oPuRdEdmG4kO2ReQJFB/yREQeQPEhDzSyfKXOoKREZySsyl2oF3mgpUigZVeU7xZZRx/2VsCa7pdg5VW9D3/92P0AncYl2IOAeXdagpVn3iHYSD9WtkV3aDfxYe+I2COYAX7skYj9Ur0swX4ZsNJelWDltdaAfn6sbH3vwZMfe0/E3oeaHyvvUcfQ4sceB+wY0xJsU8Q+UK9LsCFWf1aCle1+C+yKHyvvU23o78eGWNNFCVa2p4/Ag/Fj5d3qMbT6sY9F7BP1tgT7RMR+Bdbdj/0qYId9X4I1e+wl2kGG5I/EsGKrqHXzVYm1KVDrCvyTfG9JyDfuQbuEGeaYIWHGIuIwRxwGIho5ohEsV5bb0Yz8XZlLK0e0AhG9fG/C2lzsP8j7Yy0JQOzniP01RJVHiu/ajOWMvAvTIiHn+c6FtZAxpbn9xlqs50O15TWI4wKC5/YLmvnXKFrCCAo1VUXtRb7HMzKi5yrEG4rezCgNDxk3z62Ci3oronoeVE9EvfOg3omohQe1EFFnHtSZiLIr38V1AmaA1T++iyU98QxgH7m8ROAV7MKucwfWaATzpwle4ENqOYa/LYq9pVIlGUbzuE9iluNpwRLPoLZUW9Buo8J9iq8TWmExSMY9j3WMj0+Y21jqNcdWeJXv5FGeMQmnMyJ5hjkd9BYjWk/16NyllhV5d1yrh7+Tr3tTq4c/II2vyIvnWj38XEs/P4fsbY1tnwPbgtU01dq39bo0OP/CNEz9Eu26aHHxrY71nEF6b2vSP9Jv5ugc72WPaqwfW69HI3PGlxXGV4eG1XPm6LkeFfSe2Os1taj2SCY67rX1ujKktItOtBz2qe6bwT4D/WZMvR6NJnhcexRzL5163dk7zUdj6/VoPFKc91yRJ2/q9WgM6Zn1Yev1aGC2pavjfFuva9lRAxw723pdqz6hLDDmgHjOc4v1imbkJy00tRH5B9XZGtfn39zHMGfzLI8RqilZ37acTi/fy6olMv5CDFZtXlMO9C8Wjg9WpLFUN8T4imWYF/b3TTp2j0fNN0CLEax+PgOQcuYJSGhyEmi9E6C4I0ZdxZEZ3A0Rh7PkdA3V0a1z0Vu0fDlrVGx7Tq1SXGZHa/XYIXud0dybkk/YIM1KemiUvuEyipKGGgUNyfTq6O69Xq9F7W+LuOkaYprPtD6dCPFJWnWc6tN6y9HxFX3KM4fCZz52/mK2+VRbG4x5UrJFKEsVT7efySO5bbivXlM2x82fRfRG0V6dkdUY0YlUJkahJlvM3viSni3tEzqTQx5Mow/vMdJUpopPzTCLjvn0iCyqa28l3qgvk6HjekZW19jjavTQQQ896Poxzh7sGPeh1oaY4QSe2gFRzqVcVylpfKZ+m5+OpvQGqyP6pGAhDQ22N3HBQlZF2S8KVN4AGmcDR+nhNNbpGHxng5Ic9fvksbFr0fJfoZNbc77dpTlePpvLMzED4nqDuEa0avhUl5/WObAES+8nN8h/rR4l8qvDEW2oxPWZw5n1MqET/5gi2Cl5xgmtNml1FHu7+an1TwynpjJn53ianZKFjMj+RbA/pTQnI/px7w6YE3S2CAnZyBC7M8q9G5+vMxLnmPXjRopvNdj5FpMtWxB/Q9ddXRnNRY4YeB9Yrc1to5MG+YIxcZ1p627XdvXug0h7T8KdJUzRzpWrxP8z+m1+zDzZ2pgRqGF8A5m2db73kVLMgjrq0i5fbYNMX1fKT3MZnmmp7f5nZfq0INk+RVwoD+7WA+Dcp2fmhbNkRnJnG314H63K5iLl6ZoecbSnFMWz3R/qHRjlvka75BatuQ7NkiHMgnkeRZi+UhZ5nW81ryL1MNrZ/4W61XVRa0gxUjaDyxqS8vsxRWuulAnMap6/r2g1+bU+W+tVzWdCc3HsrOVvofWX8NvIbZ7D6PQKVuEWzQGmYJ+sRrgl2ugRxutWgZeZmYaWfbb87Jw0vdyW88TXbN1sjH1Wm0qTZs1bnbUw9fPQeOnQeBmowzadNVotmnZjiZ6LsUVbn1aG8qvDrV2D8kKkLHtkBjUKkNKNpcKoDkSqcoxvUO9FWtsirS6sVvc0wF3zIUj/Wl9f3d/mu3ukbpNv0ycPjOOXAa3SEflcprU6UmMKyPmmtq/u6u9QC3LvkQVFynyPE1cMnzr1qaxySX+td7aU7Ly1CObe0hvdx9jYDtV/v4Ec05rIaF0axE3qEWv5XTmiNYt03fE5Isr8d8mnYr+jOmZ2e9t3EhX8CRtv8qqyvDhSmJD+pczb0Ub0euTErxHFhAvtXfeAVv03jBQYYzIJfs8yozeEuxyfJLBH2yP7uWmn+BRv4kh0naReqj8H2BiOeu1cd+eWGbEZ22+gJ2rdvnVfD5lfEsxR4neeE70u7Wpj7aMu157PR6urd7nic5UeFmt8rT4W1MeNLGyUV8R01OfBXFiielwYE8Kl3ijqyF9P8joy8+lUKGXT21AuZhrYxrygeEm6B4oIn3d31evNfSaMo7dBr0dYlxq3SJQwG5fq/IBraTErdXFjH+LWi5W7UeLsRGU7haHu7hbWfrOFjMn6JUrK2XBvV/ZOIUqRszBMoa/4Rm9ZfOjS/BwK/o6ULzo0HENyhy3wb3fVnjr4ALchXus6ZzQjakFbMFiLvbt6nMUe1Tp67VB36YdwCOcxAl1L0o9oJ60rO1OWJXeph9N/Q1ZgpmJRetuz/hhcLvJINjnVGc+ILJs8mpEy38WpOxbDIWQkRS7hfPhcQxrFqTLfaao3BkNdHkGRQx0e5h5D2Du3vevzcjlV62uTSygP3gXMiYvB4clfeaxi+4VYqJnzRj48B7QOpxXUzW7xv47D8LGc6vMK5ZbRd81eBrx17hfrjCz6w/XXjOUWMpvLOYbzTPPRWW/Jz4/9vqjWm0qd0Xx4+uiP2jlgeC0V50Fl6RjvziIrbygVPBfwyZCq/6h/XJC/jfA6p1EmRx1K5pyinJrpIVMz37j0jc58FiKTpVMmU5GajSNadCN2Tx2p2/Czl3uAdW+H8ncp+S9i/d+fHUDrKVkPk0XnzEGH2mLKfthTtAE92/uzZRLjXV6+29uGFjwLb1Ar3vO9T/3xrm+7MLbyb5DwWr+nUjUoRCTrp3t2XfVgBMWTN84Bme/5RnSXnrNYfPNsHHC2aO5PrUu0pE/kmwW9UnzPkbJPc3Wqz+rx5ABv2Hfz/FCkfkdtXW3ncc+VODdLOTfXOGeknSKHt85n1XezyrjsOVwGee7sTPdLKc6253nVudH9Ui58B70aP6zADx0pW6T9VxQJz1R1Nm9RQXOhZXJPWCfKZCJZDxhndvP3XR3ZnlXwOgsY/91S9F1H0kOQpUf574hO2GZEL9G6OSDp+aZjdSb1ToW0+nuUzy9v7az/XwablZMb1/90fefBza0vbun/5uBj9Qv1K3UVlvgf1BdArKlOyNz+Vf1N/X33eHexu9xdcdePLmjMz1Xh3+5f/gv3TaX0</latexit><latexit

