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

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

Overview• Entropic Regularization and Sinkhorn• Convergence Analysis• Sinkhorn Divergences• Generative Model FittingEntropic Regularization
Schrödinger’s problem:min

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Pi,j Pi,j xi ,yj⇣[1931]
Pi,ja i bj⌘Erwin
Schrödinger↵⇡"<latexit sha1_base64="GFDRy6M/TlDbNJvkPgJDOhXXs88=">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
sha1_base64="GFDRy6M/TlDbNJvkPgJDOhXXs88=">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</latexit>"Entropic Regularization
def.

Relative-entropy: KL(⇡|↵ ⌦ ) =<latexit sha1_base64="1Ecr7XBZCK0ooJx2WSYSxRXyWmM=">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</latexit><latexit
sha1_base64="1Ecr7XBZCK0ooJx2WSYSxRXyWmM=">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</latexit>

Schrödinger’s problem: Z
def.
pW",p (↵, ) = min⇡1 =↵,⇡2 =<latexit sha1_base64="9fViwbMwb16e+5fGKDRerpzqm1o=">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
sha1_base64="9fViwbMwb16e+5fGKDRerpzqm1o=">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</latexit>

Z

log
X2

✓

◆

d⇡
(x, y) d⇡(x, y)d↵d[1931]pX2

d (x, y)d⇡(x, y) + "KL(⇡|↵ ⌦ )Erwin
Schrödinger↵↵⇡"<latexit sha1_base64="tPJGuvugksii+MqmxGAbxV7kM5s=">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</latexit><latexit
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def.

Relative-entropy: KL(⇡|↵ ⌦ ) =<latexit sha1_base64="1Ecr7XBZCK0ooJx2WSYSxRXyWmM=">AABB0nictVxbcxPJFW42t4XcIHnMyyReUmyKdYyzVblspWqNbYwXAQbJwO4KKF3GQjDSCI0kA1o/pPKav5DX5Gfkd+QfJE/5CzmX7ukeqWdOj0Post3T6u+c02e6T59zukV3kgyz2dbWPy988K1vf+e73/vw4qXv/+CHP/rx5Ss/eZSl82kvPu6lSTp90u1kcTIcx8ez4SyJn0ymcWfUTeLH3Ve7+PnjRTzNhum4NXs7iZ+OOoPx8GTY68yg6fnlKw/jBKqL+JN4PJumk7d/eH55Y2tzi/5F65UburKh9L+j9Ep0rNqqr1LVU3M1UrEaqxnUE9VRGZSv1Q21pSbQ9lQtoW0KtSF9HqszdQmwc+gVQ48OtL6C3wN4+lq3juEZaWaE7gGXBH6mgIzUVcCk0G8KdeQW0edzooytZbSXRBNlewt/u5rWCFpn6gW0SjjTMxSHY5mpE/U7GsMQxjShFhxdT1OZk1ZQ8sgZ1QwoTKAN6334fAr1HiGNniPCZDR21G2HPv8X9cRWfO7pvnP1b5LyKpRINfXo05xCRy2IfkRvcw6fsTwJcB4AhViPEWunpOsRjX4M/ZfQfg/KGdWMTrpQltR6VoncheJD7orIAyg+5IGIbEDxIRsi8giKD3mkkYidks79+CYUH74pcn4AxYd8ICIfQvEhH4rIR1B8yEci8isoPuRXIvIWFB/yloi8A8WHvCMiW1B8yJaIPIbiQx6LyH0oPuS+Rpav1CmUlOgMhVW5A/UiD7QUCbTsiPLdJOvow94MWNO9Eqy8qvfgrx+7F6DTuAS7HzDvTkqw8sw7ABvpx8q26DbtJj7sbRF7CDPAjz0UsV+olyXYLwJW2qsSrLzWGtDPj5Wt71148mPvith7UPNj5T3qPrT4sfcDdoxJCfZIxD5Qr0uwIVZ/WoKV7X4T7IofK+9TLejvx4ZY03kJVranj8CD8WPl3eoxtPqxj0XsE/WmBPtExH4J1t2P/TJgh31XgjV77CXaQQbkj8SwYquodfJVibUJUOsI/JN8b0nIN+5Cu4QZ5JgBYUYi4iBHHAQiGjmiESxXltvRjPxdmUszRzQDEd18b8LaTOzfz/tjLQlA7OWIvRVElUeK79qMZUHehWmRkLN858JayJjS3H5jLdbzodryGsT9AoLn9gua+dcpWsIICjVVRe1FvsczMqLnKsQpRW9mlIaHjJvlVsFFvRFRXQ+qK6LeelBvRdTcg5qLqIUHtRBRduW7uHbADLD6x3expCeeAewjl5cIvIId2HVuwxqNYP4cgRf4kFruw98mxd5SqZIMo3ncJzHL8bRgiadQW6oNaLdR4R7F1wmtsBgk4573dYyPT5jbWOo1x1b4LN/JozxjEk5nSPIMcjroLUa0nurRuUMtZ+Tdca0e/na+7k2tHn6fNH5GXjzX6uFnWvrZOWRvaWzrHNgmrKaJ1r6t16XB+RemYeqXaNdFi4tvdaTnDNJ7U5P+oX4zh+d4L7tUY/3Yej0amTO+rDC+OjSsnjNHz/WooPfEXq+pRbVHMtZxr63XlSGlXXSs5bBPdd8M9unrN2Pq9Wgcgce1SzH30qnXnb2TfDS2Xo/GI8V5zzPy5E29Ho0BPbM+bL0eDcy2dHScb+t1LTtqgGNnW69r1ceUBcYcEM95brFe0ZT8pLmmNiT/oDpb4/r86/sY5mye5TFCNSXr25bT6eZ7WbVExl+IwarNasqB/sXc8cGKNJZqW4yvWIZZYX9fp2P3eNR8A7QYwernMwApZ56AhCYngdY7AYo3xKirODKD2xZxOEtOVlBt3ToTvUXLl7NGxbbn1CrFZXa0Vo9tstcZzb0J+YQN0qykh0bpGy6jKGmoUdCQTK+O7t7p9VrU/paIm6wgJvlM69GJEJ+kVcepPq03HR1f1ac8Myh85mPnL2abT7S1wZgnJVuEslTxdPuZPJLbhvvqdWVz3PxZRG8U7dWCrMaQTqQyMQo12WL2xpf0bGkf05kc8mAaPXiPkaYyUXxqhll0zKdHZFFdeyvxRn2ZDB3XM7K6xh5XowcOeuBB149xdmHHuAe1FsQMx/DUCohyLuW6SknjU/VJfjqa0husjuiTgoU0NNjexAULWRVlvyhQOQU0zgaO0sNprNIx+PYaJTnq98ljY9ei5b9KJ7fmfLtDc7x8NpdnYvrEdZu4RrRq+FSXn1Y5sARL7yfb5L9WjxL51eGINlTi+szhzHoZ04l/TBHshDzjhFabtDqKvd381OonhtORMmfneJqdkoWMyP5FsD+lNCcj+nHvDpgTdLYICdnIELszzL0bn68zFOeY9eOGim812PkWky2bE39D111dGc1Fjhh4HzhbmdtGJw3yBWPiOtXW3a7t6t0HkfaehDtLmKKdK9eI/8f02/yYebKxNiNQw/gGMm3rfO8jpZgFddShXb7aBpm+rpQf5TI801Lb/c/K9FFBsj2KuFAe3K37wLlHz8wLZ8mU5M7W+vA+WpXNRcqTFT3iaE8oime7P9A7MMp9nXbJDVpzbZolA5gFszyKMH2lLPIq32peRephtLP/C3Wr66LWkGKkbAaXNSTl92OK1lwpE5jVPH9f0Wrya3260quaz5jm4shZy99A68/ht5HbPIfR6Raswk2aA0zBPlmNcEu01iOM180CLzMzDS37bPnZOWl6uS3nia/ZutkYe1GbyhHNmjc6a2Hq56Hx0qHxMlCHLTprtFo07cYSPRdji5Y+rQzlV4dbqwbluUhZ9sgMahggpRtLhVHti1TlGN+g3om0tkRaHVit7mmAu+ZDkP61vrq6v8l390jdIt+mRx4Yxy99WqVD8rlMa3WkxhSQ86favrqrv00tyL1LFh
sha1_base64="1Ecr7XBZCK0ooJx2WSYSxRXyWmM=">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</latexit>

Schrödinger’s problem: Z
def.
Wp",p (↵, ) = min⇡1 =↵,⇡2 =

Z

log
X2

✓

◆

d⇡
(x, y) d⇡(x, y)d↵d[1931]X2

dp (x, y)d⇡(x, y) + "KL(⇡|↵ ⌦ )min {E(c(X, Y )) + "I(X, Y ) ; X ⇠ ↵, Y ⇠ }(X,Y )
Mutual information<latexit sha1_base64="uua73HSTlgNcDKopCGGMZIfysco=">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</latexit><latexit
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Erwin
Schrödinger

Christian
LéonardProbabilistic Interpretation
def.

Relative-entropy: KL(⇡|↵ ⌦ ) =<latexit sha1_base64="1Ecr7XBZCK0ooJx2WSYSxRXyWmM=">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
sha1_base64="1Ecr7XBZCK0ooJx2WSYSxRXyWmM=">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</latexit>

Schrödinger’s problem: Z
def.
Wp",p (↵, ) = min⇡1 =↵,⇡2 =

Z

log
X2

✓

◆

d⇡
(x, y) d⇡(x, y)d↵d[1931]X2

dp (x, y)d⇡(x, y) + "KL(⇡|↵ ⌦ )Erwin
Schrödinger

min {E(c(X, Y )) + "I(X, Y ) ; X ⇠ ↵, Y ⇠ }(X,Y )
Mutualinformation
7.6. Dynamic Formulation
over the Paths
Space

Christian
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Á=0

Á = .05

Á = 0.2

Á=1

Impact of Regularization
↵

10

"

10 3

0.1

↵

⇡"

⇡" = argmin⇡⇢Z

Theorem:

R2

✓

||x

⇡"

y||2 + " log
"!+1

✓

! ↵⌦

⇡"

d⇡
(x, y)d↵d◆◆d⇡(x, y)+ ; ⇡1 = ↵, ⇡2 ="!0⇡" ! ⇡OT

Impact of Regularization
↵

10

"

10 3

0.1

↵

⇡"

⇡" = argmin⇡⇢Z

Theorem:

R2

✓

||x

⇡"

y||2 + " log
"!+1

✓

! ↵⌦

⇡"

d⇡
(x, y)d↵d◆◆d⇡(x, y)+ ; ⇡1 = ↵, ⇡2 ="!0⇡" ! ⇡OT

Impact of Regularization
Cumulative:
Copula:def.

C⇡ (x, y) =11d⇡(x, y)s
" = .1 " = 10 3

1

10 3

0.1

10

1

1
(s,
t)
=
C
(C
(s),
C
(t))
⇡
⇡
↵
def.

" ↵
" = 10Rx Ry⇡"

Theorem:⇡""!0⇡" (s, t) " !
+1

0
1 t
min(s, t) (dependence)(independence)stImpact of Regularization
Cumulative:
Copula:def.

C⇡ (x, y) =11d⇡(x, y)s
" = .1 " = 10 3

1

10 3

0.1

10

1

1
(s,
t)
=
C
(C
(s),
C
(t))
⇡
⇡
↵
def.

" ↵
" = 10Rx Ry⇡"

Theorem:⇡""!0⇡" (s, t) " !
+1

0
1 t
min(s, t) (dependence)(independence)
st

min

Sinkhorn’s Algorithm ⇣P2U(a,b)

Proposition:<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit><latexit
sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit>Ppi,j d(xi , yj ) Pi,j + "Pi,j log

Pi,j = ui Ki,j vj

Pi,ja i bjdef.

⌘

Ki,j = e<latexit sha1_base64="+2swvmn1CRFRsgsGG2A0werotFo=">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d(xi ,yj )p"min

Sinkhorn’s Algorithm ⇣P2U(a,b)P

pi,j d(xi , yj ) Pi,j + "Pi,j log

Pi,j = ui Ki,j vj

Proposition:<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit><latexit
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⌘

Ki,j = evd(xi ,yj )p"(K> u) = b

min

Sinkhorn’s Algorithm ⇣P2U(a,b)P

pi,j d(xi , yj ) Pi,j + "Pi,j log

Pi,j = ui Ki,j vj

Proposition:<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit><latexit
sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit>uPi,ja i bjKi,j = e(Kv) = av<latexit 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u

def.

⌘

a
Kv

Theorem: [Sinkhorn 1964]v

d(xi ,yj )p"(K> u) = bb
K> u(u, v) converges.

minSinkhorn’s Algorithm ⇣P2U(a,b)

Proposition:Ppi,j d(xi , yj ) Pi,j + "Pi,j log

Pi,j = ui Ki,j vj<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit><latexit
sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit>uPi,ja i bjdef.

Ki,j = e(Kv) = a
u

⌘

v
a
Kv

Theorem: [Sinkhorn 1964]v

d(xi ,yj )p"(K> u) = bb
K> u(u, v) converges.

Only matrix/vector multiplications.

Matrix-vectors

Matrix-matrix

parallelization
K

,...,

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1

K

GPU
v

q

V
,...,

K
v1

! Convolution on regular grids, separable kernels.

vq

Sinkhorn, IPFP, RAS, …

Richard Dennis Sinkhorn

Yule Udny

Edwards Deming

Frederick Stephan

Many names:

Sinkhorn algorithm
DAD scaling
Iterative proportional fitting
Biproportional fitting
RAS algorithm
Matrix scaling

Udny 1912
Kruithof, 1937
Deming and Stephan in 1940
Sinkhorn 1964

Sinkhorn Evolution

Sinkhorn Evolution

constraints P1m = a and P€ 1n = b (one can afford to ignore the no
Other Regularizations
straint using entropy⇢Z because that penalty incorporates a logarithmic t
2 positive orthant). This implies that the s
the entries of
P to staythemin||x in y||d⇡(x, y) + "R(⇡) ; ⇡1 = ↵, ⇡2 =⇡2R
is no longer affineanditerative Bregman projections do not converge
solution. A workaround is to use instead Dykstra’s algorithm (1983[Bauschke and Lewis, 2000]), as detailed in [Benamou et al., 2015].
↵
uses projections according to the Bregman divergence associated toRemark 8.1 for more details regarding Bregman divergences. An issu
eral these projections cannot be computed explicitly. For the squared ncorresponds to computing the Euclidean projection on (Ca1 , Cb2 ) (withtivity constraints), which can be solved efficiently using projection alg
plices [Condat, 2015]. The main advantage of the quadratic regularizat
is that it produces sparse approximation of the optimal coupling, yet t
expense of a slower algorithm that cannot be parallelized as efficientl
compute several
optimaltransports simultaneously
in §
◆2
✓
◆
Z ✓(as discussedZd⇡d⇡contrasts
theapproximationachievedbyentropicandquadratic
regulR(⇡) =
dxdy
R(⇡) = logd⇡(x, y)dxdy
dxdy

constraints P1m = a and P€ 1n = b (one can afford to ignore the no
Other Regularizations
straint using entropy⇢Z because that penalty incorporates a logarithmic t
2 positive orthant). This implies that the s
the entries of
P to staythemin||x in y||d⇡(x, y) + "R(⇡) ; ⇡1 = ↵, ⇡2 =⇡2R
is no longer affineanditerative Bregman projections do not converge
solution. A workaround is to use instead Dykstra’s algorithm (1983[Bauschke and Lewis, 2000]), as detailed in [Benamou et al., 2015].
↵
uses projections according to the Bregman divergence associated toRemark 8.1 for more details regarding Bregman divergences. An issu
eral these projections cannot be computed explicitly. For the squared ncorresponds to computing the Euclidean projection on (Ca1 , Cb2 ) (withtivity constraints), which can be solved efficiently using projection alg
plices [Condat, 2015]. The main advantage of the quadratic regularizat
is that it produces sparse approximation of the optimal coupling, yet t
expense of a slower algorithm that cannot be parallelized as efficientl
compute several
optimaltransports simultaneously
in §
◆2
✓
◆
Z ✓(as discussedZd⇡d⇡contrasts
theapproximationachievedbyentropicandquadratic
regulR(⇡) =
dxdy
R(⇡) = logd⇡(x, y)dxdy
dxdy

Unbalanced OT
W⌧,p
p (↵,
<latexit sha1_base64="/CtRAoR7STbT84YpDUiMH8ZSmmU=">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

def.