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Divergence on a graph:def. Pdiv(s)i = (i,k)2G si,kP(k,i)2G sk,iMin-cost flow:min {hs, wi ; div(s) = h}s>0Network Flowi w i, j<latexit sha1_base64="C3HSIygLm7jMq8Lhb9ojN1NPja8=">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</latexit><latexit

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Divergence on a graph:def. Pdiv(s)i = (i,k)2G si,kP(k,i)2G sk,iMin-cost flow:min {hs, wi ; div(s) = h}s>0Optimal transport: bi-partite graph.
h = ( a, b) wi,j = Ci,j
s=P

a

b

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Divergence on a graph:def. Pdiv(s)i = (i,k)2G si,kP(k,i)2G sk,iMin-cost flow:min {hs, wi ; div(s) = h}s>0Optimal transport: bi-partite graph.
h = ( a, b) wi,j = Ci,j
s=P

a

b

Theorem: on a graph with E edges and V vertices,
9 a network simplex algorithm of complexity

Robert Endre
TarjanO(V E log V log(V ||C||1 )) if Ci,j 2 Z.

OT simplex: n = m, complexity O(n3 log(n)2 ).

James Orlin

I n
ÿ

J

W1 as
a b)ReducedFlows
W1 (a,= max
fi (aMin-cost
E, |(Òf)i,j | Æ wi,j .
i ≠ bi ) : ’(i, j) œ
fœR
n