R

) = min dp d⇡ + ⌧ KL(⇡ 1 |↵) + ⌧ KL(⇡ 2 | )[Liereo, Mielke, Savaré 2015]⇡

See also:[Chizat, Schmitzer, Peyré, Vialard 2015]<latexit 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Monsaingeon, Vorotnikov 2015]<latexit 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</latexit>Unbalanced OT
W⌧,p
p (↵,
<latexit sha1_base64="/CtRAoR7STbT84YpDUiMH8ZSmmU=">AABFaXictVx7bxu5Eaevr2v6Snr/FH0AmzopktZNbfeKFjgUuER2HJ+VxIlkJ3dRYmillbLJSqvoFecU/d2P1c/RD1CgRVH0K3RmSC65EneH66ZZ2OZy+ZsZzpLDmSE34SiJJ9Pt7b9tfPSNb37r29/5+LuXvvf9H/zwR5ev/Ph0ks7Gneikkybp+GnYnkRJPIxOpvE0iZ6OxlF7ECbRk/B1DZ8/mUfjSZwOm9N3o+j5oN0fxr24055C1dnlv1xrTaPz6eLJ8mwxWr5YtKbt2dZoeaPVTmpbrTCq3Qxa0Ztu1Atas0E8XLRGcW0ZtK62rgateDgNui9GQauLtcFvggDRgSR4VAciUH228x5p3YTHrqe774nJtbPLm9u3tulfsF7YUYVNof4dp1eu/l20RFekoiNmYiAiMRRTKCeiLSZwPRM7YluMoO65WEDdGEoxPY/EUlwC7AxaRdCiDbWv4Xcf7p6p2iHcI80JoTvAJYGfMSADcR0wKbQbQxm5BfR8RpSxtoj2gmiibO/gb6hoDaB2Kl5CLYfTLX1x2Jep6Ik/UR9i6NOIarB3HUVlRlpByQOrV1OgMII6LHfh+RjKHUJqPQeEmVDfUbdtev4Paom1eN9RbWfinyTldbgC0VC9TzMKbTEn+gG9zRk8k/IkwLkPFCLVRyy9JV0PqPdDaL+A+gdwLamkdRLCtaDaZSmyBpcLWWORB3C5kAcssg6XC1lnkcdwuZDHConYMencjW/A5cI3WM6P4HIhH7HIx3C5kI9Z5ClcLuQpi/wKLhfyKxZ5Fy4X8i6LPILLhTxikU24XMgmizyBy4U8YZH7cLmQ+wpZPFPHcKVEJ2Zm5W0o53mgpUig5jYr3x2yji7sHY853SnA8rN6D/66sXseOo0KsPse465XgOVH3gHYSDeWt0X3aDVxYe+x2EMYAW7sIYv9QrwqwH7hMdNeF2D5uVaHdm4sb33vw50be5/FPoCSG8uvUQ+hxo196LFijAqwxyz2kXhTgPWx+uMCLG/3G2BX3Fh+nWpCezfWx5rOCrC8PT0FD8aN5VerJ1Drxj5hsU/FeQH2KYv9Eqy7G/ulxwr7dQFWr7GXaAXpkz8SwYwto9bOZiWWRkCtzfBPsrUlId84hHoO088wfcIMWMRBhjjwRNQzRN1brklmRyfk7/JcGhmi4YkIs7UJS1O2fTdrj6XEA7GXIfZWEGUeKb5r3Zc5eRe6hkNOs5ULSz59SjP7jaVIjYdyy6sRD3MIObZf0sjfomgJIyjUVBm1l9kaL5EB3Zch3lL0pnupefC4aWYVbNQ5iwodqJBFvXOg3rGomQM1Y1FzB2rOoszMt3EtjxFg9I/vYkF3cgRIH7n4CsAruA2rzj2YowGMn2PwAh9TzUP426DYm7vKJMNoHtdJzHI8z1niMZQWYhPqTVS4R/F1QjMsAslky4cqxsc7zG0s1JyTVniZreRBljHxpxOTPP2MDnqLAc2nanSOqGZJ3p0sVcPfy+a9LlXD75PGl+TFy1I1/FRJP72A7E2FbV4A24DZNFLaN+WqNGT+RdLQ5Uu06qLFxbc6UGMG6Z1XpH+o3szhBd5LjUpSP6ZcjcbE6t8k178qNIyeJ5aeq1FB70l6vboUVO7JUMW9plxVhpRW0aGSw9xVfTPYpqvejC5Xo3EMHleNYu6FVa46ekdZb0y5Go1TIfOeS/LkdbkajT7dS32YcjUamG1pqzjflKtadtSAjJ1NuapVH1IWGHNAcszLGuMVjclPmilqMfkH5dka2+dfX8cwZ/MiixHKKRnftphOmK1l5RJpfyECqzatKAf6FzPLB8vTWIhdNr6SMkxz6/s6HbPGo+broMUAZr/cA+By5glIqHMSaL0ToLjDRl35nmncLovDUdJbQbVU7ZT1Fg1fmTXK151RLReXmd4aPbbIXk9o7I3IJ6yTZjk91AvfcBFFTkP1nIZ4elV097War3ntb7O40QpilI20Du0IyZ208jjVpfWGpePrapdnCpfc8zHjF7PNPWVtMOZJyRahLGU87XY6j2TX4bq6JUyOWz4L6I2ivZqT1YhpR2rCRqE6Wyy98QXdG9ontCeHPCSNDrzHQFEZCblrhll0zKcHZFFte8vxRn3pDJ0sT8jqantcju5b6L4DXT3GqcGK8QBKTYgZTuCu6RHlXMp0lZLGx+K32e5oSm+wPKJPchZS05D2JspZyLIo+2WOyltA42iQUbo/jVU6Gt9ao8RH/S55TOyat/zXaedW72+3aYwXj+biTEyXuO4S14BmjdzVlXerHKQEC+eTXfJfy3uJ/KpwRBvKcX1hcZZ6GdKOf0QR7Ig844RmGzc78q3t/NTqE83pWOi9c9zNTslCBmT/AlifUhqTAf3YZwf0Drq0CAnZSB+7E2fejcvXidkxZvy4WMhTDWa8RWTLZsRf07Vn14TGoowY5DqwXBnbWid18gUj4jpW1t3M7fLVB5HmnIQ9SiRFM1ZuEP+b9Fv/6HGyuTYiUMP4BibK1rneR0oxC+qoTat8uQ3SbW0pr2UyvFBSm/XPyHQtJ9keRVwoD67WXeDcoXvJC0fJmOSerLWR62hZNhcpj1b0iL3tURQv7X5frcAo9xatkps051o0SvowCqZZFKHbclnkVb7lvPLU/WhP/i/Uja7zWkOKgTAZXKkhLr8fUbRmS5nAqJbj9zXNJrfWxyutyvkMaSwOrLn8Hmqvwm8tt773oxPmrMIdGgOSgrkzGpE1wVoLP153crz0yNS0zL3hZ8akbmXXXCS+ltbNxNjzylSOadScq6yFLl+ExiuLxitPHTZpr9FoUddrS3TGxhZNtVvpy68Kt2YFyjOWMu+RaVTsIaUdS/lR7bJU+Rhfo75maW2ztNowW+3dAHvO+yDdc311dr/PVvdA3CXfpkMemIxfujRLY/K5dG15pCYpIOdPlX21Z3+LapB7SBYUKctznDhj5K5Th65lJumv1MqWkp03FkGfW3qr2mgb26Ly79eQA5oTE5qXGvEptYiU/LYcwYpFumX5HAFl/tvkU0m/ozxmtlubdxLk/AkTb8pZZXjJSGFI+ucyb4dr0euhFb8GFBPOlHcdAq3qbxgpSIzOJLg9ywm9IVzl5E6C9GhDsp/rdkru4g0tiW6R1AvxZw8bI6NeM9btsaV7rPv2a2iJWjdv3dWC55d4c+T4XWRHr02r2kD5qIuV+4vRaqtVLn9fpofZCl+jjxm1sSMLE+XlMS3xmTcXKVE1LhLjw6VaL6rIX03yKjLL3Slfyrq1ppzPNEgb85LiJe4cKCJc3t0Npzd3k+lHuEYvJKxNTdZwlDAbl6r8gG1pMSsVrERIdj23JiXWelS0Xhge9qph7Li0lBFZwURwuRvZ2u5DKxet8NkYSaEj5MneojjRpvkZXPg7EK4oUXP0ySE2wM+9LWpi/wOcinijyjKzGVAN2oTuSgzeVv3MtyjX0RuLuk3fh4M/jxh0zUkf04paVXZJmZfcpu5P/y1Zg7GIWOlNy+p9sLnwPVnnVKU/MVk4vjex0N/kVO2L5uDTkzwXfz5yf4PrRU/ob5uq9UFT53uQ51CFhz7P4PfOTevqvGxO5fpa5+LLQ64DeudF43AHsDhmMe18LNTYeiMfngNah14Jdb1a/K/90HwMp+q8fLlN6JuzVx5vXbaLVGYW/eLqc8Zw8xnNxRz9eaZZ74zX5OYn/b+g0ptKrd58eProl5oxoHkthMyH8tJJvD2KjLy+VHB/wCVDKv4j/rrBf5XwJqNRJEcVSnq/opiabsFT019eunqnn/nIZOgUyZSnZuKJBp2MrYlDcRd+apkHWPWUqPymUv5FrPs72i7U9sh66Gy6zCC0qC6iLIjZTevSvTlHWyQxnumVZ3ybUIN74nWqxfO+D6g9nvlt5vpW/CWJnOv3RSq6uchkdZfPzKsQepDfgZO5IP29b0Bn6mU2S55AG3jsMcpzVDJS0l8/LwjRpbhwVdIF

def.

R

) = min dp d⇡ + ⌧ KL(⇡ 1 |↵) + ⌧ KL(⇡ 2 | )⇡[Liereo, Mielke, Savaré 2015]

See also:[Chizat, Schmitzer, Peyré, Vialard 2015]<latexit 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Monsaingeon, Vorotnikov 2015]<latexit 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</latexit>

R p
( ↵

p

Hellinger

)2

⌧ !0

↵

W⌧,p
p (↵, )

Unbalanced⌧ ! +1

Wpp (↵, )BalancedUnbalanced OT
W⌧,p
p (↵,
<latexit sha1_base64="/CtRAoR7STbT84YpDUiMH8ZSmmU=">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

def.

R

) = min dp d⇡ + ⌧ KL(⇡ 1 |↵) + ⌧ KL(⇡ 2 | )⇡[Liereo, Mielke, Savaré 2015]

See also:[Chizat, Schmitzer, Peyré, Vialard 2015]<latexit 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Monsaingeon, Vorotnikov 2015]<latexit 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Ifw7xH8demXa+pI5+FZ1DOi6NOvi1LoideTW1euNFdKOKp7oLbVQ3UAbXxLvW1Ux/PSMyW79FNl33jFtWRSFup9wof814R0FEtz1aLpozqE+L5419khyIO1vmTVkj13UM59RrbCPlDy18taR7WreG5GTiGU65Dj+OYPSsVzSDzvzuNg3K3JJ1WKN8NlBflkJqQsm13SotbvQNthvbvadZc4Sb9hr+j11cyy9moJikhbRnMzxoJWXspoQ8PkLNdVvbW6tFG+Ga+phdl2KytVTW3t+2Jp/9/trE7+ZYf836aFcWTVzJfdqrGw39jO/7btvI2HuvVWFnRdX/UbW/q/akvX9UO2Hfl3YfSphvlGL9d5k6x0y1xH3zInd2+FxoFFDrLnPbVippaVKiv+ZoWsiC/GIr49rtfjZ6Krpncyx4zB+cSB3LVp7sQd6hUojt+6NM8g9rygeG5GrYPv4O0oOQX7oZK7keuRkwrqg0J0XMZ0iNuMNFpGvp+PJ4v3keK+aJxBsE9vyOxhok9i4e10aKlJAXk3pyf2FaJbpsa3d8vs3aSkYT7dYW4QTgrlK+aVvbYcPSa5REZG2Tl9kUuXeMrN+6JEKqFe5M93I7cpJ1K/k9O3ZTMS9GjFsk5XCVm1T5vmxMu64r3nA6pjPoEyV7irdu7gWM/p7Tm0SSc8j2X7UZk3KGvY5K96XqbIfnRMniGGoslfpVg3i1QuN7aPde2dk1qUW5s2tp5jXZ1eWfFziHMMDdGim06otpDSWhSlUC0VKbm8caLrEWXpUyvvklXyvr0ZjRky7Wd5xxBqN3yP+lVtvsOCT+S7/kekR9wN0dFlruNRzll/zoPPV8wIZd7NGTxpY/fVMe3cTfRsFM71XnhKOaTZtpnq63Uc0eyueqAeqr8GWn3I/bW3FDOdR7BJZT+Wfz/Ffn6bn71DOHw70oZ6nt/RWLePUN55lPJc3/Pmp7sdoFqlu53T9VMOyeuT2E/5gXpK44Ym2mCpjyN2FD+ku9d8lPGmuWPK1arckF9P9z7I8KBS95Ia2pEke6X4JmiRxr4N8AadIOR7F9AOZ42k61BaHBeT8zqc9vTMrev8lvkupMtnWpc9fQuQL/eF3hUZ2gt5QaXgnK+U7MudBulnSs5jyhj7gs5my4jaxAthKsUzQ+YkJZfBpinj9hDdi1w+n01zHr7zbKHK94m5LD2Gp9yJxPooUi1rOIbaToEe2y3vzS5LHE+dae/lJTe0ixRXcm4LKldcfe6piUV3BZA3I+1J/FvZBmK48rki+/OKpatLuq2rTndh+eQ3g8Z0e6Kx1jrbwtsXXa2hidcwPK8qMixyOytbbRzFe06atr6uvLYW3k1aL7P0Rz4KB9DX8I3HBxF9F+c0/RTeJVpsDeGxhlDZoDMXhw4aR0Eq93IqaAF9TalI5yii9Dw2LZ47cp/J9p9W7eR3reLtp8dUA7cb1+aekt9qMJ/wex89P0Xeuc1jWmnzPeuU75qOkDi+Maek8PzHHUp7CQ9b6bE+C56ov6pIYqPnNPJ4pW6pJPcp/jrgO6GkJ7XTVixKOL66GWVftnU9jrQmY0v+vJt53mcq7oZDRAjGvm2gSCFe2l0Y1bnpsfTxlA40nVDL3835FUdUE+L2OID+XNXdNfu5unL2hmIxPF8kvwBk/zrPhBDmNAvzLubHXwzClCFY8oji37lq9utNSIF38iBqi976etcP+4YxjZ4xQpM7IczvD/Hpc2xpOGvlP5uGp87HyjVLXP4tEp734bM5KCWfMzuEMo3prPtH2mMdRXiHEVEo345jc+jQHBPvLMCx8wP190Gqr0k37dLJrJD8j3NUUipD6AwEems+axDiZ5fmENqlja1ylduf1sB+dsDvNTs11dN6lRgifGL1tCC/8cXmFohj7a06ETVbd591zO817OXtvXr3S92N0SuACY1a9yp+xKYRQ2GuyvdHc/uYB0sltxiWkeFdhz1VvoOd66UXYZ0DGtcNSm2hKME9KvuqHsViTd/OIzof9Zd0/zT757Ngv/XSyo9vx9CLuM+8Fr+PmT2QaICj9rKV+tCXOnZcofGNK1bwY881uln8EY6r5Dc8yrGgq69yx25+uxgpOR9cvsnZd9JJVou2cg7uecSeMr+wyVT5LG+L0lKqbWOFPfqsz/4fv7u8Vv6l1+rLi/W7a6t31z5dX/753+lfgf2R+jP1F6CLNfWh+rl6BDZ5ADL9k/pX9e/qP87/8fxfzv/t/Jec9Yc/0Jg/VYX/zv/zfwC0uEjt</latexit>

R p
( ↵

p

⌧ !0

)2

↵

HellingerW⌧,p",p (↵,

def.

R

W⌧,p
p (↵, )

⌧ ! +1

Wpp (↵, )Balanced

Unbalanced) = min dp d⇡ + ⌧ KL(⇡ 1 |↵) + ⌧ KL(⇡ 2 | ) + "KL(⇡|↵ ⌦ )⇡Sinkhorn’s algorithm:u⇣ a ⌘1+ ⌧"
Kv

v

✓

b
K> u

◆1+ ⌧"

Wasserstein Barycenters
P

Barycenters of measures (↵s )s :s s = 1P?
2 argmin s s Wpp (↵s , )<latexit 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</latexit>
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Guillaume Carlier<latexit sha1_base64="70KW2R/z064lRPY5iuKzB5+V95U=">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</latexit><latexit
sha1_base64="70KW2R/z064lRPY5iuKzB5+V95U=">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

?

↵1

↵3
[Solomon et al, SIGGRAPH 2015]

Martial Agueh

Wasserstein Barycenters
P

Barycenters of measures (↵s )s :s s = 1P?
2 argmin s s Wpp (↵s , )<latexit 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</latexit>
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Guillaume Carlier

Martial Agueh<latexit sha1_base64="70KW2R/z064lRPY5iuKzB5+V95U=">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</latexit><latexit
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[Solomon et al, SIGGRAPH 2015]

Sinkhorn’s algorithm:

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and 4.14, theconvergence(asof Sinkhorn’salgorithm
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Sinkhorn’s algorithm def.
will often fail toPi,jterminate as soon as some of theKL(P|K)=P
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Remark
4.22,
andtherelatedto a direct<latexit sha1_base64="hyTeRiv4gE55ee1xx7LmitI6vfw=">AABBzXictVzbchu5EYU3t7Vz8yaPeZlE65Q35TiyslW5bKVqZUmWteLatEnZ3l3aLl5GNO0hh+aQ9IVWXvMLeU1+JN+RP0ie8gvpCzDAkJhpjOIYJQkD4nQ3eoBGdwN0b5qMsvn29j8vfPCtb3/nu9/78OKl7//ghz/68eWPfvIgSxezfnzST5N09qjXzeJkNIlP5qN5Ej+azuLuuJfED3sv9vDzh8t4lo3SSXv+Zho/HneHk9HpqN+dQ9OT40Y0GMHnw3jSj//49PLW9vVt+hdtVm7oypbS/5rpR9GJ6qiBSlVfLdRYxWqi5lBPVFdlUL5RN9S2mkLbY7WCthnURvR5rM7UJcAuoFcMPbrQ+gJ+D+HpG906gWekmRG6D1wS+JkBMlJXAJNCvxnUkVtEny+IMraW0V4RTZTtDfztaVpjaJ2rZ9Aq4UzPUByOZa5O1e9pDCMY05RacHR9TWVBWkHJI2dUc6AwhTasD+DzGdT7hDR6jgiT0dhRt136/F/UE1vxua/7LtS/ScorUCLV0qNPcwpdtST6Eb3NBXzG8iTAeQgUYj1GrL0iXY9p9BPov4L2O1DOqGZ00oOyotazSuQeFB9yT0QeQvEhD0VkA4oP2RCRTSg+ZFMjETsjnfvxLSg+fEvkfA+KD3lPRN6H4kPeF5EPoPiQD0Tk11B8yK9F5C0oPuQtEXkMxYc8FpFtKD5kW0SeQPEhT0TkARQf8kAjy1fqDEpKdEbCqtyFepEHWooEWnZF+W6SdfRhbwas6X4JVl7V+/DXj90P0Glcgj0ImHenJVh55h2CjfRjZVt0m3YTH/a2iD2CGeDHHonYL9TzEuwXASvtRQlWXmsN6OfHytb3S3jyY78UsXeg5sfKe9RdaPFj7wbsGNMSbFPE3lMvS7AhVn9WgpXtfgvsih8r71Nt6O/HhljTRQlWtqcPwIPxY+Xd6iG0+rEPRewj9boE+0jEfgXW3Y/9KmCHfVuCNXvsJdpBhuSPxLBiq6h181WJtSlQ6wr8k3xvScg37kG7hBnmmCFhxiLiMEccBiIaOaIRLFeW29GM/F2ZSytHtAIRvXxvwtpc7D/I+2MtCUDs54j9NUSVR4rv2oxlSd6FaZGQ83znwlrImNLcfmMt1vOh2vIaxN0Cguf2M5r51yhawggKNVVF7Vm+xzMyoucqxCuK3swoDQ8ZN8+tgot6LaJ6HlRPRL3xoN6IqIUHtRBRSw9qKaLsyndxnYAZYPWP72JFTzwD2EcuLxF4Bbuw69yGNRrB/GmCF3ifWu7C3xbF3lKpkgyjedwnMcvxuGCJZ1BbqS1ot1HhPsXXCa2wGCTjnnd1jI9PmNtY6TXHVvgs38mjPGMSTmdE8gxzOugtRrSe6tE5ppYz8u64Vg9/O1/3plYPf0AaPyMvnmv18HMt/fwcsrc1tn0ObAtW01Rr39br0uD8C9Mw9Uu066LFxbc61nMG6b2uSf9Iv5mjc7yXPaqxfmy9Ho3MGV9WGF8dGlbPmaPnelTQe2Kv19Si2iOZ6LjX1uvKkNIuOtFy2Ke6bwb7DPSbMfV6NJrgce1RzL1y6nVn7zQfja3Xo/FAcd7zjDx5U69HY0jPrA9br0cDsy1dHefbel3Ljhrg2NnW61r1CWWBMQfEc55brFc0Iz9poamNyD+ozta4Pv/mPoY5myd5jFBNyfq25XR6+V5WLZHxF2KwavOacqB/sXB8sCKNldoR4yuWYV7Y3zfp2D0eNd8ALUaw+vkMQMqZJyChyUmg9U6A4g0x6iqOzOB2RBzOktM1VEe3zkVv0fLlrFGx7Sm1SnGZHa3VY4fsdUZzb0o+YYM0K+mhUfqGyyhKGmoUNCTTq6O7t3q9FrW/LeKma4hpPtP6dCLEJ2nVcapP6y1Hx1f0Kc8cCp/52PmL2eZTbW0w5knJFqEsVTzdfiaP5LbhvnpN2Rw3fxbRG0V7tSSrMaITqUyMQk22mL3xFT1b2id0Joc8mEYf3mOkqUwVn5phFh3z6RFZVNfeSrxRXyZDx/WMrK6xx9XooYMeetD1Y5w92DHuQK0NMcMJPLUDopxLua5S0vhM/To/HU3pDVZH9EnBQhoabG/igoWsirKfFai8AjTOBo7Sw2ms0zH4zgYlOer3yWNj16Llv0Int+Z8u0tzvHw2l2diBsR1h7hGtGr4VJef1jmwBCvvJzvkv1aPEvnV4Yg2VOL6xOHMepnQiX9MEeyUPOOEVpu0Ooq93fzU+ieGU1OZs3M8zU7JQkZk/yLYn1KakxH9uHcHzAk6W4SEbGSI3Rnl3o3P1xmJc8z6cSPFtxrsfIvJli2Iv6Hrrq6M5iJHDLwPnK3NbaOTBvmCMXGdaetu13b17oNIe0/CnSVM0c6Vq8T/E/ptfsw82dqYEahhfAOZtnW+95FSzII66tIuX22DTF9Xyo9zGZ5oqe3+Z2X6uCDZPkVcKA/u1gPg3Kdn5oWzZEZyZxt9eB+tyuYi5emaHnG0pxTFs90f6h0Y5b5Gu+QWrbkOzZIhzIJ5HkWYvlIWeZ1vNa8i9TDa2f+FutV1UWtIMVI2g8sakvL7MUVrrpQJzGqevy9oNfm1PlvrVc1nQnNx7Kzld9D6c/ht5DbPYXR6Batwk+YAU7BPViPcEm30CON1s8DLzExDyz5bfnZOml5uy3nia7ZuNsZe1qbSpFnzWmctTP08NJ47NJ4H6rBNZ41Wi6bdWKKnYmzR1qeVofzqcGvXoLwQKcsemUGNAqR0Y6kwqgORqhzjG9Rbkda2SKsLq9U9DXDXfAjSv9bXV/e7fHeP1C3ybfrkgXH8MqBVOiKfy7RWR2pMATl/qu2ru/o71ILce2RBkTLf48QVw6dOfSpnuaS/1DtbSnbeWgRzb+mV7mNsbIfqv91AjmlNZLQuDeJT6hFr+V05ojWLdN3xOSLK/HfJp2K/ozpmdnvbdxIV/Akbb/Kqsrw4UpiQ/qXM29FG9HrkxK8RxYQL7V33gFb9N4wUGGMyCX7PMqM3hLscnySwR9sj+7lpp/gUb+JIdJ2kXqk/BdgYjnrtXHfnlhmxGduvoCdq3b51Xw+ZXxLMUeJ3nhO9Lu1qY+2jrtaez0erq3e54nOVHhZrfK0+FtTHjSxslFfEdNRnwVxYonpcGBPCpd4o6shfT/I6MvPpVChl09tQLmYa2MY8o3hJugeKCJ93d9XrzX0ijKO3Qa9HWJcat0iUMBuX6vyAa2kxK3VxYx/i1ouVu1Hi7ERlO4Wh7u4W1n6zhYzJ+iVKytlwb1f2TiFKkbMwTKGv+EZvWXzo0vwMCv6OlC86NBxDcoct8G931Z46eA+3IV7qOmc0I2pBWzBYi727epzFHtU6eulQd+mHcAjnMQJdS9KPaCetKztTliV3qYfTf0VWYKZiUXrbs/4YXC7ySDY51RnPiCybPJqRMt/FqTsWwyFkJEUu4Xz4XEMaxaky32mqNwZDXR5BkUMdHuYeQ9g7t73r83I5Vetrk0soD94FzImLweHJX3msYvuFWKiZ80bePwe0DqcV1M1u8b+Ow/CxnOrzCuWW0XfNnge8de4X64ws+sP114zlFjKbyzmG80zz0Vlvyc+P/b6o1ptKndG8f/roj9o5YHitFOdBZekY784iK28oFTwX8MmQqv+of1yQv43wMqdRJkcdSuacopya6SFTM9+49I3OfBYik6VTJlORmo0jWnQjdk8dqVvws5d7gHVvh/J3KfkvYv3fnx1A6ylZD5NF58xBh9piyn7YU7QBPdv7s2US411evtvbhhY8C29QK97zvUP98a5vuzC28m+Q8Fr/UqVqUIhI1k/37LrqwQiKJ2+cAzLf843oLj1nsfjm2TjgbNHcn1qXaEWfyDcLeqX4niNln+bqVJ/V48kB3rDv5vmhSP2G2rrazuOeK3FulnJurnHOSDtFDq+dz6rvZpVx2XO4DPLc2VL3SynOtud51bnR/VIufAe9Gj+swA8dKVuk/RcUCc9UdTZvUUFzoWVyT1gnymQiWQ8YZ3bz910d2S4reC0Dxn9cij52JD0EWXqU/47ohG1G9BKtmwOSnm86VmdSb1dIq79H+fTy1o31/8tgs3Kyc/0P12/c29n6/Kb+bw4+VD9Tv1BXYYn/Tn0OxJrqhA4n/6r+pv6+29xd7r7b/TN3/eCCxvxUFf7t/uW/9xOlhQ==</latexit><latexit
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P2U(a,b)ated to the cost matrix C as
Ci,j
def.
def.
KL
≠
(K) ="projections).
argmin KL(P|K).
min
KL(P|K)Á =Ki,j = e ÁRemark
4.7 P(Relationwithiterative
Denoting
U(a,b)
hP, Ci+ " KL(P|a
⌦Proj
b),
P2U(a,b)PœU(a,b)

Ó

Ô

def.
2 def.
T
Indeedusing
theCdefinitionabove
Ca1 =one{P has
: P1that
and
=
P
:
P
1m = bm = a}b<latexit sha1_base64="g6aAhDZlOKcl5B2wugAi4F8lGYU=">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KL
P
=
Proj
argmin1 ﬂ C=
2 . OneÁ
the rows and columns constraints, one has U(a,U(a,b)b) = C(K)can uaiterative projections [Bregman, 1967]

Iterative projections:def.

b PœU(a,b)

(¸)
(¸+2)
KL
(¸+1)
P(¸+1) = ProjKL
(P
)
and
P
=
Proj
(P
).
1
2
Ca
C
def.

def.

b

Since the sets Ca1 and Cb2 are affine, these iterations are known to converge to(`)
Theorem: of (4.7),Psee! P? =1967].
argmin[Bregman,
These KL(P|K)
iterates are equivalent to Sinkhorn iter
1
P2Ca1 \Cb
For affinesince(Ca1defining, Cb2 ),def.
P(2¸) = diag(u(¸) )K diag(v(¸) ),one hasthekernel
kernelbecomenegligible
be stored
in memory
as posit
theKK
becometootoonegligible
to betostored
in memory
as positivenu
becomeinstead
insteadnull.
Thisresulta matrix
product
becomenull.
This
can canthenthenresultin a inmatrixproduct
Kv orKv
KT u
smallerentries
entriesthatbecomeresultin a division
by
0 in
the
Ci,j
smallerthatbecomenullnulland andresultin a divisionby 0 intheSinkh
Ki,jby=carrying
eby "carrying
aouthP, Ci +of"ofEquation
KL(P|a⌦b)=Such" KL(P|K)+cancstpartlyi bjcoEquation(4.15).
Such
issuesbewherepartlyresolved(4.15).
issuescanberesolved
themultipliers
multipliersu andin thelog domain.
approachis careonontheu andv invthelog domain.
ThatThatapproachis carefullypRemark
4.22,
andandrelatedto atodirect
resolutionof theofdual
problem(4.2
Remark
4.22,
relateda directresolution
the ofdualof problemin
KL(P|K)
Shrödinger
problem:

Bregman Iterative Projections<latexit sha1_base64="BKDev5VQxrfCyjh/WSsW5fgsAQ4=">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</latexit><latexit
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P2U(a,b)Remark
4.74.7(Relation
withwith
iterative
projections).
Denoting
Remark(Relation
iterative
projections).
Denoting
Ó
Ô
Ó T
def.
def.
1
2
def.
: P1
= a}= a}and andCb =C 2Pdef.
= {P
: mP1
=: P
P 1:mP=Tb
1 =b
U(a, b) = C 1 [ C 2 Ca C=1 {P<latexit sha1_base64="c9gBd5gWodkBDryKhwuTlNgEJlA=">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</latexit><latexit
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a

a

b

b

m

m

the rows and columns constraints, one has U(a, b) = Ca1 ﬂ Cb21. One2 can us
the rows and columns constraints, one has U(a, b) = Ca ﬂ Cb . One
iterative projections [Bregman, 1967]

iterative projections [Bregman, 1967]

KL
(¸)
(¸+2)
KL
(¸+1)
P(¸+1)
=
Proj
(P
)
and
P
=
Proj
(P
1
2
def.
def.
Ca KL
(¸+1)
(¸)
(¸+2) C
KL ). (¸+P

def.