i=1

GeodDist(i,analogousj)i,j =The associatedCdual
problem,which iswtheof Formula (6.4), is then<latexit sha1_base64="TXfcToHAWoD2Yv1jStLf2Nzu3hg=">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</latexit><latexit
sha1_base64="TXfcToHAWoD2Yv1jStLf2Nzu3hg=">AABB73ictVzNchu5EYY3f2vnz5scc5lE65Sd8iqy46oktbVVK1OyrLVsyyZle3dpq/gzommPODSHlGRz+RY55JTKNa+QSw7JS+QNklNeIf0DDDAkZhqjOEZJwoD4uhs9QKO7Abo7TobZdGPjnxc++Na3v/Pd73148dL3f/DDH/348kc/eZKls0kvPuilSTp51u1kcTIcxQfT4TSJn40ncee4m8RPu68b+PnTk3iSDdNRa/p2HD8/7gxGw6NhrzOFpsPL6+3G4Xx4/dUi+ixqT+Oz6XwnTvtbwHhxOOeGdvcoOl0srkKva4eX1zbWN+hftFq5oStrSv/bTz+KDlRb9VWqemqmjlWsRmoK9UR1VAbla3VDbagxtD1Xc2ibQG1In8dqoS4Bdga9YujRgdbX8HsAT1/r1hE8I82M0D3gksDPBJCRugKYFPpNoI7cIvp8RpSxtYz2nGiibG/hb1fTOobWqXoJrRLO9AzF4Vim6kj9jsYwhDGNqQVH19NUZqQVlDxyRjUFCmNow3ofPp9AvUdIo+eIMBmNHXXboc//RT2xFZ97uu9M/ZukvAIlUk09+jSn0FEnRD+itzmDz1ieBDgPgEKsx4i1U9L1MY1+BP3n0P4AyoJqRiddKHNqXVQiG1B8yIaI3IHiQ+6IyD0oPuSeiNyH4kPuayRiJ6RzP74JxYdvipwfQfEhH4nIx1B8yMci8gkUH/KJiPwKig/5lYi8A8WHvCMi70HxIe+JyBYUH7IlIg+g+JAHInIbig+5rZHlK3UCJSU6Q2FVbkK9yAMtRQItm6J8t8k6+rC3A9Z0rwQrr+ot+OvHbgXoNC7BbgfMu6MSrDzzdsBG+rGyLbpLu4kPe1fE7sIM8GN3RewX6lUJ9ouAlfa6BCuvtT3o58fK1vc+PPmx90XsA6j5sfIe9RBa/NiHATvGuAS7L2IfqTcl2BCrPynByna/CXbFj5X3qRb092NDrOmsBCvb0yfgwfix8m71FFr92Kci9pk6K8E+E7FfgnX3Y78M2GHflWDNHnuJdpAB+SMxrNgqap18VWJtDNQ6Av8k31sS8o270C5hBjlmQJhjEbGTI3YCEXs5Yi9Yriy3oxn5uzKXZo5oBiK6+d6EtanYv5/3x1oSgNjKEVtLiCqPFN+1GcsJeRemRUJO850LayFjSnP7jbVYz4dqy2sQDwsIntsvaeZfp2gJIyjUVBW1l/kez8iInqsQpxS9mVEaHjJumlsFF3UmoroeVFdEvfWg3oqomQc1E1EnHtSJiLIr38W1A2aA1T++izk98QxgH7m8ROAVbMKucxfWaATzZx+8wMfU8hD+Nin2lkqVZBjN4z6JWY7nBUs8gdpcrUG7jQq3KL5OaIXFIBn3fKhjfHzC3MZcrzm2wot8J4/yjEk4nSHJM8jpoLcY0XqqR+cetSzIu+NaPfzdfN2bWj38Nml8QV481+rhp1r66Tlkb2ls6xzYJqymsda+rdelwfkXpmHql2jXRYuLb/VYzxmkd1aT/q5+M7vneC8NqrF+bL0ejcwZX1YYXx0aVs+Zo+d6VNB7Yq/X1KLaIxnpuNfW68qQ0i460nLYp7pvBvv09Zsx9Xo09sHjalDMPXfqdWfvOB+Nrdej8URx3nNBnryp16MxoGfWh63Xo4HZlo6O8229rmVHDXDsbOt1rfqIssCYA+I5zy3WK5qQnzTT1IbkH1Rna1yff3Ufw5zNizxGqKZkfdtyOt18L6uWyPgLMVi1aU050L+YOT5YkcZc3RTjK5ZhWtjfV+nYPR41vwdajGD18xmAlDNPQEKTk0DrnQDFG2LUVRyZwd0UcThLjpZQbd06Fb1Fy5ezRsW2Q2qV4jI7WqvHNtnrjObemHzCPdKspIe90jdcRlHS0F5BQzK9Orp7p9drUfsbIm68hBjnM61HJ0J8klYdp/q03nR0fEWf8kyh8JmPnb+YbT7S1gZjnpRsEcpSxdPtZ/JIbhvuq9eVzXHzZxG9UbRXJ2Q1hnQilYlRqMkWszc+p2dL+4DO5JAH0+jBe4w0lbHiUzPMomM+PSKL6tpbiTfqy2TouJ6R1TX2uBo9cNADD7p+jNOAHeMB1FoQMxzAUysgyrmU6yoljU/UJ/npaEpvsDqiTwoW0tBgexMXLGRVlP2yQOUU0DgbOEoPp7FMx+DbK5TkqN8nj41di5b/Cp3cmvPtDs3x8tlcnonpE9ebxDWiVcOnuvy0zIElmHs/uUn+a/UokV8djmhDJa4vHM6slxGd+McUwY7JM05otUmro9jbzU8tf2I47Stzdo6n2SlZyIjsXwT7U0pzMqIf9+6AOUFni5CQjQyxO8Pcu/H5OkNxjlk/bqj4VoOdbzHZshnxN3Td1ZXRXOSIgfeBxdLcNjrZI18wJq4Tbd3t2q7efRBp70m4s4Qp2rlylfhfo9/mx8yTtZUZgRrGN5BpW+d7HynFLKijDu3y1TbI9HWl/DiX4YWW2u5/VqaPC5JtUcSF8uBu3QfOPXpmXjhLJiR3ttKH99GqbC5SHi/pEUd7RFE82/2B3oFR7uu0S67RmmvTLBnALJjmUYTpK2WRl/lW8ypSD6Od/V+oW10XtYYUI2UzuKwhKb8fU7TmSpnArOb5+5pWk1/rk6Ve1XxGNBePnbX8DbT+HH4buc1zGJ1uwSrcpjnAFOyT1Qi3RCs9wnjdLvAyM9PQss+Wn52Tppfbcp74mq2bjbFPalPZp1lzprMWpn4eGq8cGq8Cddiis0arRdNuLNGhGFu09GllKL863Fo1KM9EyrJHZlDDACndWCqMal+kKsf4BvVOpLUh0urAanVPA9w1H4L0r/Xl1f1NvrtH6g75Nj3ywDh+6dMqHZLPZVqrIzWmgJxvafvqrv42tSD3LllQpMz3OHHF8KlTj8oil/SXemdLyc5bi2DuLZ3qPsbGtqn+mxXkMa2JjNalQdyiHrGW35UjWrJI647PEVHmv0M+Ffsd1TGz29u+k6jgT9h4k1eV5cWRwoj0L2Xedlei110nfo0oJpxp77oLtOq/YaTAGJNJ8HuWGb0h3OX4JIE92i7Zz1U7xad4I0eidZJ6rj4LsDEc9dq57s4tM2Iztl9BT9S6feu+HjK/JJijxO88J3od2tWOtY86X3o+H62O3uWKz1V6mC3xtfqYUR83srBRXhHTVp8Gc2GJ6nFhTAiXeqOoI389yevIzKdToZRNb0O5mGlgG/OS4iXpHigifN7dVa83d00YR3eFXpewLjVukShhNi7V+QHX0mJW6uLKPsStFyt3o8TZicp2CkPd3S2s/WYLGZP1S5SUs+HeruztQpQiZ2GYQk/xjd6y+NCl+SkU/B0pX3RoOIbkDpvg326qhtp+D7ch3ug6ZzQjakFb0F+KvTt6nMUe1Tp641B36YdwCOcxBF1L0g9pJ60rO1OWJXeph9M/JSswUbEove1ZfwwuF3kkq5zqjGdIlk0ezVCZ7+LUHYvhEDKSIpdwPnyuIY3iSJnvNNUbg6Euj6DIoQ4Pc48h7J3b3vV5uZyq9bXKJZQH7wLmxMXg8OSvPFax/UIs1MR5I++fA1qHowrqZrf4X8dh+FhO9XmFcsvou2avAt4694t1Rhb94fprxnILmc3lHMN5pvnorLfk58d+X1TrTaXOaN4/ffRH7RwwvOaK86CydIx3Z5GVN5QKngv4ZEjVf9TfLsjfRniT0yiTow4lc05RTs30kKmZb1z6Rmc+C5HJ0imTqUjNxhFNuhHbULvqDvw0cg+w7u1Q/i4l/0Ws//uzfWg9IuthsuicOWhTW0zZD3uK1qdne3+2TGK8y8t3e1vQgmfhe9SK93wfUH+869sqjK38GyS81u+rVPULEcny6Z5dV10YQfHkjXNA5nu+Ed2l5ywW3zw7DjhbNPenliWa0yfyzYJuKb7rSNmjuTrWZ/V4coA37Dt5fihSv6a2jrbzuOdKnPdLOe8vcc5IO0UOZ85n1Xezyrg0HC79PHd2ovulFGfb87zq3OhWKRe+g16NH1TgB46UTdL+a4qEJ6o6mzeroDnTMrknrCNlMpGsB4wzO/n7ro5sTyp4nQSM/14p+p4j6Q7I0qX8d0QnbBOil2jdbJP0fNOxOpN6t0Ja/T3Kw8trN5b/L4PVysHN9d+v33h0a+3z2/q/OfhQ/Uz9Ql2FJf5b9TkQ21cHwOAP6q/q7+ofm5PNP27+afPP3PWDCxrzU1X4t/mX/wK8FrME</latexit>

Proposition:[Beckmann]<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit><latexit
sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit>

W1 (a, b) = min

sœRE
+

Y
] ÿ
[

(i,j)œE

Z^

wi,j si,j : div(s) = a ≠ b .
\This is a linear program and more precisely an instance of min-cost flow pro
ai 6=efficient0
Highlydedicated simplex solvers have been devised to solve it, seestance [Ling and Okada, 2007]. Figure 6.1 shows an example of primal and dua
tions. Formulation (6.6) is the so-called Beckman formulation [Beckmann, 1952
has been used and extended to define and study traffic congestion models, seestance [Carlier et al., 2008].
bi = 0<latexit sha1_base64="l2tSCwsm77Z/Pg2/VJaoxwKR7Ss=">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Martin Beckmann

si,j = 0(a, b) and s

mputation of W1 (a, b) on a planar graph with uniform weights wi,j = 1.
6.5) (increasing value from red to blue). The green color of the edges is
ht: flow s solution of (6.6), where bold black edges display non-zero si,j ,

I n
ÿ

J

W1 as
a b)ReducedFlows
W1 (a,= max
fi (aMin-cost
E, |(Òf)i,j | Æ wi,j .
i ≠ bi ) : ’(i, j) œ
fœR
n

i=1

GeodDist(i,analogousj)i,j =The associatedCdual
problem,which iswtheof Formula (6.4), is then<latexit sha1_base64="TXfcToHAWoD2Yv1jStLf2Nzu3hg=">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</latexit><latexit
sha1_base64="TXfcToHAWoD2Yv1jStLf2Nzu3hg=">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</latexit>

Proposition:[Beckmann]<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit><latexit
sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit>

W1 (a, b) = min

sœRE
+

Y
] ÿ
[

(i,j)œE

Z^

wi,j si,j : div(s) = a ≠ b .
\This is a linear program and more precisely an instance of min-cost flow pro
ai 6=efficient0
Highlydedicated simplex solvers have been devised to solve it, seestance [Ling and Okada, 2007]. Figure 6.1 shows an example of primal and dua
tions. Formulation (6.6) is the so-called Beckman formulation [Beckmann, 1952
has been used and extended to define and study traffic congestion models, seestance [Carlier et al., 2008].
bi = 0<latexit sha1_base64="l2tSCwsm77Z/Pg2/VJaoxwKR7Ss=">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Martin Beckmann

si,j = 0(a, b) and s
Network simplex, E,
V = O(n) (e.g. regular graph):

mputation of W1 (a, b) on a planargraph2with uniform weights wi,j = 1.
3
2
Wpvaluein fromO(nredlog(n)) green color ofWthe1 edgesin O(nlog(n)2 )6.5) (increasing
to blue). Theisht: flow s solution of (6.6), where bold black edges display non-zero si,j ,Hungarian and Auction Algorithms! Primal-dual algorithms.
! Hungarian only works for n = m, a = b = 1.
<latexit sha1_base64="OpXBGwZQrMRyphZ1pFZDlsTy03A=">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</latexit><latexit
sha1_base64="OpXBGwZQrMRyphZ1pFZDlsTy03A=">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</latexit>

James Munkres! Auction is approximate alternate c-transforms.
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Harold Kuhn

Dimitri Bertsekas

Overview• Linear Programming• PDE-based• Semi-discrete• Entropic Regularizations

Monge-Ampère equation

29

Z

↵
having Hausdorff dimension smaller than
d
≠
1
(e.g.
hypersurfaces).
min||x T (x)||2 d↵(x)Xonsider costs of the
c(x,
y)
=
h(x
≠
y)
where
h
is
a
strictly=T] ↵form
X
Gaspard
T
n.
Monge