= ProjCa1 (P ) and P

def.

= bProjC 2 (PbSince the sets Ca1 and Cb2 are affine, these iterations are known to converge to t
1
2

Ca and1967].
Cb areTheseaffine,these areiterations
areto
known
to conveofSince(4.7),thesee sets[Bregman,iterates
equivalent
Sinkhornitera
of (4.7),see [Bregman, 1967]. These iterates are equivalent to Sinkhor
sincedefining
def.
since defining
P(2¸) = diag(u(¸) )K diag(v(¸) ),one hasone hasP(2¸) = diag(u(¸) )K diag(v(¸) ),def.

P(2¸+1) = diag(u(¸+1) )K diag(v(¸) )def.

[Bregman, 1967]

thekernel
kernelbecomenegligible
tostored
bestoredstoredin memorypositive
as posit
theKK
becometootoonegligible
to be
in in
memorynu
the kernel
K become
too negligible
to bememoryasaspositivenu
becomeinstead
null.
This
then
resulta matrix
product
becomeinstead
null.
This
can
resulta matrix
product
KvororKv
KTTu
becomeinsteadnull.
This
can canthenthenresultin ain in
matrixproduct
Kv
K
smaller
entriesthatbecomeandresultina division
by
0 in
the
smaller
entriesthatbecomenull
resulta division
ini,j
the
Sinkh
C
smaller
entriesthatbecomenullnullandandresultin ain division
byby0 0in
the
Sinkh
where
Ki,jbyby=carryingecarryingaout
bjco
hP, Ci +of"ofEquation
KL(P|a⌦b)=Such" KL(P|K)+cancstpartlyof Equation(4.15).
Such
issuesberesolvedcoiout
Equation(4.15).
Such
issuescanbepartly
partlyresolvedby "carrying(4.15).
issuescanberesolved
onmultipliers
themultipliers
andin thetheThatapproachcarefully
themultipliersu uandinlogdomain.
domain.
That
approachis careonontheu andv invvthe
log logdomain.
That
approachisiscarefullypp
Remark4.22,
and relatedtoresolution
dualofofproblem(4.2
Remark
4.22,
andandrelatedto atodirectresolution
of ofthethe(4.2
Remark
4.22,
relatedaa direct
directresolution
ofdualthedualproblem
of problemin
KL(P|K)
Shrödinger
problem:

Bregman Iterative Projections

P2U(a,b)Remark 4.7 (Relation with iterative projections). Denoting

Remark
4.74.7(Relation
withwith
iterative
projections).
Denoting
Remark(Relation
iterative
projections).
DenotingÓÔ
1 def.
2 def.
TÓ P Ó: P 1m = bÔand 2 Cdef.
=
def.Ca = {P : P1m = a}
b
1
T
def.
2P
1
2 Ca C=1 def.
{P
:
P1
=
a}
and
C
=
:
P
m
b
= {P : P1 = a} and C = P 1:mP=Tb
1 =b
U(a, b) = C [ C
a

b

a

b

m

m

the rows and columns constraints, one has U(a, b) = Ca1 ﬂ Cb2 . One can u

1 ﬂ C 2 . One can us
the rows
andcolumns
constraints,
one
has
U(a,
b)
=
C
a
iterative
projections
1967]one has U(a, b) = Cb1 ﬂ C 2 . One
the rows
andcolumns[Bregman,constraints,ab
iterative projections [Bregman,
1967]
def.
def.

(¸)
projections[Bregman,1967]
P(¸+1)= ProjKL
Iterativeiterative
projections:
Ca1 (P ) and(¸+1)
P(¸+2) = ProjKL
(P
).
2
C

KL
(¸)
(¸+2)
KL
(¸+1)
P(¸+1)
=
Proj
(P
)
and
P
=
Proj
(P
). (¸+
1
2
def.
def.
C
C
(¸+1)
KL
(¸)
(¸+2)
KL
a
b
) iterations
and Pare known= Proj
1 (P
2 (P toSince the setsPCa1 and =Cb2 Proj
are affine,these
to converge
C
C
a
b (0)
(`)
1
?
Theorem:
P
!
1
2
K
=
P iteraCP=
argminKL(P|K)of (4.7),[Bregman,iterates
are areequivalent
toSinkhornSince thesets Csee
Cb are 1967].
affine,These
these
iterations
known
toconverge
to t
a
a and
1
1
1
2
1defining
2 C and C are
P2C
the
sets
affine,a \Cb these iterations are known to conve
since
For affine
(C
,
C
),
a
b
ofSince(4.7),
see[Bregman,1967].
These
iterates are equivalent
to Sinkhorn itera
(1)
a b
P(¸)
2
(2¸) def.
(¸)= diag(u),
of (4.7),see [Bregman, 1967].PThese
iterates)K
arediag(vequivalent
toSinkhorCsincedefining
(3)
b
P
def.
sinceonedefining
P(2)has
P(2¸) = diag(u(¸) )K diag(v(¸) (4)
),
def.

b

def.

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sha1_base64="vfF3Fmv2QosO4uooZuk2vF87K5I=">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</latexit>(¸)(¸) P(5)P(2¸)=
diag(u)Kdiag(v),(2¸+1)(¸+1)(¸)<latexit sha1_base64="m4yt+a3Val3SpfuPtTQ7FzcDFn8=">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_base64="m4yt+a3Val3SpfuPtTQ7FzcDFn8=">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</latexit>

one hasone hasdef.

P= diag(udef.

P)K diag(v)(2¸+2)(¸+1)
def.
(2¸+1)(¸+1)(¸)(¸+1) )andP=
diag(u)Kdiag(vPP= diag(u)K diag(v )def.

?

[Bregman, 1967]

thekernel
kernelK
becometoo
negligible
tobe
bestoredstoredinamemory
as posit
practice,
however,that
slowdown
is rarely
observed
inpracticeforsimpler
reason:theK
becometoo
negligible
tobestoredinmemoryaspositivenu
thekernelK
becometoo
negligible
toinmemoryaspositivenul Stability ofSinkhorn’s
Sinkhorn Iterations).
As
wediscussinRemarks
4.13
algorithm
will null.
often This
fail tocan
terminateassoon inas asomeof theproductelements Kv
of TT
becomeinsteadthen
resultmatrix
becomeinstead
null.
This
can
then
resultinamatrixproduct
Kv
orKu
becomeinstead
null.
This
can
then
resultinamatrixproduct
Kv
orK
nce of Sinkhorn’salgorithm
deteriorates
as
Á
æ
0.
In
numerical
the kernel K become too negligible to be stored in memory as positive numbers, andsmaller
entriesthatbecomenull
andresultina division
by
0 in
the
smaller
entriesthatbecomenull
andresultin divisiona division
by
0in
ini,j
the
Sinkh
C
smaller
entriesthatbecomenull
andresult
in
a
by
0
the
Sinkh
T
t slowdownbecome
is rarely
observed
inpracticeforasimpler
reason:instead null. This can then result in a matrix product Kv or K u with
ever
" a
where
K
=
e
bjco
hP,
Ci
+
"
KL(P|a⌦b)="
KL(P|K)+
cst
ofEquation(4.15).
Such
issuescanbepartlyresolvedbycarrying
co
i,j
iout
of
Equation(4.15).
Such
issuescanbepartlyresolvedbycarryingofEquation(4.15).
Such
issuescanbepartlyresolvedbycarrying
outwill often failtoterminate
as
soon
as
some
of
the
elements
ofsmaller entries that become null and result in a division by 0 in the Sinkhorn update
onmultipliers
theandin thethelogdomain.
That
approachcarefully
oo negligible
toononbethe
storedin multipliers
memoryas
numbers,andthemultipliers
u upositive
and
vvthe
in
log
domain.
That
approachis careandv inbelog resolved
domain.
approachisiscarefullyppof Equation(4.15).
Such uissues
canpartlybyThatcarrying
out
computations
Ta direct resolution of the dual of problem (4.2
4.22,
and
relatedtoThis can thenresultinRemark
a 4.22,matrixproduct
Kvto
oradomain.
K
with
ever
on themultipliers
u
and
v
in
the
log
That
approach
is ofcarefully
presented
in(4.2
Remarkandrelateddirectresolutionof thedual
problemRemark
4.22,
andrelatedtoresolution
the ofdualof problemin
KL(P|K)
Shrödinger problem: au directcome null Remark
and resultin aanddivision
byto0aindirect
the Sinkhorn
updateP2U(a,b)4.22,
relatedresolution
the dual of problem
Remark4.7 (Relation
withiterativeofprojections).
Denoting(4.2).
ch issues can be
partly resolved
by carryingout iterative
computations
Remark
4.74.7(Relation
withwith
iterative
projections).
Denoting
Remark(Relation
projections).
DenotingÓÔ
4.7 (Relation
with iterative
projections).
Denoting
1isdef.
2 def.
Tnd v in theRemark
log domain.
That approach
carefully
presentedinÓ P Ó: P 1m = bÔ=def.Ca = {P : P1m = a} Óand 2 C
def.
b
1
T
def. Ô
1ofdef.
2P
1 of12the
C
=
{P
:
P1
=
a}
and
C
=
:
P
1 =Tb
ted toadirectresolution
problem(4.2).
def. dual
def.
m
2
T
a
b
=m{P
U(a, b) = Ca [ C
Cab= {P C: aP1= a}: P1
andm =Cb a}= Pand: P 1Cmb==b P :mP 1m = b
the rows and columns constraints, one has U(a, b) = Ca1 ﬂ Cb2 . One can uwith iterative projections).
Denoting
1 ﬂ C 2 . One can us
1
2
the
rows
and
columns
constraints,
one
has
U(a,
b)
=
C
a
the rows
columns
constraints,one
has U(a,Ca ﬂ U(a,Cb . One
Bregmaniterative1967]b)theandrows
andcolumns[Bregman,constraints,
one= has
b)can
= use
Cba1 ﬂCb2 . OneÓprojections
Ô
2 def. [Bregman,
T[Bregman,1967]
iterative
projections
1967]
P : P1m =a}iterative
andCprojectionsbdef.
b = P : P 1m =def.

Bregman Iterative Projections(¸)
projections[Bregman,1967]
P(¸+1)= ProjKL
Iterativeiterative
projections:
Ca1 (P ) and
(¸+1)
KL
(¸)
(¸+2)

(¸+1)
P(¸+2) = ProjKL
(P
).
2
C

def.
KL def.
(¸+1)(¸+1)
KLand(¸)P
(¸+2)
(¸+1)
P
=
Proj
(P
)
=
Proj
).def.KL(4.16)
1
2
1
2 (P
P
=
Proj
(P
)
and
P
=
Proj
(P
). (¸+
1
2
C
def. canconstraints, one has U(a, b) = Ca ﬂ(¸+1)
CbC.a Oneuse
Bregman
C
C
KL
(¸)
(¸+2)
KL
b
a
b
) iterations
and Pare known= Proj
1 (P
2 (P toSince the setsPCa1 and =Cb2 Proj
are affine,these
to converge
CCa
Bregman, 1967]b (0)1 the solution? these iterations are known to converge to
1 and(`)
2 are affine,
Since the sets
C
C
Theorem:
P
!
1
2
K
=
P iteraCP=
argminKL(P|K)abandof (4.7),
see[Bregman,1967].
These
iterates
are areequivalent
toSinkhorn
Since
the
sets
C
C
are
affine,these
iterations
known
toconverge
to t
a
a
b
f.
def.
1
1
KL of(¸)(4.7), see [Bregman,
(¸+2)
KL
(¸+1)
1
2
1967].
These
iteratesareequivalent
toSinkhorniterations(4.15)
1defining
P2C= ProjCa1 For(P )affine
and
P
=2 ),
Proj
).
(4.16) arethesets
Ca andCb areaffine,iterationsare(1)toknown
to conve
2 (Pa \Cb these
since
C
(C
,
C
ofSince(4.7),
see[Bregman,1967].
These
iterates
equivalent
Sinkhorn
itera
b
a
b
P(¸)
since defining2(2¸) def.
(¸)
P=
diag(u)Kdiag(v),of(4.7),
see[Bregman,1967].
These
iteratesareequivalent
toSinkhor
C
def.
since
defining
(2¸)
(¸)
(¸)
2
(3)
b
P converge= diag(u)Kdiag(v ),to thesolutionPb are affine, these iterations are known to
def.
(2¸)
(¸)
sincearedefining
P(2)
one
has
Piterations= diag(u)K diag(v(¸) (4)),
1967]. These
iterates
equivalent
toSinkhorn(4.15)
one hasdef.
PP(2¸) = def.
diag(u(¸) )K diag(v(¸) ),P(5)
def.

def.

b

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</latexit>P(2¸) def.

one has(¸)(¸)= diag(u )K diag(v ),one hasP

and P(2¸+1) def.

(¸+1)
(¸)
P(2¸+1)(¸+1)= diag(u (¸) )K diag(v )= diag(u(2¸+2))K diag(v )

(¸+1)

P?

(¸+1)

def.
def. (2¸+1)
(¸+1)
(¸)
= diag(udiag(v(2¸+2)and(¸+1)
P P (¸+1)= diag(u)K)Kdiag(v)= diag(udef.

)K diag(v))

Hilbert Projective MetricdH (u, u ) = || log(u)<latexit sha1_base64="YX5RigRyUj3zJruhrbetrBPkcTg=">AABFOnictVzrbxy3EafTV+q+nPZjv2yquLEDxZVcFy0QFIitkyXFsi1bJ9mJzxbusXc6e+/2fA/5cbn/rGj/jn4tWqBB0fZL/4DOg1xy77g7XNX1QhKXy9/McJYczgy5bo2S/mS6sfHnC+9969vf+e733v/+xR/88Ec//smlD356PEln43Z81E6TdPy41ZzESX8YH0370yR+PBrHzUEriR+1Xmzh80dn8XjST4f16ZtR/HTQ7A373X67OYWqk0uPOyeN3dMrjeO4Pb2bduL5bLHu3nx8NWrELztxN2oM0/Fg3kjSXq711U9Xqj6+ujg5Prm0tnFtg/5Fq4VNXVhT+t9B+sGHf1MN1VGpaquZGqhYDdUUyolqqglcT9Sm2lAjqHuq5lA3hlKfnsdqoS4CdgatYmjRhNoX8LsHd0907RDukeaE0G3gksDPGJCRugyYFNqNoYzcIno+I8pYW0R7TjRRtjfwt6VpDaB2qk6hVsKZlqE47MtUddXvqA996NOIarB3bU1lRlpBySOnV1OgMII6LHfg+RjKbUIaPUeEmVDfUbdNev53aom1eN/WbWfqG5LyMlyROtS9TzMKTXVG9CN6mzN4xvIkwLkHFGLdRyy9Il0PqPdDaD+H+ntwLahkdNKCa061i1LkFlw+5JaI3IHLh9wRkftw+ZD7IvIALh/yQCMROyad+/GHcPnwhyLnB3D5kA9E5EO4fMiHIvIYLh/yWER+BZcP+ZWIvA2XD3lbRN6By4e8IyLrcPmQdRF5BJcPeSQit+HyIbc1snimjuFKiU5fmJU3oZzngZYigZqbony3yDr6sLcC5nS7ACvP6hr89WNrATqNC7DbAeOuW4CVR94O2Eg/VrZFu7Sa+LC7InYPRoAfuydiv1DPC7BfBMy0FwVYea7tQzs/Vra+d+HOj70rYu9ByY+V16j7UOPH3g9YMUYF2AMR+0C9LMCGWP1xAVa2+4dgV/xYeZ2qQ3s/NsSazgqwsj09Bg/Gj5VXq0dQ68c+ErGP1esC7GMR+yVYdz/2y4AV9m0B1qyxF2kF6ZE/EsOMLaPWzGYllkZArSnwT7K1JSHfuAX1EqaXYXqEGYiInQyxE4jYzxD7wXJNMjs6IX9X5nKYIQ4DEa1sbcLSVGzfydpjKQlA1DJEbQlR5pHiuzZ9OSPvwtRIyGm2cmEppE9pZr+xFOvxUG55DeJ+DsFj+5RG/jpFSxhBoabKqJ1mazwjI7ovQ7yi6M300vCQcdPMKrio1yKq5UG1RNQbD+qNiJ
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||f ||V = max(f )

log(u0 )||V

!
u0

0 )

def.