Y
d↵
d
Densities:=measures
⇢↵ ,
=with
⇢ densities, using (2.8),
Monge-Ampère
equation). Fordx
dx<latexit sha1_base64="+Vdl2Cm9sPiNvDWtQV5eT4YVB/Y=">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</latexit><latexit
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UniqueT = r' solving
Theorem:[Brenier]e following
Monge-Ampère-type
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det(ˆ 2 Ï(x))ﬂ— (ÒÏ(x)) = ﬂ– (x)' convex(2.31)

Yann
Brenierœ Rd◊d is the Hessian of Ï. The Monge-Ampère operator det(ˆ 2 Ï(x))
tood as a non-linear degenerate Laplacian. In the limit of small
Ï = Id + ÁÏ, one indeed recovers the Laplacian as a linearization
h mapsdet(ˆ 2 Ï(x)) = 1 + Á Ï(x) + o(Á).
2

s

29

Monge-Ampère equationZ↵
having Hausdorff dimension smaller than
d
≠
1
(e.g.
hypersurfaces).
min||x T (x)||2 d↵(x)Xonsider costs of the
c(x,
y)
=
h(x
≠
y)
where
h
is
a
strictly=T] ↵form
X
Gaspard
T
n.
Monge

Y
d↵
d
Densities:=measures
⇢↵ ,
=with
⇢ densities, using (2.8),
Monge-Ampère
equation). Fordx
dx<latexit sha1_base64="+Vdl2Cm9sPiNvDWtQV5eT4YVB/Y=">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</latexit><latexit
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UniqueT = r' solving
Theorem:[Brenier]e following
Monge-Ampère-type
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det(ˆ 2 Ï(x))ﬂ— (ÒÏ(x)) = ﬂ– (x)' convex(2.31)

Yann
Brenierœ Rd◊d is the Hessian of Ï. The Monge-Ampère operator det(ˆ 2 Ï(x))! asFinite-elements/ finite-di↵erences
tood
a non-linear degenerate
Laplacian. discretization
In the limit of small
coneindeedof convex
functions.
Ï = Idof+ theÁÏ, onerecovers
the Laplacian as a linearization
h maps! non-classicalboundary
conditions.
det(ˆ 2 Ï(x))= 1 + Á Ï(x) + o(Á).
2Brittany
FroeseJean-David
Benamou

Adam
Oberman...

Displacement Interpolation
Optimal assignement:min ||x0
def.

x0,i

xt,i

x1, (i)

Displacement interpolation: ↵t =<latexit sha1_base64="wSZIdWVlUd/e1Ck7WsSz1n74IuU=">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</latexit><latexit
sha1_base64="wSZIdWVlUd/e1Ck7WsSz1n74IuU=">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</latexit>

xt = (1t)x0 + tx1

x1
1
n

P

||
i xt,iDisplacement Interpolation
Optimal assignement:min ||x0
def.

x0,i

x1, (i)xt,iDisplacement interpolation: ↵t =xt = (11nt)x0 + tx1

Monge map ] ↵ = :def.
↵t = ((1 t)Id + t )] ↵ = (tId + (1

||

x1

P

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def.

Optimal coupling ⇡ 2 U(↵, ): ↵t = ((1997.1. Continuous Formulation<latexit sha1_base64="HkcklliKP9H/pizN9q2mP/JWrS0=">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</latexit><latexit
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t –=
0 0

–1/4

t–1/2= 1/2–3/4

t–1= 1t)1)]t)PX + tPY )] ⇡Robert McCannL (–t )= Lc (–, —) as definedin (2.15)
Rd can be computed by looking for a minimal
1th(–t )leads
these twominimal-pathmeasures. This
path is describedby advecting
nitionto the following
reformulationof W 2 , original
introt=0 between
ureusing a vector
field v[2000]
Benamou
and Breniert defined at each instant. The vector field vt and theGeodesic
1⁄
must
satisfy the formulation:conservation of mass ⁄formula,resulting in
2
W 2 (–0 , –1 ) =minÎvt (x)Î2 d–t (x)dt,(7.2)
d(–t ,vt )t sat. (7.1) 0Rˆ–t+ div(––t=0 = –0 , –t=1
= –1 Figure 7.1 shows(7.1)
is a scalar-valued
measureandandvt a vector-valued
measure.
t vt ) = 0
ˆt
mples of such paths of measures.
ormulation (7.2) is a non-convex101
formulation in the variables (–t , vt )t because
nstraint (7.1) involving the product –t vt . Introducing a vector-valued measurelled the “momentum”)
def.
Jt = –t vt ,Benamou-Brenier Formulation<latexit sha1_base64="C0HQ+n/y+AryOqMHF/KlHNr69jM=">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</latexit><latexit
sha1_base64="C0HQ+n/y+AryOqMHF/KlHNr69jM=">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</latexit>u and Brenier showed in their landmark paper [2000] that it is instead convex
riable (–t , Jt )t when writing
W 22 (–0 , –1 ) =min(–t ,Jt )t œC(–0 ,–1 )

⁄ 1⁄
0

Rd

◊(–t (x), Jt (x))dxdt,

e define the set of constraints as
;
<
ˆ–t
def.
C(–0 , –1 ) = (–t , Jt ) :+ div(Jt ) = 0, –t=0 = –0 , –t=1 = –1 ,
ˆt

(7.3)

(7.4)

L (–t )= Lc (–, —) as definedin (2.15)
Rd can be computed by looking for a minimal
1th(–t )leads
these twominimal-pathmeasures. This
path is describedby advecting
nitionto the following
reformulationof W 2 , original
introt=0 between
ureusing a vector
field v[2000]
Benamou
and Breniert defined at each instant. The vector field vt and theGeodesic
1⁄
must
satisfy the formulation:conservation of mass ⁄formula,resulting in
2
W 2 (–0 , –1 ) =minÎvt (x)Î2 d–t (x)dt,(7.2)
d(–t ,vt )t sat. (7.1) 0Rˆ–t+ div(––t=0 = –0 , –t=1
= –1 Figure 7.1 shows(7.1)
is a scalar-valued
measureandandvt a vector-valued
measure.
t vt ) = 0
ˆt
mples of such paths of measures.
ormulation (7.2) is a non-convex101
formulation in the variables (–t , vt )t because
nstraint (7.1) involving the product –t vt . Introducing a vector-valued measurelled the “momentum”)
def.
Jt = –t vt ,Benamou-Brenier Formulation<latexit sha1_base64="C0HQ+n/y+AryOqMHF/KlHNr69jM=">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</latexit><latexit
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riable (–t , Jt )t when writing
W 22 (–0 , –1 ) =min(–t ,Jt )t œC(–0 ,–1 )

⁄ 1⁄
0

Rd

◊(–t (x), Jt (x))dxdt,

e define the set of constraints as
;
<
ˆ–t
def.
C(–0 , –1 ) = (–t , Jt ) :+ div(Jt ) = 0, –t=0 = –0 , –t=1 = –1 ,
ˆt

(7.3)

(7.4)

L (–tt ) t t
0
Lc (–, —) as definedin and
(2.15)
beR computedmeasure.
by ⁄looking
for a7.1
minimala=scalar-valued
measurevRtd cana vector-valuedshows⁄ Figure1–t is a scalar-valued
measureandvavector-valued
measure.
Figure
7.1
shows
1
t
2 intro2minimal-pathth(–suchbetweenthese Wtwomeasures.
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amples of such paths of measures.
d(–t ,vt )t sat. (7.1) 0
R field v and theureusing (7.2)
a vectorfield
v[2000]at each instant.
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vectorBenamou
andisBreniert definedmulationanon-convex
formulationinthevariables(–vt )tt because
tt ,because
e formulation
(7.2)
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formulationinthevariables
(–
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)
t
t
Geodesic
formulation:⁄formula,
where–t is a scalar-valued
measureandvt a vector-valued
measure. Figure 7.1
1⁄
must
satisfy
theconservationof mass
resultinginaint (7.1)(7.1)
involvingtheproduct–v.
Introducingavector-valued
measureconstraint
involvingtheproduct–v.
Introducingavector-valued
measuret tt
2
Wtwo, –1 ) = of suchmin paths
Îvt (x)Î2 d–t (x)dt,(7.2)
ofmeasures.
2 (–0examples
dcalledthe“momentum”)
the “momentum”) (–t ,vt )t sat. (7.1) 0 R
ˆ–t The formulationJ def.
(7.2)
ist , a non-convex formulation in the variables (–t , vt )t b
=
–
v
t
t
+
div(–
v
)
=
0
and
–t=0 = –0 , –t=1
= –1 Figure 7.1 shows
(7.1)
J
=
–
v
is a scalar-valued measureandvtt at ,vector-valued
measure.
t ttˆttheshowedofconstraint(7.1)
involvingthe[2000]
product
Introducingmou
andBrenierin
theirlandmark
paperthat –
itt v
ist .instead
convex a vector-valued m
mples
of
such
paths
of
measures.
d Brenier(often
showedin theirlandmark paper [2000] that it is instead convex
calledthe“momentum”)
variable
(–
,
J
)
when
writingt (7.2)t tormulation
iswritinga non-convex101
formulationin thedef.variables (–t , vt )t because
ble
(–
,
J
)
when
⁄ 1⁄
t t t
Momentumchangeofvariable:
Jt = –t vt ,2
nstraint(7.1)
involving theIntroducing
a vector-valued
measure⁄ 1–⁄
t vt .◊(–
uous
Formulation
103
W
minproduct(7.3)
t (x), Jt (x))dxdt,
2 (–0 , –1 ) =(–t ,JBrenier) 0
Benamouand2t )t œC(–0 ,–1showed
Rd in their landmark paper [2000] that it is instead
lledWthe“momentum”)
(–
,
–
)
=
min
◊(–t (x), Jt (x))dxdt,
(7.3)
1
2 0
d
def.
(–
t ,Jt )t œC(–0
Rwriting
in thevariable(–t , 0asJ,–
when
we define theset ofconstraintst )1t)=
J
–
v
,
t
t
t
+
; is the following lower-semi<function:æ R ﬁ {+Œ}
continuous ⁄convex⁄
ˆ–1t
fineC(–
theset of constraints
as div(Jt ) = 0,paper(–t , Jt ) in: their–t=0 =[2000]–0 , –t=1=–, instead(7.4)convex
2 +landmark
0 , –1 );= showedu andBrenierthat
it1 is
<(x),
W
(–
,
–
)
=
min
◊(–
Jt (x))dxdt,
ˆt
0
1
t
2
Y (–t2,Jt )t œC(–0 ,–1 ) 0 Rd
ˆ–
def.
t
ÎbÎ
)t twhen–riable=t , Jt(–, Jt ) :writing+ div(J_(7.4)
0 , –1 )(–t ) =a 0, –ift=0 a=>–0,
0 , –t=1 = –1 ,]d ˆtthe set of ⁄defineconstraints
as = 0,1⁄’ (a,2whereb) œ Rwe
◊
R
,
◊(a,
b)
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(7.5)
0
if
(a,
b)
+
;
<
_
W 2 (–0 , –1 ) =mindef.
◊(–(7.3)[
tt (x), Jt (x))dxdt,ˆ–+Œd otherwise.
(–t ,J) œC(– ,–1 ) , 0JC(–
+ div(Jt ) = 0, –t=0 = –0 , –t=1 = –1 ,0 , t–t1 ) = 0 (–t t ) R:ˆte define the set of constraints as<
on might seem;complicated,
but
it
is
crucialtoimposethatthemomentum
ˆ–t
def.
C(–
,
–
)
=
(–
,
J
)
:
+ div(J= 0, is–t=0= –0in, –an= –1 , way (7.4)
0 when1
t =t 0. Note alsot )(7.3)
t=1 informal
vanish–t (x)
thatwrittenˆt

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sha1_base64="C0HQ+n/y+AryOqMHF/KlHNr69jM=">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</latexit>

def.