,u

0

8 (u, u0 ) 2 (Rn+,⇤ )2(udH

Hilbert’s projective metric:min(f )

dH is a distance on the set of rays !
u.

!
u

ru
{
=

;

0
r>

}

This can be shows to be a distance on the projective cone R+,ú / ≥,where u ≥ uÕ means that ÷s > 0, u = suÕ (the vector are equal
up to rescaling, hence the naming “projective”). This means thatdH satisfies
the triangularand)2dH (u, uÕ ) = 0 if andHilbert’s projective
metric:8 (u,inequalityu0 ) 2 (Rn+,⇤
only if u ≥ def.
uÕ . This is a projective version of Hilbert’s original
!
0
0
0
u
d
(u,
u
)
=
||
log(u)
log(u)||
distance
on bounded open convex setsV Hilbert [1895]. The projecH
def.
0}tive cone Rn+,ú
/ ≥ is a complete metric space for this distance.
>r||f ||V = max(f ) min(f )u;It was introduced independently by [Birkhoff, 1957] and{r[Samel=
!
!
u
son
et
al.,
1957].
They
provedthefollowing
fundamental
theorem,dH is a distance on the set of rays u .
which shows that a positive matrix is a strict contraction on theof positivetheorem:
vectors.
Birkho↵ ’s cone
contraction

Hilbert Projective Metric

(u
dH

0 )

,u

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Õm 2
Theorem 1.1. Let K œ Rn◊m+,ú , then for (v, v ) œ (R+,ú )Ô
Y
÷(K)≠1def.
_] ⁄(K) = Ô<1ÕÕ÷(K)+1dH (Kv, Kv ) Æ ⁄(K)dH (v, v ) where
def.
K K
_
[ ÷(K) = max Ki,k Kj,¸ .
j,k i,¸i,j,k,¸

R2+

Remark 1.11 (Perron-Frobenius). A typical application of Theorem 1.1 is to provide a quantitative proof of Perron-Frobenius
2+
K
K 1.13, is linked2 toR
2 in Remark
theorem, which,as explained
a loKR+
cal linearization of Sinkhorn’s iterates. A matrix K œ K
Rn◊n
with
+
K€ 1n = 1n maps n into n . If furthermore K > 0, then according to Theorem 1.1, it is strictly contractant for the met-Perron Frobenius
Simplex: ⌃k =p 2 Rk+ ;Pi pi = 1

Stochastic matrix: A 2 Rn+ , A> 1k = 1k
Theorem: [Perron-Frobenius]A : ⌃k ! ⌃k

If A > 0, 9!p? , Ap? = p? .
9⇢ 2 [0, 1[, ||Ak p

p? || 6 ⇢k

p?

⌃3

A⌃3

A2 ⌃ 3A1 ⌃3 = {p? }p € Rothblum, 1999] andreferences therein). An intuitive
way to handle these2¸
K= solve
13 . themKiteratively,
K Sinkhorn{K}¸ it satisfies the left
3 1is3 to
3
3that
by
modifying
first
u
so
under
Hilbert’s
Metric(¸) , v(¸) ) æ (uı , vı ) andTheorem 4.2. One
has
(u
2
¸
2
of
Equation(4.14)
and
then
v
to
satisfyitsright-hand
side.
These
two
upd
K
K
{K
3
3 K 3
3K
3
{}
3◊3
¸
ı3
3

tion of K

3

æ {pSinkhorn’s} the invariant
probability distribution of K œ R+,ú withalgorithm:(¸)
ı 4.2. One has
2¸ (u(¸) , v(¸)(¸)ı(uı , vı ) and 2¸
Theorem
)
æ
d
(u
,
u
)
=
O(⁄(K)),d(v,v) = O(⁄(K)). of K œ¸ıHHFigureFigure
4.8: Evolutionof K ofprobability
distribution
3 æ
4.8: EvolutionK¸ {p3 }æthe
{pıinvariant} theinvariantprobability
distribution
of
a
b

K

€

13 =
131.3 = 13 .
K€

(¸+1)
u(¸)
=
ı

def.

and 2¸v(¸+1) =, u )Kv= (¸)
O(⁄(K)

def.

,
T (¸)
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dH (u
),
One also has(0)
One has (u(¸) , v(¸)initialized) æ (uıwith
, vı )anand
arbitrary positive vector(¸) v = 1m . The division operatordıH (P1m, a) Note that a different in
One
also
has
(¸) , v(¸)ı ) ıand
(¸)
ıæ
(¸)
(¸)
ı
betweentwo
vectorsistobeunderstood
entry-wise.
Theorem
4.2.
One
has
(u
)
(u
,
v
) Æ ) æ (u , 2¸
4.2.2¸ Onedhasv )2 andH (u(u(¸) TheoremıTheorem:(¸), u,ıv), leaddH (v, v ) = O(⁄(K)
1 ≠ ⁄(K)will likely
to a different
solutionfor ).
u, v, since (4.22)u,(¸)
v are only define
H (u , u ) = O(⁄(K)(P
1mv/⁄, a)for 2¸any(¸)ı (¸) constant2¸(¸)ı thenı (¸)dH
2¸
(¸)
ı
2¸
ı
multiplicative(ifu,v
satisfy(4.13)
so
do
⁄u,
(¸),€
dH (u dH, u),dHd(P) =, vO(⁄(K)).
uHH(u) =, uO(⁄(K)) = O(⁄(K)),(v, d(v
) = O(⁄(K)
).
1, )v
,Æb)
n
H(u
2
(¸)
ı
1
≠
⁄(K)
turns out however
these
iterations converge (see Remark 4.7 for a justificadH (vthat, v
)Æ
2
14.13
≠ for
⁄(K)
iterative
projections,andRemarkastrictcontraction(¸),€ result) and a
One also(¸)
d(P
1n , b)Onehasalso has(¸)ı FigureH4.5, top row,dH (P coupling
1m , a)diag(u)K(¸)the ısame def.
optimaldiag(v).
shows the
d
(v
,
v
)
Æ
H
d
(u
,
u
)
Æ
(¸)
(¸)
(¸)
2
H
(¸)
(¸))K
(¸) ) computed
1 ≠has⁄(K)
2 diag(vwhere we denoted
P coupling= 1diag(udiag(v).
Lastly,one(¸)
of thediag(u)K
bySinkhorniterations. It ev
≠
⁄(K)
d
(P
1
,
a)
m
H
d
(P
1
,
a)
(¸)
ı (¸)
m
H
ı
(4.23)dH (uÆ, uoptimaldH, u(u) def.
) Æ coupling
the Gibbs kernel
K
towards
thesolving(4.2) by progressive
2
(¸),€
2 (¸)
1
≠
⁄(K)
(¸)
(¸)
(¸)
ı
(¸)
ı
(¸)
ı
1
≠
⁄(K)
d
(P
1
,
b)
ndiagonal.
H log(Pwhere
wedenotedP=
diag(u)Kdiag(v).
Lastly,
one
(¸)
ımass
Î
log(P
)
≠
)Î
Æ
d
(u
,
u
)
+
d
(v
,
v
)
the
away
from
the
Œ
H
H
dH (v , v ) Æ(¸),€ (¸),€
21 ≠ ⁄(K)d(P
1n , b)1n , b)
H
d
(P
(¸)
ı(¸)
H
(¸)
ı
ı
(¸)
ı
ı
dH (v
)Æ
Remark
4.4(Historical
Perspective).
algorithm
wasoriginallyintrodudofH, v(v
v log(P) ÆThis)Î
Î log(P
),≠
Æ
d
(u
,
u
)
+
dH (v
where Pistheuniquesolution(4.2).
2
Œ
H
2
1
≠
⁄(K)
1
≠
⁄(K)
def.
(¸) )Kofdiag(v(¸) ). byconvergence
Sinkhorn[1964]ed P(¸) = diag(uproof
Lastly,one
haswith later contributions [Sinkhorn aıdef.
1967,that
Sinkhorn,1967].
Itvwasused
as a hasheuristic to scale a matrixÕ ) solution
mearlier
2 , one
(¸)
def.
where
P
is
the
uniqueof(4.2).
(¸)(¸)
Proof.
One
notices
for
any
(v,
œ
(R
)
(¸)
(¸)
(¸)
+,ú
where
we denoted
P =P
diag(u).ıandLastly,
one
has
where
we fits
denoted= diag(u)Kdiag(v). Lastly,one
has propo(¸)
ı desired(¸)ı )K diag(v(¸)
marginals(typicallyuniform)known
as theiterativelog(P ) ≠ log(P )ÎŒ Æ dH (u , u ) + dH (v , v )(4.24)
ting procedure(IPFP) ı[Deming
Stephan,and[Bacharach,1
Õ 1940]m RAS
Õ noticesÕ forandÕ 2 , one
(¸)
(¸)
ı
(¸)
ı
Proof.
One
that
any
(v,
v
)
œ
(R
)
has
d
(v,
v
)
=
d
(v/v
,
1
)
=
d
(1
/v,
1
/v
).
(¸)
ı
(¸)
ı
(¸)
ı
m
m) +
H Î log(P
H
Îods
log(P
) ≠ log(P)Îextended
(v
,v
Œ Æ
Hin(u
H)+,ú
)H≠
log(P
)ÎdŒ
Æinfinite
dH, u
(u
, udm
+bydH
(v) , v )[19
[Idel, 2016],
and
later
dimensions
Ruschendorf
unique solution of (4.2).

adopted very early as well in the
precisely to obtain ap
Õ field of economics,Õsolution and performs
a largenumberofone
iterations),
one usuallyratherthesolutionandperforms
a largenumberstudies
of iteration
coupling
solving(4.2),
has thefollowing
Jacobian
local convergence rate. One can write
Sinkhorn
updateas iterationsof a fixed-point
local
convergence
rate.
One
can
write
Sinkhorn
up
≠1
≠1
T
(¸+1)
(¸)
ˆ (f)
= diag(a)P § diag(b) § P .
map f Sinkhorn= (f ) wheremapf(¸+1)= (f(¸) ) §where
fixed point:
4.3. Speeding-up Sinkhorn’sIIterationsITwithf/Á )ˆ≠ (f)1
This
Jacobianisapositivematrix=1n ,(f)
and
thu
n
(f)
=
Á
log
K
(e
log(b),
def.
1
=
Á
lo
def.
1
= Frobeniuswhere
2 § 1 theorem,where
2 §g/Á1) ≠it hasa single
eigenvector
1m(g)
with
ass
= Á=
logdominantK(elog(a).
2 (g)
=
Á lo
2
coupling solving (4.2), one has the following Jacobian
value 1. Since f is defined up to a constant, it is actually the secof/Á )K diag(eg/Á ) the≠1
≠1and
For optimal(f,g)
solving(4.30),
denotingP=
diag(e
1
≠
Ÿ
<
1
which
governsthelocal
linear
rate,
this
shows
For
optimal(f,g)
solving(4.30),
P = thdiPT. optimal
Proposition: ˆ (f) = diag(a) § P § diag(b) §denoting
enough,This For
Jacobian
is a positive
matrixn , and thus byÎf(¸)with≠ fÎˆ= (f)1O((1n ≠=Ÿ)1¸ ).
` largeenough,Frobenius theorem, it has a single dominant eigenvector 1m with associate
Numerically, in “simple cases” (such as when there exists a smoot
value 1. Since f is defined up to a constant, it is actually the second eig
when Á = 0), this rate scales like Ÿ ≥ Á. We refer to [Knight, 2008] fo
1 ≠ Ÿ < 1 which governs the local linear rate, and this shows that forin the bistochastic (assignment) case.
enough,Îf(¸) ≠ fÎ = O((1 ≠ Ÿ)¸ ).

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</latexit>

4.3 Speeding-up
Sinkhorn’s
Numerically,in “simple cases”(such Iterations
as when there exists a smooth Mon
when Á = 0), this rate scales like Ÿ ≥ Á. We refer to [Knight, 2008] for mor
The main computational bottleneck of Sinkhorn’s iterations is the vectin the bistochastic (assignment) case.€
plication against kernels K and K , with complexity O(nm) if impleme
now detail several important cases where the complexity can be improv

4.3

Speeding-up Sinkhorn’s Iterations

Remark 4.15 (Parallel and GPU Friendly Computation). The simplici
algorithm
yields anbottleneck
extremelyofefficientapproachto compute
simulta
The maincomputational
Sinkhorn’s
iterations
is the vector-masolution and performs
a largenumberofone
iterations),
one usuallyratherthesolutionandperforms
a largenumberstudies
of iteration
coupling
solving(4.2),
has thefollowing
Jacobian
local convergence rate. One can write
Sinkhorn
updateas iterationsof a fixed-point
local
convergence
rate.
One
can
write
Sinkhorn
up
≠1
≠1
T
(¸+1)
(¸)
ˆ (f)
= diag(a)P § diag(b) § P .
map f Sinkhorn= (f ) wheremapf(¸+1)= (f(¸) ) §where
fixed point:
4.3. Speeding-up Sinkhorn’sIIterationsITwithf/Á )ˆ≠ (f)1
This
Jacobianisapositivematrix=1n ,(f)
and
thu
n
(f)
=
Á
log
K
(e
log(b),
def.
1
=
Á
lo
def.
1
= Frobeniuswhere
2 § 1 theorem,where
2 §g/Á1) ≠it hasa single
eigenvector
1m(g)
with
ass
= Á=
logdominantK(elog(a).
2 (g)
=
Á lo
2
coupling solving (4.2), one has the following Jacobian
value 1. Since f is defined up to a constant, it is actually the secof/Á )K diag(eg/Á ) the≠1
≠1and
For optimal(f,g)
solving(4.30),
denotingP=
diag(e
1
≠
Ÿ
<
1
which
governsthelocal
linear
rate,
this
shows
For
optimal(f,g)
solving(4.30),
P = thdiPT. optimal
Proposition: ˆ (f) = diag(a) § P § diag(b) §denoting
enough,This For
Jacobian
is a positive
matrixn , and thus byÎf(¸)with≠ fÎˆ= (f)1O((1n ≠=Ÿ)1¸ ).
` largeenough,Frobenius theorem, it has a single dominant eigenvector 1m with associate
Numerically, in “simple cases” (such as when there exists a smoot
value 1. Since f is defined
up to a constant, it is actually the second eig
1
when
Á
=
0),
this
rate
scales like Ÿ ≥Local
Á. We refer
to [Knight,
2008] fo"rate:
⇠ " that
Global1 ≠rate:⇠ e governs
Ÿ < 1 which
the local linear
rate, andthisshowsforthe bistochastic (assignment) case.
[Franklinenough,and inLorenz,
1989][Robert Berman 2017]
Îf(¸) ≠ fÎ = O((1 ≠ Ÿ)¸ ).

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</latexit>

4.3 Speeding-up
Sinkhorn’s
Numerically,in “simple cases”(such Iterations
as when there exists a smooth Mon
when Á = 0), this rate scales like Ÿ ≥ Á. We refer to [Knight, 2008] for mor
The main computational bottleneck of Sinkhorn’s iterations is the vectin the bistochastic (assignment) case.€
plication against kernels K and K , with complexity O(nm) if impleme
now detail several important cases where the complexity can be improv

4.3

Speeding-up Sinkhorn’s Iterations

Remark 4.15 (Parallel and GPU Friendly Computation). The simplici
algorithm
yields anbottleneck
extremelyofefficientapproachto compute
simulta
The maincomputational
Sinkhorn’s
iterations
is the vector-masolution and performs
a largenumberofone
iterations),
one usuallyratherthesolutionandperforms
a largenumberstudies
of iteration
coupling
solving(4.2),
has thefollowing
Jacobian
local convergence rate. One can write
Sinkhorn
updateas iterationsof a fixed-point
local
convergence
rate.
One
can
write
Sinkhorn
up
≠1
≠1
T
(¸+1)
(¸)
ˆ (f)
= diag(a)P § diag(b) § P .
map f Sinkhorn= (f ) wheremapf(¸+1)= (f(¸) ) §where
fixed point:
4.3. Speeding-up Sinkhorn’sIIterationsITwithf/Á )ˆ≠ (f)1
This
Jacobianisapositivematrix=1n ,(f)
and
thu
n
(f)
=
Á
log
K
(e
log(b),
def.
1
=
Á
lo
def.
1
= Frobeniuswhere
2 § 1 theorem,where
2 §g/Á1) ≠it hasa single
eigenvector
1m(g)
with
ass
= Á=
logdominantK(elog(a).
2 (g)
=
Á lo
2
coupling solving (4.2), one has the following Jacobian
value 1. Since f is defined up to a constant, it is actually the secof/Á )K diag(eg/Á ) the≠1
≠1and
For optimal(f,g)
solving(4.30),
denotingP=
diag(e
1
≠
Ÿ
<
1
which
governsthelocal
linear
rate,
this
shows
For
optimal(f,g)
solving(4.30),
P = thdiPT. optimal
Proposition: ˆ (f) = diag(a) § P § diag(b) §denoting
enough,This For
Jacobian
is a positive
matrixn , and thus byÎf(¸)with≠ fÎˆ= (f)1O((1n ≠=Ÿ)1¸ ).
` largeenough,Frobenius theorem, it has a single dominant eigenvector 1m with associate
Numerically, in “simple cases” (such as when there exists a smoot
value 1. Since f is defined
up to a constant, it is actually the second eig
1
when
Á
=
0),
this
rate
scales like Ÿ ≥Local
Á. We refer
to [Knight,
2008] fo"rate:
⇠ " that
Global1 ≠rate:⇠ e governs
Ÿ < 1 which
the local linear
rate, andthisshowsforthe bistochastic (assignment) case.
[Franklinenough,and inLorenz,
1989][Robert Berman 2017]
Îf(¸) ≠ fÎ = O((1 ≠ Ÿ)¸ ).

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4.3 Speeding-up
Sinkhorn’s
P" ||1 ) Mon
Numerically,in “simple cases”(such Iterations
as when therelog(||Pexists(`)a smooth0
when Á = 0), this rate scales like Ÿ ≥ Á. We refer to [Knight, 2008] for mor
The main computational bottleneck of Sinkhorn’s
iterations is the vect
-0 5
in the bistochastic (assignment) case.€
plication against kernels K and K , with complexity
O(nm) if impleme`
now detail several important cases where the-1 complexity can be improv

4.3

P"

Speeding-up Sinkhorn’s Iterations-1 5

Remark 4.15 (Parallel and GPU Friendly Computation). The simplici`-2
1000to 2000
3000
4000
5000
algorithm
yields anbottleneck
extremelyofefficientapproachcomputesimulta
The maincomputational
Sinkhorn’sis thevector-ma" iterations

Overview• Entropic Regularization and Sinkhorn• Convergence Analysis• Sinkhorn Divergences• Generative Model Fitting

Sinkhorn Divergences
OT ( OT, ) ( OT<, <).( OT, () OT, )(OT<, ).( , ).

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existsa measurefromsuchfromthat": >: therein general,
existsa di↵ering
measuredi↵eringsuch thatin 0general,
there there
existsa measuredi↵eringfromsuch that

Wp",p (↵,

Z

"

"

"

"

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(2)

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(2)

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minimizing
the cost)" (↵,
withrespect
to ↵respect
does
allow
Consequently,
thecostOT")(↵,
withto ↵doesallow
not all
Consequently,
minimizing
theOT
cost
OT
with) respectto ↵ not
does
not
" (↵,
p minimizing
us to recover↵to=recover, but↵ towardsa↵“medianaxis”-likestructure↵, but= rather, drives
but rather
drivestowardsa “medianaxis”-likestructu
us to usrecover↵= ratherdrives↵ towards
a “medianaxis”-likestructureof "-localizedmeans
constructedfrom from[FCVP17]–[FCVP17]
see Figure
1.
of Fréchet"-localizedFréchetmeans
constructedfromsee Figure
1.
of "-localizedmeans
constructed[FCVP17]– see –Figure1.
⇡ 1 =↵,⇡
2=X 2 Fréchet

def.

) =

min

d (x, y)d⇡(x, y) + "KL(⇡|⇠)

Problem: W" (↵, ↵) = 0<latexit sha1_base64="6+uaoGEaT2UGT2pHQhLHgztd+tI=">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</latexit>

min Wp",p (↵, )<latexit 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min

d (x, y)d⇡(x, y) + "KL(⇡|⇠)

Problem: W" (↵, ↵) = 0<latexit sha1_base64="6+uaoGEaT2UGT2pHQhLHgztd+tI=">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</latexit>

min Wp",p (↵, )<latexit 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sha1_base64="6+uaoGEaT2UGT2pHQhLHgztd+tI=">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</latexit>

min Wp",p (↵, )"↵(a) " =(a)
0.01.
(a)
" = 0.01.
"=
0.01.

p
Wp," (↵,

Theorem:(c) " =(c)1.00.
" = 1.00.
" =(c)
1.00.

ptheinp1Entropic
1from
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1: Entropic
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(↵,
)
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(
,
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thatup
distribution↵ are ↵ are ↵
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"

optimized
tooptimized
fit anempirical(in blue)
we blue)minimize
to fitandistribution
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distribution(in
weOT
minimize
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optimized
to fitanempirical
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blue)
as
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" (↵,
" (↵,
[Ramdas,Garcı́aTrillos,
Cuturi,
2017]
TheseThese
registrations
are performed
theinunit
square,withsquare,an
EarthMover’sTheseregistrations
are inperformed
the square,
unit
with
an EarthMoveregistrations
are performed
thein
unit
with
an
Earth
Mover’scost C(x,
= kx
cost
C(x,
y) = yk.
kx yk.
costy)C(x,
y)
=yk.
kx
5
5
5
p
2

"!0

Wpp (↵, )W",p (↵, ) " ! +1 ||↵[Léonard 2012][Carlier et al 2017]

2

def.

Kernel norms (MMD): ||⇠|| dp =<latexit sha1_base64="r1eLjjM0nIYTfox/yjuqVbdKFEU=">AABEWnictVzrchPJFW7IbWFzYZP8y5/ZGFKwBcR2tiqp2krVgmWMFwMGycDuCrt0GQmBpBG6YaNV5X3yEnmF/N1UHiD5lVfIuXRP90g9c3ocQpftnlZ/55w+0336nNMtmqN+bzLd3Pz+wsUf/PBHP/7JR5cuf/zTn/38F1c++eWzSTIbt+KjVtJPxi+ajUnc7w3jo2lv2o9fjMZxY9Dsx8+bb3bw8+fzeDzpJcPa9GwUvxw0usNep9dqTKHp5Erlan2YjAeL+mlvebK41T4eLY+3o3r8th13olv13nB6sqi/OD3eXkbt66c3z24cj6J6G3pfP73Bf89uXD25srF5e5P+ReuVLV3ZUPrfYfLJp39RddVWiWqpmRqoWA3VFOp91VATKN+qLbWpRtD2Ui2gbQy1Hn0eq6W6DNgZ9IqhRwNa38DvLjx9q1uH8Iw0J4RuAZc+/IwBGalrgEmg3xjqyC2iz2dEGVvzaC+IJsp2Bn+bmtYAWqfqFbRKONMzFIdjmaqO+hONoQdjGlELjq6lqcxIKyh55IxqChRG0Ib1Nnw+hnqLkEbPEWEmNHbUbYM+/xf1xFZ8bum+M/VvkvIalEhV9eiTlEJDzYl+RG9zBp+xPH3g3AUKsR4j1t6Rrgc0+iH0X0D7IyhLqhmdNKEsqHVZiNyB4kPuiMg9KD7knog8gOJDHojIQyg+5KFGInZMOvfjq1B8+KrI+QkUH/KJiHwKxYd8KiKfQfEhn4nIb6D4kN+IyHtQfMh7IvIBFB/ygYisQfEhayLyCIoPeSQid6H4kLsamb9Sx1ASotMTVuUdqGd5oKXoQ8sdUb67ZB192LsBa7qVg5VXdQX++rGVAJ3GOdjdgHnXycHKM28PbKQfK9ui+7Sb+LD3Rew+zAA/dl/EfqVe52C/Clhpb3Kw8lo7gH5+rGx9H8KTH/tQxD6Cmh8r71GPocWPfRywY4xysIci9ol6m4MNsfrjHKxs96tgV/xYeZ+qQX8/NsSaznKwsj19Bh6MHyvvVs+h1Y99LmJfqNMc7AsR+zVYdz/264Ad9n0O1uyxl2kH6ZI/EsOKLaLWSFcl1kZArSHw76d7S5984ya0S5huiukSZiAi9lLEXiDiIEUcBMs1Se3ohPxdmUs1RVQDEc10b8LaVOzfTvtjrR+AqKSIygqiyCPFd23GMifvwrRIyGm6c2EtZExJar+xFuv5UGx5DeJxBsFz+xXN/JsULWEEhZoqovYq3eMZGdFzEeIdRW9mlIaHjJumVsFFnYqopgfVFFFnHtSZiJp5UDMRNfeg5iLKrnwXVw+YAVb/+C4W9MQzgH3k/BKBV3AHdp37sEYjmD+H4AU+pZbH8LdKsbdUiiTDaB73ScxyvMxY4jHUFmoD2m1UWKH4uk8rLAbJuOdjHePjE+Y2FnrNsRVepjt5lGZMwun0SJ5uSge9xYjWUzk6D6hlSd4d18rh76fr3tTK4XdJ40vy4rlWDj/V0k/PIXtNY2vnwFZhNY209m29LA3OvzANU79Muy5aXHyrAz1nkN5pSfr7+s3sn+O97FCN9WPr5WhMnPFNMuMrQ8PqeeLouRwV9J7Y6zW1qPRIhjrutfWyMiS0iw61HPap7JvBPm39Zky9HI1D8Lh2KOZeOPWys3eUjsbWy9F4pjjvuSRP3tTL0ejSM+vD1svRwGxLQ8f5tl7WsqMGOHa29bJWfUhZYMwB8ZznFusVjclPmmlqPfIPirM1rs+/vo9hzuY4jRGKKVnfNp9OM93LiiUy/kIMVm1aUg70L2aOD5alsVDbYnzFMkwz+/s6HbvHo+YPQIsRrH4+A5By5n2Q0OQk0Hr3geKWGHVlR2Zw2yIOZ0lnBVXXrVPRW7R8OWuUbTuhVikus6O1eqyTvZ7Q3BuRT3hAmpX0cJD7hvMoSho6yGhIpldGd+/1es1qf1PEjVYQo3SmtehEiE/SiuNUn9arjo6v6VOeKRQ+87HzF7PNHW1tMOZJyBahLEU83X4mj+S24b56U9kcN38W0RtFezUnq9GjE6mJGIWabDF74wt6trSP6EwOeTCNFrzHSFMZKT41wyw65tMjsqiuvZV4o75Mho7rE7K6xh4Xo7sOuutBl49xdmDHeAS1GsQMR/BUC4hyLqe6SkjjY3UrPR1N6A0WR/T9jIU0NNjexBkLWRRlv8pQeQdonA0cpYfTWKVj8PU1SnLU75PHxq5Zy3+NTm7N+XaD5nj+bM7PxLSJ6zZxjWjV8KkuP61yYAkW3k+2yX8tHiXyK8MRbajE9djhzHoZ0ol/TBHsiDzjPq02aXVke7v5qdVPDKdDZc7O8TQ7IQsZkf2LYH9KaE5G9OPeHTAn6GwR+mQjQ+xOL/VufL5OT5xj1o/rKb7VYOdbTLZsRvwNXXd1TWgucsTA+8ByZW4bnRyQLxgT17G27nZtF+8+iLT3JNxZwhTtXLlO/G/Qb/Nj5snG2oxADeMbmGhb53sfCcUsqKMG7fLFNsj0daW8mspwrKW2+5+V6WpGsgpFXCgP7tZt4NyiZ+aFs2RMck/W+vA+WpTNRcqjFT3iaDsUxbPd7+odGOW+SbvkBq25Os2SLsyCaRpFmL5SFnmVbzGvLPUw2pP/C3Wr66zWkGKkbAaXNSTl92OK1lwp+zCref6+odXk1/p4pVcxnyHNxYGzlr+D1k/ht5HbPIfRaWaswl2aA0zBPlmNcEu01iOM190MLzMzDS37bPnZOWl6uS3nia/ZutkYe16ayiHNmlOdtTD189B47dB4HajDGp01Wi2admOJTsTYoqZPK0P5leFWK0F5JlKWPTKD6gVI6cZSYVTbIlU5xjeo9yKtTZFWA1arexrgrvkQpH+tr67u79LdPVL3yLdpkQfG8UubVmmPfC7TWhypMQXk/Lm2r+7qr1MLcm+SBUXKfI8TVwyfOrWoLFNJf6d3toTsvLUI5t7SO93H2Ng61f+whhzQmpjQujSIz6lHrOV35YhWLNJtx+eIKPPfIJ+K/Y7imNntbd9JlPEnbLzJq8ry4khhSPqXMm/7a9HrvhO/RhQTzrR33QRa5d8wUmCMyST4PcsJvSHc5fgkgT3aJtnPdTvFp3hDR6LbJPVC/TnAxnDUa+e6O7fMiM3YPoOeqHX71n09ZH79YI4Sv/Oc6DVoVxtoH3Wx8nw+Wg29y2Wfi/QwW+Fr9TGjPm5kYaO8LKauvgjmwhKV48KYEC7lRlFG/nKSl5GZT6dCKZvehnI208A25hXFS9I9UET4vLvrXm/uhjCO5hq9JmFdatwiUcJsXKLzA66lxazUpbV9iFsvFe5GfWcnytspDHV3t7D2my1kTNavr6ScDfd2Za9nohQ5C8MUWopv9ObFhy7NL6Dg70j5okPDMSR3WAX/9o7aUbsf4DbEW13njGZELWgL2iuxd0OPM9ujWEdvHeou/RAO4Tx6oGtJ+h7tpGVlZ8qy5C71cPrvyAqMVSxKb3uWH4PLRR7JOqcy4+mRZZNH01Pmuzhlx2I4hIwkyyWcD59rSKPoKPOdpnJjMNTlEWQ5lOFh7jGEvXPbuzwvl1Oxvta5hPLgXcCcuBgcnvzlxyq2X4iFGjtv5MNzQOvQKaBudov/dRyGj+VUnlcotwl91+x1wFvnfrHOyKI/XH7NWG4hszmfYzjPJB2d9Zb8/Njvi0q9qcQZzYenj/6onQOG10JxHlSWjvHuLLLyhlLBcwGfDIn6j/rbBfnbCG9TGnlylKFkzinyqZkeMjXzjUvf6MxnITJZOnkyZanZOKJKN2J31L66Bz87qQdY9nYof5eS/yLW//3ZNrR2yHqYLDpnDurUFlP2w56itenZ3p/Nkxjv8vLd3hq04Fn4AbXiPd9H1B/v+tYyY8v/Bgmv9YcqUe1MRLJ6umfXVRNGkD154xyQ+Z5vRHfpOYvFN88GAWeL5v7UqkQL+kS+WdDMxTcdKVs0V0f6rB5PDvCGfSPND0Xq99TW0HYe91yJ82Eu58MVzhPSTpbDqfNZ8d2sPC47Dpd2mjub634Jxdn2PK84N1rJ5cJ30Ivx3QJ815GyStp/Q5HwWBVn82YFNGdaJveEdahMJpL1gHFmI33fxZHtvIDXPGD8D3LRDxxJ90CWJuW/IzphGxO9vtbNLknPNx2LM6n3C6Q136Nkmvauo50H5tZicZa+r+cdZy7M/0awoPWAu6m5mShlT+IcOk26ERgTJb4jKZ0byPLI0oTIIlGZiZLMAr69Ow+QZR5ApyNI0xEpdEVJ9Ao+ubKxtfr/Y6xXnm3f3tq8vfVke+PLu/r/zvhI/Ub9Vl2HfeOP6kuYoYfqCDj9Vf1dfa/+sfPPysXKpcrH3PXiBY35lcr8q/z6v3LNML0=</latexit><latexit sha1_base64="OIvAMZ30f1Hst1b7I9ROaJAFG3Y=">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</latexit>(b) " =(b)
0.10.
(b)
" = 0.10.
"=
0.10.