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def.

asures (–t , Jt ) were density functions, but this is acceptable because ◊ is aous function (and hence defined even if the measures do not have a density
to Lebesgue measure), and can thus be extended in an unambiguous way
y to functions.

L (–tt ) t t
0
Lc (–, —) as definedin and
(2.15)
beR computedmeasure.
by ⁄looking
for a7.1
minimala=scalar-valued
measurevRtd cana vector-valuedshows⁄ Figure1–t is a scalar-valued
measureandvavector-valued
measure.
Figure
7.1
shows
1
t
2 intro2minimal-pathth(–suchbetweenthese Wtwomeasures.
This
path
is
described
byadvectingnitionto thefollowingreformulationofW,
originalt )leads
2
(–
,
–
)
=
min
Îv
(x)Î
d–t (x)dt,
t=0 pathssofofmeasures.
1t
2 0
amples of such paths of measures.
d(–t ,vt )t sat. (7.1) 0
R field v and theureusing (7.2)
a vectorfield
v[2000]at each instant.
The
vectorBenamou
andisBreniert definedmulationanon-convex
formulationinthevariables(–vt )tt because
tt ,because
e formulation
(7.2)
is
a
non-convex
formulationinthevariables
(–
,
v
)
t
t
Geodesic
formulation:⁄formula,
where–t is a scalar-valued
measureandvt a vector-valued
measure. Figure 7.1
1⁄
must
satisfy
theconservationof mass
resultinginaint (7.1)(7.1)
involvingtheproduct–v.
Introducingavector-valued
measureconstraint
involvingtheproduct–v.
Introducingavector-valued
measuret tt
2
Wtwo, –1 ) = of suchmin paths
Îvt (x)Î2 d–t (x)dt,(7.2)
ofmeasures.
2 (–0examples
dcalledthe“momentum”)
the “momentum”) (–t ,vt )t sat. (7.1) 0 R
ˆ–t The formulationJ def.
(7.2)
ist , a non-convex formulation in the variables (–t , vt )t b
=
–
v
t
t
+
div(–
v
)
=
0
and
–t=0 = –0 , –t=1
= –1 Figure 7.1 shows
(7.1)
J
=
–
v
is a scalar-valued measureandvtt at ,vector-valued
measure.
t ttˆttheshowedofconstraint(7.1)
involvingthe[2000]
product
Introducingmou
andBrenierin
theirlandmark
paperthat –
itt v
ist .instead
convex a vector-valued m
mples
of
such
paths
of
measures.
d Brenier(often
showedin theirlandmark paper [2000] that it is instead convex
calledthe“momentum”)
variable
(–
,
J
)
when
writingt (7.2)t tormulation
iswritinga non-convex101
formulationin thedef.variables (–t , vt )t because
ble
(–
,
J
)
when
⁄ 1⁄
t t t
Momentumchangeofvariable:
Jt = –t vt ,2
nstraint(7.1)
involving theIntroducing
a vector-valued
measure⁄ 1–⁄
t vt .◊(–
uous
Formulation
103
W
minproduct(7.3)
t (x), Jt (x))dxdt,
2 (–0 , –1 ) =(–t ,JBrenier) 0
Benamouand2t )t œC(–0 ,–1showed
Rd in their landmark paper [2000] that it is instead
lledWthe“momentum”)
(–
,
–
)
=
min
◊(–t (x), Jt (x))dxdt,
(7.3)
1
2 0
d
def.
(–
t ,Jt )t œC(–0
Rwriting
in thevariable(–t , 0asJ,–
when
we define theset ofconstraintst )1t)=
J
–
v
,
t
t
t
+
; is the following lower-semi<function:æ R ﬁ {+Œ}
continuous ⁄convex⁄
ˆ–1t
fineC(–
theset of constraints
as div(Jt ) = 0,paper(–t , Jt ) in: their–t=0 =[2000]–0 , –t=1=–, instead(7.4)convex
2 +landmark
0 , –1 );= showedu andBrenierthat
it1 is
<(x),
W
(–
,
–
)
=
min
◊(–
Jt (x))dxdt,
ˆt
0
1
t
2
Y (–t2,Jt )t œC(–0 ,–1 ) 0 Rd
ˆ–
def.
t
ÎbÎ
)t twhen–riable=t , Jt(–, Jt ) :writing+ div(J_(7.4)
0 , –1 )(–t ) =a 0, –ift=0 a=>–0,
0 , –t=1 = –1 ,]d ˆtthe set of ⁄defineconstraints
as = 0,1⁄’ (a,2whereb) œ Rwe
◊
R
,
◊(a,
b)
=
(7.5)
0
if
(a,
b)
+
;
<
_
W 2 (–0 , –1 ) =mindef.
◊(–(7.3)[
tt (x), Jt (x))dxdt,ˆ–+Œd otherwise.
(–t ,J) œC(– ,–1 ) , 0JC(–
+ div(Jt ) = 0, –t=0 = –0 , –t=1 = –1 ,0 , t–t1 ) = 0 (–t t ) R:ˆte define the set of constraints as<
on might
seem;complicated,
but
it
is
crucialtoimposethatthemomentumNon-smoothconvexoptimization.
ˆ–t
def.
C(–
,
–
)
=
(–
,
J
)
:
+ div(J= 0, is–t=0= –0in, –an= –1 , way (7.4)
0 when1
t =t 0. Note alsot )(7.3)
t=1 informal
vanish–t (x)
thatwrittenˆt
Finite elements/di↵erences
discretization.

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sha1_base64="C0HQ+n/y+AryOqMHF/KlHNr69jM=">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</latexit>

def.