[Ramdas, Garcı́a Trillos,
Cuturi, 2017]

R

|| dp

p
d(x,
y)
d⇠(x)d⇠(y)
X2

Proposition: || · || ||·||p is a norm for 0 < p < 2.

Arthur
Gretton

"!+1

1
S" (↵,) ) "!+1! !12 h ↵? (↵ ) i;) i;
S" (↵,? (↵2 h ↵ , ,C C

Sinkhorn Divergences(4)(4)

which
if C(x,a well-known
quantity:thethe
EnergyDistancewhich
is, is,if C(x,y) y)==
kx kx yk,yk,
a well-known
quantity:
EnergyDistance
MMD.
Interpolating
betweenstandard
familiesof divergences,
SinkhornMMD.
betweentwotwostandard
familiesof divergences,thetheSinkhornpInterpolatingdef.
p to top
1 of pof
1
divergences
could
expected
share
some
their
desirable
properties.
divergences
could
expected
share
some
their
desirable
properties.
Wp," (↵,)be=be
W
(↵,
)
W
(↵,
↵)
W
p,"
p,"
p," ( , )2
2The
present
paper
shows
that
this
indeed,what
happens:
prove
that
The
present
paper
shows
that
this
is, is,indeed,what
happens:
wewe
prove
that
a convex
function
each
input
and
that,
measureS" Sis" ais convex
function
of of
each
of of
its its
input
and
that,
forforanyanymeasure↵, ↵,concaveconcave<latexit 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sha1_base64="NGCGb9j9PjaGiCt5eqRJdYYEdaU=">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</latexit>S" (↵,). ).
0 0= =S" (S",( ), )6 6S" (↵,(5)(5)

Theorem: [Feydy, Séjourné, P, Vialard, Trouvé, Amari 2018]

dp divergences.
Information
theoretic
Sinkhorndivergences.Before
BeforemovingInformation
theoretic
Sinkhornmovingon on
to toourour"If e[AKO17]
is positive:
contributions,
mention[AKO17]
definition
anotherpositive
contributions,
letletus usmentionforforthethedefinition
of of
anotherpositive
p overdivergence
based
Sinkhorn’s
iterations,overdiscreteprobability
simplex.
divergence
based
on on
Sinkhorn’s
iterations,
thethediscreteprobability
simplex.
W> a0rather
Wdi↵erent) isof convex.
",p",p (·,point
The
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opts
aand
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view,
forward
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opts
forfordi↵erent
view,
as as
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putputforwardweak⇤
geometric
structurefeaturespaceandhandlecontinuous
probability
thethegeometric
structureof ofourourfeaturespaceandhandlecontinuous
probability
W
(↵
,
)
!
0
()
↵
*
",p
n
n
densities.
densities.

min Wp",p (↵, )↵p

min W",p (↵, )↵Sample Complexity↵ˆlog(||↵
ˆ

d=2

↵

n

↵
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log(Wp (ˆ↵, ↵ˆ 0 ))d=3

E(|Wp (ˆ↵, ˆ)Theorem:E(|||↵
ˆ ˆ||k<latexit sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">AABCR3ictVxLcxu5EYY2r7Xy8m6Om8MkWqfsLduRFFclqa2tWr0sa821ZJOyvbu0VUNyRNEacugZUn7QvOZv5Z5/kH+QVKUqOaVyTD+AAYbEDDCKY5QkDIivu9EDNLoboDvjeJBN1tf/uvLB977/gx/+6MMrqz/+yU9/9vOrH338OEumaTc67iZxkj7thFkUD0bR8WQwiaOn4zQKh504etI538HPn1xEaTZIRq3Jm3H0bBj2R4PTQTecQNPJ1T+19/auv2uPknQ4m7WJ3iyNevP2WTgJ2mE8v7Xc2onm85Pz4FbAsED16MTTaF4EySZABAB5dyNY/SJYPbw+ej671T5Nw+5sYz7bnM9vnFxdW7+9Tv+C5cqGrKwJ+e8o+Sg4Fm3RE4noiqkYikiMxATqsQhFBuU7sSHWxRjanokZtKVQG9DnkZiLVcBOoVcEPUJoPYfffXj6TraO4BlpZoTuApcYflJABuIaYBLol0IduQX0+ZQoY2sZ7RnRRNnewN+OpDWE1ok4g1YXTvX0xeFYJuJU/IHGMIAxjakFR9eVVKakFZQ8MEY1AQpjaMN6Dz5Pod4lpNJzQJiMxo66Denzv1FPbMXnruw7FX8nKa9BCURTjj7JKYTigugH9Dan8BnLEwPnPlCI5Bix9op0PaTRj6D/DNofQJlTTemkA2VGrfNK5A4UG3LHidyHYkPuO5ENKDZkw4k8gmJDHkkkYlPSuR3fhGLDN52cH0KxIR86kY+g2JCPnMjHUGzIx07kt1BsyG+dyLtQbMi7TuR9KDbkfSeyBcWGbDmRx1BsyGMncg+KDbknkeUrNYWSEJ2BY1VuQb3IAy1FDC1bTvm2yTrasNsea7pbgnWv6l34a8fueug0KsHuecy70xKse+btg420Y9226B7tJjbsPSf2AGaAHXvgxH4lXpRgv/JYaeclWPdaa0A/O9Ztfb+GJzv2ayf2AdTsWPcedQgtduyhx44xLsEeObEPxcsSrI/VT0uwbrvfBLtix7r3qRb0t2N9rOm0BOu2p4/Bg7Fj3bvVE2i1Y584sU/F6xLsUyf2G7Duduw3Hjvs2xKs2mNXaQfpkz8SwYqtohbmqxJrY6AWOvjH+d4Sk2/cgXYXpp9j+oQZOhH7OWLfE9HIEQ1vubLcjmbk77q5NHNE0xPRyfcmrE2c/Xt5f6zFHojdHLG7gKjySPFdq7FckHehWlzISb5zYc1nTEluv7EWyflQbXkV4rCA4Ll9RjP/JkVLGEGhpqqoneV7PCMDeq5CvKLoTY1S8XDjJrlVMFGvnaiOBdVxot5YUG+cqKkFNXWiLiyoCydKr3wT1/aYAVr/+C5m9MQzgH3k8hKAV7AFu849WKMBzJ8j8AIfUcsh/G1S7O0qVZJhNI/7JGY5nhUscQq1mViDdh0V7lJ8HdMKi0Ay7nkoY3x8wtzGTK45tsLzfCcP8oyJP50BydPP6aC3GNB6qkfnPrXMybvjWj38vXzdq1o9/B5pfE5ePNfq4SdS+sklZG9JbOsS2CasprHUvq7XpcH5F6ah6qu066LFxbc6lHMG6b2uSf9AvpmDS7yXHaqxfnS9Ho3MGF9WGF8dGlrPmaHnelTQe2KvV9WC2iMZybhX1+vKkNAuOpJy6Ke6bwb79OSbUfV6NI7A49qhmHtm1OvO3nE+Gl2vR+Ox4LznnDx5Va9Ho0/PrA9dr0cDsy2hjPN1va5lRw1w7Kzrda36iLLAmAPiOc8t2itKyU+aSmoD8g+qszWmz7+8j2HO5nkeI1RT0r5tOZ1OvpdVS6T8hQis2qSmHOhfTA0frEhjJjad8RXLMCns78t09B6Pmm+AFgNY/XwG4MqZxyChykmg9Y6B4oYz6iqOTOE2nTicJacLqLZsnTi9Rc2Xs0bFthNqdcVlerRaj22y1xnNvTH5hA3SrEsPjdI3XEbRpaFGQUNuenV091au16L215248QJinM+0Lp0I8UladZxq03rT0PE1ecozgcJnPnr+Yrb5VFobjHkSskUoSxVPs5/KI5ltuK/eFDrHzZ8F9EbRXl2Q1RjQiVTmjEJVtpi98Rk9a9rHdCaHPJhGF95jIKmMBZ+aYRYd8+kBWVTT3rp4o75Uho7rGVldZY+r0X0D3beg68c4O7BjPIBaC2KGY3hqeUQ5q7muEtJ4Km7lp6MJvcHqiD4uWEhFg+1NVLCQVVH2WYHKK0DjbOAo3Z/GIh2Fby9Rckf9Nnl07Fq0/Nfo5Fadb4c0x8tnc3kmpkdcN4lrQKuGT3X5aZEDSzCzfrJJ/mv1KJFfHY5oQ11cnxucWS8jOvGPKIIdk2cc02pzrY5ibzM/tfiJ4nQk1Nk5nmYnZCEDsn8B7E8JzcmAfsy7A+oEnS1CTDbSx+4Mcu/G5usMnHNM+3EDwbca9HyLyJZNib+ia66ujOYiRwy8D8wX5rbSSYN8wYi4ptK667VdvfsgUt+TMGcJU9Rz5Trxv0G/1Y+aJ2tLMwI1jG8gk7bO9j4SillQRyHt8tU2SPU1pfw0l+G5lFrvf1qmTwuS7VLEhfLgbt0Dzl16Zl44S1KSO1vqw/toVTYXKY8X9IijPaUonu1+X+7AKPdN2iXXaM21aZb0YRZM8ihC9XVlkRf5VvMqUvejnf1fqGtdF7WGFAOhM7isIVd+P6JozZQyhlnN8/ecVpNd6+lCr2o+I5qLQ2Mtv4PWX8FvJbd69qPTKViFbZoDTEE/aY1wS7DUw4/XdoGXmpmKln7W/PScVL3MlsvE12zddIx9UZvKEc2a1zJroeqXofHCoPHCU4ctOmvUWlTtyhKdOGOLljyt9OVXh1urBuWpk7LbI1OogYeUZizlR7XnpOqO8RXqrZPWupNWCKvVPA0w17wP0r7WF1f3u3x3D8Rd8m265IFx/NKjVTogn0u1VkdqTAE535H21Vz9bWpB7h2yoEiZ73HiiuFTpy6VeS7pb+TOlpCd1xZB3Vt6JfsoG9um+u+WkENaExmtS4W4Qz0iKb8pR7BgkW4bPkdAmf+QfCr2O6pjZrO3fidBwZ/Q8SavKs2LI4UR6d+VeTtYil4PjPg1oJhwKr3rDtCq/4aRAmNUJsHuWWb0hnCX45ME9mg7ZD+X7RSf4o0MiW6T1DPxhYeN4ahXz3VzbqkRq7F9Bj1R6/qt23q4+cXeHF38LnOiF9KuNpQ+6mzh+XK0QrnLFZ+r9DBd4Kv1MaU+ZmSho7wipi0+9+bCEtXjwhgfLvVGUUf+epLXkZlPp3wpq96KcjHTwDbmjOIl1z1QRNi8u+tWb+6GYxydJXodwprUuMVFCbNxicwPmJYWs1JXlvYhbr1SuRvFxk5UtlMo6uZuoe03W8iIrF8sXDkb7m3K3i5EKe4sDFPoCr7RWxYfmjQ/h4K/A2GLDhVHn9xhE/zbLbEj9t7DbYiXss4ZzYBa0Bb0FmLvUI6z2KNaRy8N6iZ9Hw7+PAaga5f0A9pJ68rOlN2Sm9T96b8iK5CKyCm97ll/DCYX90iWOdUZz4Asm3s0A6G+i1N3LIqDz0iKXPz58LmGaxSnQn2nqd4YFHX3CIoc6vBQ9xj83rnuXZ+XyalaX8tcfHnwLqBOXBQOT/7KYxXdz8dCpcYbef8c0DqcVlBXu8X/Og7FR3Oqz8uXW0bfNXvh8da5XyQzsugP118zmpvPbC7n6M8zyUenvSU7P/b7glpvKjFG8/7poz+q54DiNROcB3VLx3hzFml5fanguYBNhkT8U/x5xf1thJc5jTI56lBS5xTl1FQPNzX1jUvb6NRnPjJpOmUyFanpOKJJN2J3xIG4Cz87uQdY93Yof5eS/yLW/v3ZHrSekvVQWXTOHLSpLaLshz5F69Gzvj9bJjHe5eW7vS1owbPwBrXiPd8H1B/v+rYKYyv/Bgmv9a9FInqFiGTxdE+vqw6MoHjyxjkg9T3fgO7ScxaLb54NPc4W1f2pRYlm9In7ZkGnFN8xpOzSXB3Ls3o8OcAb9mGeHwrEb6ktlHYe91wX56NSzkcLnDPSTpHDa+Oz6rtZZVx2DC69PHd2IfslFGfr87zq3OhuKRe+g16N71fg+4aUTdL+OUXCqajO5k0raE6lTOYJ60ioTCTrAePMMH/f1ZHtRQWvC4/x3y9F3zck3QdZOpT/DuiELSV6sdTNHknPNx2rM6n3KqSV36M8ubq2sfh/GSxXjjdv//H2xsM7a19uy//m4EPxifi1uA5L/PfiSyB2JI6BwT9WPl75ZOWXW3/Z+tfWv7f+w10/WJGYX4jCv+2V/wLgpdNy</latexit><latexit
sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">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</latexit>

Wp (↵, )|) = O(n||↵||k |) = O(n

1
d

1
2

)

)

Optimal transport: su↵ers from curse of dimensionality.
! Adapt to support dimensionality [Weed, Bach 2017]

log(n)

Sample Complexity↵ˆlog(||↵
ˆ

d=2

↵

n

↵
ˆ 0 ||k )<latexit sha1_base64="tOLVHuCcODYvL/AduiUQnUlHORs=">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</latexit><latexit<latexit
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log(n)↵
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sha1_base64="iqmR2w6gptZhqD4tCYOzKsFQvt8=">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</latexit>

log(Wp (ˆ↵, ↵ˆ 0 ))d=3

E(|Wp (ˆ↵, ˆ)Theorem:E(|||↵
ˆ ˆ||k<latexit sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">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</latexit><latexit
sha1_base64="bag9QVdCb/ToHSbCpc63UssOMBE=">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</latexit>

1dWp (↵, )|) = O(n

||↵

1
2

||k |) = O(nlog(n)))

log(W",p (ˆ↵, ↵ˆ 0 ))

Optimal transport: su↵ers from curse of dimensionality.
! Adapt to support dimensionality [Weed, Bach 2017]

W",p (↵, )|) = O("log(n)

log(W",p (ˆ↵, ↵ˆ 0 ))

Theorem: [Genevay, Bach, P, Cuturi]

E(|W",p (ˆ
↵, ˆ)

d=2

d
2

n

1
2

)

d=5

log(n)

Overview• Entropic Regularization and Sinkhorn• Convergence Analysis• Sinkhorn Divergences• Generative Model Fitting

Density Fitting and Generative Models
Observations:

def.

=

1
n

Pn

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i=1 xiParametric model: ✓ 7! ↵✓<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><latexit
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Observations:

def.

=

1
n

Pn

↵✓
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Parametric model: ✓ 7! ↵✓<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><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=">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><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

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Stochastic gradient descent

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1
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min E(✓) = W",p (↵✓ ,
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def.

x1 ↵✓
<latexit sha1_base64="TKa8+ATXQYu13dSUNb6eZuTq9gM=">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
sha1_base64="TKa8+ATXQYu13dSUNb6eZuTq9gM=">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</latexit>

z1

g✓

X

y1

...

Generative model...

(y1 , . . . , yn )...

(x1 , . . . , xm )(z1 , . . . , zm )<latexit sha1_base64="TjoBBEODG1m4HabJKUjPHdtFO3E=">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</latexit><latexit
sha1_base64="TjoBBEODG1m4HabJKUjPHdtFO3E=">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</latexit>

C

✓2

✓1
...

def.

(dp (xi , yj ))i,j

randn⌧ rÊ(✓)✓Pp
1
Ê(✓) = W",p ( m i g✓ (zi ) ,yn<latexit sha1_base64="7H1S8QwhorAsfW/ryppJL5U4Ekg=">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</latexit><latexit
sha1_base64="7H1S8QwhorAsfW/ryppJL5U4Ekg=">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</latexit>

Input data

Z

Stochastic gradient descent

xm

✓

zm

)

e C/"
K

1m

h(C

1/·

1/·

b

K)b, ai

⇥mK >

⇥nK

)

b

Ê(✓)

a

Sinkhorn, ` = 0, . . . , L

1

Automatic Differentiation
Setup: E : Rn ! R computable in K operations.
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sha1_base64="EGEItvC57NMEhoT3ZTnfdA7Jy/Y=">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</latexit><latexitzufunction y = E(x)a = xi1*xb = rho(a)z = xi2*b
y = 1/2*norm(z-u)^2<latexit sha1_base64="USnAlRyitZVd89K4qnfd5zcfb4E=">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</latexit><latexitp

Hypothesis: elementary operations (a ⇥ b, log(a), a . . . )and their derivatives cost O(1).

Question: What is the complexity of computing rE : Rn ! Rn ?

Automatic Differentiation
Setup: E : Rn ! R computable in K operations.
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sha1_base64="EGEItvC57NMEhoT3ZTnfdA7Jy/Y=">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</latexit><latexituzfunction y = E(x)a = xi1*xb = rho(a)z = xi2*b
y = 1/2*norm(z-u)^2<latexit sha1_base64="USnAlRyitZVd89K4qnfd5zcfb4E=">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</latexit><latexitp

Hypothesis: elementary operations (a ⇥ b, log(a), a . . . )and their derivatives cost O(1).

Question: What is the complexity of computing rE : Rn ! Rn ?
Finite di↵erences:1
rE(✓) ⇡ (E(✓ + " 1 ) E(✓), . . . E(✓ + " n ) E(✓))"
K(n + 1) operations, intractable for large n.

Automatic Differentiation
Setup: E : Rn ! R computable in K operations.
⇠1⇢ ⇠2<latexit sha1_base64="Vu4PmXFB/J+C+8LYNg1Wa7z209g=">AAA9l3ictVvrchu3FYbTS2L15qS/OpnpbCu7k2Rcj6Rmpp3JaBpbsi3Fii2ZlOwktD28rKi1l1x6l5QvDH/0afq3fYI+R9+g/dVX6DkHwAJLYvcAqiuMJCyI75yDA+BcgGVvkibFdGPjn5fe+8EPf/Tj9z+4vPaTn/7s57+48uFHJ0U2y/vxcT9Ls/xxr1vEaTKOj6fJNI0fT/K4O+ql8aPeix38/NF5nBdJNm5P30ziJ6PucJycJv3uFJqeXfn4auc07/bnm4v51qIzzvLRvNMfZNPF062rz66sb9zYoJ9otbKpKutC/RxmH/56S3TEQGSiL2ZiJGIxFlOop6IrCijfiU2xISbQ9kTMoS2HWkKfx2Ih1gA7g14x9OhC6wv4O4Sn71TrGJ6RZkHoPnBJ4TcHZCSuASaDfjnUkVtEn8+IMrbW0Z4TTZTtDfzvKVojaJ2KM2jlcLqnLw7HMhWn4k80hgTGNKEWHF1fUZmRVlDyyBrVFChMoA3rA/g8h3qfkFrPEWEKGjvqtkuf/4t6Yis+91Xfmfg3SXkNSiRaavRZSaErzol+RLM5g8+kPClwHgKFWI0Ra69I1yMa/Rj6z6H9PpQF1bROelDm1LpoRO5AcSF3WORdKC7kXRZ5AMWFPGCRh1BcyEOFRGxOOnfjW1Bc+BbL+QiKC3nEIh9CcSEfssgTKC7kCYv8FooL+S2LvAPFhbzDIu9BcSHvscg2FBeyzSKPobiQxyzyNhQX8rZC1u/UHEpGdBJmV96EepUHWooUWm6y8t0i6+jC3vLY0/0aLL+rd+G/G7vrodO4BnvbY92d1mD5lXcXbKQby9uiPfImLuwei92HFeDG7rPYr8TzGuxXHjvtRQ2W32sH0M+N5a3v1/Dkxn7NYu9DzY3lfdQDaHFjH3h4jEkN9pDFHomXNVgfq5/XYHm73wK74sbyfqoN/d1YH2s6q8Hy9vQEIhg3lvdWj6DVjX3EYh+L1zXYxyz2G7Dubuw3Hh72bQ1W+9g18iBDikdi2LFN1LrlrsTaBKh1Gf5p6VtSio170M5hhiVmSJgRi7hbIu56Ig5KxIG3XEVpRwuKd3kurRLR8kT0St+EtSnbf1D2x1rqgdgtEbtLiKaIFOdaj+WcogvdwiGnpefCms+YstJ+Yy1W66HZ8mrEgwpCru0zWvnXKVvCDAo11UTtrPTxEhnRcxPiFWVvepSaB4+bllbBRr1mUT0Hqsei3jhQb1jUzIGasahzB+qcRZmdb+M6HivA6B/nYk5PcgXIGLm+RBAV3ASvswd7NIL1cwhR4ENqeQD/W5R7c6VJMszm0U/iKceTiiXOoTYX69BussJdyq9T2mExSCZ7PlA5Pj7h2cZc7TlphRelJ4/KExN/OgnJMyzpYLQY0X4Ko3OPWhYU3claGH6v3Pe6Foa/TRpfUBQva2H4qZJ+egHZ2wrbvgC2BbtporRv6qE05PmLpKHra+R10eLirI7UmkF6rwPp76uZ2b/AvOxQTerH1MNoFNb4isr4QmgYPReWnsOoYPQko15di4JHMlZ5r6mHypCRFx0rOcxT6Mxgn4GaGV0Po3EIEdcO5dxzqx66eiflaEw9jMaJkOeeC4rkdT2MxpCepT5MPYwGnrZ0VZ5v6qGWHTUgc2dTD7XqYzoFxjMgueZli4mKcoqTZopaQvFB82mNHfOv+jE8s3la5gjNlExsW0+nV/qyZol0vBCDVZsGyoHxxcyKwao05mKLza+kDNOKf1+lY3w8av4AtBjB7pd3ANyZeQoS6jMJtN4pUNxks67qyDRui8XhKjldQnVU65SNFg1feWpUbXtGrVxeZkZr9Nghe13Q2ptQTHhAmuX0cFA7w3UUOQ0dVDTE0wvR3Vu1X6va32BxkyXEpFxpfboRkjdpzXmqS+stS8fX1C3PFIq88zHrF0+bT5W1wZwnI1uEsjTxtPvpcyS7Df3qdWHOuOVnEc0o2qtzshoJ3UgVbBaqT4tlND6nZ0P7mO7kkIek0Yd5jBSViZC3ZniKjufpEVlU295yvFFf+oRO1guyutoeN6OHFnroQIfnODvgMe5DrQ05wzE8tT2ynLVSVxlpPBe/L29HM5rB5ow+rVhITUPam7hiIZuy7LMKlVeAxtUgs3R/Gst0NL6zQonP+l3ymNy1avmv0c2tvt/u0hqvX831JzED4rpFXCPaNfJWVz4tc5ASzJ2fbFH82jxK5BfCEW0ox/WpxVnqZUw3/jFlsBOKjFPabdzuqPa2z6eWP9GcDoW+O8fb7IwsZET2LwL/lNGajOjXfndA36BLi5CSjfSxO0kZ3bhinYRdYyaOS4R8q8Gst5hs2Yz4a7r27ipoLcqMQfqBxdLa1jo5oFgwJq65su5mbzd7H0Sa9yTsVSIpmrXyCfH/lP7qX71O1ldWBGoYZ6BQts41HxnlLKijLnn5Zhuk+9pSXi1leKqkNv7PyHS1ItkuZVwoD3rrAXDu07PkhaskJ7mLlT7Sjzad5iLlyZIecbSnlMVLuz9UHhjlvk5ecp32XIdWyRBWwbTMInRf7hR5mW8zryp1P9rF/4W60XVVa0gxEuYEV2qIO9+PKVuzpUxhVcv1+4J2k1vr+VKvZj5jWosjay9/D62/gb9abv3sR6dXsQq3aA1ICubJaES2RCs9/HjdqvDSK1PTMs+Gn1mTupfdcpH8Wlo3k2OfB1M5pFXzWp1a6PpFaDy3aDz31GGb7hqNFnW7tkTP2NyirW4rffmFcGsHUJ6xlPmITKMSDyntXMqP6oClyuf4GvWWpbXB0urCbrVvA+w974N07/Xl3f196d0jcYdimz5FYDJ/GdAuTSjm0q3NmZqkgJw/V/bV3v0dakHuPbKgSFm+x4k7Rt469aksSkl/pzxbRnbeWAT93tIr1Ufb2A7V/7CCHNGeKGhfasTn1CNW8ttyREsW6YYVc0R08t+lmErGHc05s93bzElUiSdMvil3leElM4Ux6Z87edtfyV73rfw1opxwpqLrHtAKn2GkIDH6JMEdWRY0Q+jl5E2CjGh7ZD9X7ZS8xRtbEt0gqedi28PGyKzXrHV7bekR67F9Bj1R62bWXT14fqk3R47fRW70uuTVRipGnS89X4xWV3m56nOTHmZLfI0+ZtTHzixMllfFdMQX3lykRGFcJMaHS9goQuQPkzxEZnk75UtZ99aUqycN0sacUb7EvQeKCFd094kzmvuUGUdvhV6PsDY12cJRwtO4TJ0P2JYWT6Uur/gh2Xq50Rullieq8xSauu0tjP2WFjIm65cK7sxG9rZl71SyFP4URlLoC/lGb11+aNP8Agr+jYQrO9Qcfc4OWxDf3hQ74vY7eBviparLE82IWtAWDJZy764aZ7VHs45eWtRt+j4c/HkkoGtO+oQ8aajskjIvuU3dn/4rsgK5iFnpTc/wMdhc+JGscgoZT0KWjR9NIvR3cULHojn4jKTKxZ+PvNfgRnEq9HeawsagqfMjqHII4aHfY/Cbc9M7nJfNqVlfq1x8eUgvoG9cNA5v/upzFdPPx0Ll1oy8ew5oHU4bqGtv8b+OQ/MxnMJ5+XIr6Ltmzz1mXfaL1YksxsPhe8Zw81nN9Rz9eWbl6Ey05OYn474oaKYyazTvnj7Go2YNaF5zIc9Beekk3l5FRl5fKngv4JIhE/8R/7jEfxvhZUmjTo4QSvqeop6a7sFT09+4dI1Of+Yjk6FTJ1OVmskjWvRG7I7YF3fgd6eMAEPfDpXfpZT/Eev+/uwAWk/JeuhTdHly0KG2mE4/zC3agJ7VGeOzK+uby99CXq2cbN3Y3LixebS1/uUt9Q3lD8TH4reQl2yKP4ovxR6M9xhk+ov4q/ib+Pv2r7b/vH1ne092fe+SwvxSVH62j/4L3/7e/A==</latexit>

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y

x

a b

u

z

function y = E(x)a = xi1*xb = rho(a)z = xi2*b
y = 1/2*norm(z-u)^2

function dx = nablaE(x)dz = z-u
db = xi2’*dzda = diag(dphi(a)) * db
dx = xi1’*da<latexit sha1_base64="USnAlRyitZVd89K4qnfd5zcfb4E=">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

pHypothesis: elementary operations (a ⇥ b, log(a), a . . . )and their derivatives cost O(1).

Question: What is the complexity of computing rE : Rn ! Rn ?
Finite di↵erences:1
rE(✓) ⇡ (E(✓ + " 1 ) E(✓), . . . E(✓ + " n ) E(✓))"
K(n + 1) operations, intractable for large n.

Theorem: there is an algorithm to compute rEin O(K) operations. [Seppo Linnainmaa, 1970]<latexit 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This algorithm is reverse mode automatic di↵erentiation

Seppo
Linnainmaa

Examples of Images Generation
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Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, anddatasets.

Examples of Images Generation
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h⇠
(a) WGAN MNIST(b) WGAN CelebA(a) WGAN MNISTg✓<latexit sha1_base64="lWnDUgue/pmzTJYmkYn0saVTkWY=">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Inception score"(d) toMMD
GAN theMNIST(e) MMDMMD GAN LSUN
! Needlearn
metric d(x,y) =GAN||h⇠CelebA(x) h⇠ (y)||(f)(GANs)(d) MMD GAN MNIST(e) MMD GAN CelebA(f) MMD GAN

Figure2: Generated!
Influence
ofsamples"? from WGAN and MMD GAN on MNIST, CelebA, and LSUN bedroom
datasets.
Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and! Performance evaluationof generative models is an open problem. Ian Goodfellow
datasets.

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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