<latexit sha1_base64="6paB0LzF1z3/AHLNRwOBOt3W5gs=">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</latexit><latexit
sha1_base64="6paB0LzF1z3/AHLNRwOBOt3W5gs=">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</latexit>

def.

asures (–t , Jt ) were density functions, but this is acceptable because ◊ is a
Quadratic
coneinterior
ous !
function(and hence
definedeven point.
if the measures do not have a density
to !
Lebesgueand can thus
be extended
in an unambiguous way
First measure),order proximal
methods(ADMM/DR).
y to functions.
<latexit sha1_base64="qmhW0oci/X1mczD/ecUNSpvle54=">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
sha1_base64="qmhW0oci/X1mczD/ecUNSpvle54=">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</latexit>

Overview• Linear Programming• PDE-based• Semi-discrete• Entropic Regularization

Semi-discrete Method
L1 (g)
L4 (g)
L2 (g)=
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def.

Laguerre cell: Lj (g) = x ; 8`, ||x

yj ||2! computation in O(m log(m)) in 2-D and 3-D.

Proposition:
Optimal transport:
yj ! Lj (g)gj 6 ||xy` ||2g`

Semi-discrete Method
L1 (g)=y1

L4 (g)
L2 (g)<latexit sha1_base64="t5eybmmsf+EgOiAsv/fpA0yXp0U=">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</latexit><latexit
sha1_base64="t5eybmmsf+EgOiAsv/fpA0yXp0U=">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</latexit>Pm

j=1 bj yj

Proposition:
Optimal transport:
yj ! Lj (g)y4y2<latexit sha1_base64="FfxBk0L28QlJ67dGvl5dhCzi8Qw=">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</latexit><latexit
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L3 (g)y3<latexit 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def.

Laguerre cell: Lj (g) = x ; 8`, ||xgj 6 ||xyj ||2y` ||2g`! computation in O(m log(m)) in 2-D and 3-D.

Mass conservation:

Gradient descent:

8 j,
g

R

Lj (g)(1d↵ = bj

Franz
Aurenhammer

VladimirOlikerQuentinMérigot+ Laird Prussner⌧ )g + ⌧R

Lj (g)

d↵
Bruno
Levy

Evolution of the Semi-Discrete Optimization
94

Semi-discrete Optimal Transport

¸=1

¸=3

¸ = 50¸ = 100

Matching

Figure 5.2: Iterations of the semi-discrete OT algorithm minimizing (5.8) (herea simple gradient descent is used). The support (yj )j of the discrete measure — isindicated by the red points, while the continuous measure – is the uniform measure
on a square. The blue cells display the Laguerre partition (Lg(¸) (yj ))j where g(¸) isthe discrete dual potential computed at iteration ¸.

use of a Newton solver which is applied to sampling in computer graphics is proposed in [De Goes et al., 2012], see also [Lévy, 2015] for appli-

Evolution of the Semi-Discrete Optimization
94

Semi-discrete Optimal Transport

¸=1

¸=3

¸ = 50¸ = 100

Matching

Figure 5.2: Iterations of the semi-discrete OT algorithm minimizing (5.8) (herea simple gradient descent is used). The support (yj )j of the discrete measure — isindicated by the red points, while the continuous measure – is the uniform measure
on a square. The blue cells display the Laguerre partition (Lg(¸) (yj ))j where g(¸) isthe discrete dual potential computed at iteration ¸.

use of a Newton solver which is applied to sampling in computer graphics is proposed in [De Goes et al., 2012], see also [Lévy, 2015] for appli-

Overview• Linear Programming• PDE-based• Semi-discrete• Entropic Regularization

Entropic
Regularization
ns to the original transport
problem(2.11):
LÁC (a, b) =

def.

min

PœU(a,b)P

ÈP, CÍ ≠ ÁH(P).

(

H(P) =convexi,jfunction,
Pi,j (log(P1)
he objective is a Á-strongly
problem
4.2 has a unique opti
i,j )n. The idea to regularize the optimal transport problem by an entropic t
traced back to modeling ideas in transportation theory [Wilson, 1969]: Ac
patterns in a network do not agree with those predicted by the solution ofl transport problem. Indeed, the former are more diffuse than the latter, w
o rely on a few routes as a result of the sparsity of optimal couplings for 2
igate this sparsity, researchers in transportation proposed a model, calledy” model [Erlander, 1980], that is able to form a more “blurred” predictio
given marginals and transportation costs.
ure 4.1 illustrates the effect of the entropy to regularize a linear "program overx 3 (which can thus be visualized as a triangle in 2-D). Note how the entr
the original LP solution away from the boundary of the triangle. The opti
gressively moves toward an “entropic center” of the triangle. This is fur
d in the proposition below. The convergence of the solution of that regularm towards an optimal solution of the original linear program has been stu
def.

Entropic
Regularization
ns to the original transport
problem(2.11):
LÁC (a, b) =

def.

min

PœU(a,b)P

ÈP, CÍ ≠ ÁH(P).

(

H(P) =convexi,jfunction,
Pi,j (log(P1)
he objective is a Á-strongly
problem
4.2 has a unique opti
i,j )n. The idea to regularize the optimal transport problem by an entropic t
traced back to modeling ideas in transportation theory [Wilson, 1969]: Ac
patterns in a network do not agree with those predicted by the solution ofl transport problem. Indeed, the former are more diffuse than the latter, w
o rely on a few routes as a result of the sparsity of optimal couplings for 2
igate this sparsity, researchers in transportation proposed a model, calledy” model [Erlander, 1980], that is able to form a more “blurred” predictio
given marginals and transportation costs.
ure 4.1 illustrates the effect of the entropy to regularize a linear "program over2 thex 3Sinkhorn(which canthus be visualized
as a triangle
in 2-D). inNotehow⌧ -approximate
solutionO(n⌧ 3 ).entr
algorithm:
the original LP solution 72away from the boundary of the triangle.
The opti
3
Network
simplex:O(n)(exact).
Interior points: O(n log(⌧ )).
gressively moves toward an “entropic center” of the triangle. This is fur
d in the proposition
below. The isconvergence
the solutionof that regular! Regularization
crucial inofhighdimension.
m towards an optimal solution of the original linear program has been stu
def.

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