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

Dual and Semi-discrete
Gabriel Peyré
www.numerical-tours.com
ÉCOLE NORMALES U P É R I E U R E

Overview• Dual Problem• W1• Semi-discrete Problem• Optimal Quantization

Discrete Dual Problem
Wc (a, b) = min<latexit sha1_base64="k31DWOUFIeDNkCyXKSkoKxf19Uc=">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</latexit><latexit
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P2Rn⇥m+hC, Pi : P1 = a, P> 1 = b

Discrete Dual Problem
Wc (a, b) = min

P2Rn⇥m+= min<latexit sha1_base64="uwFODPNe8xlixV6yPyT2R7XByQ4=">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sha1_base64="uwFODPNe8xlixV6yPyT2R7XByQ4=">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</latexit>

hC, Pi : P1 = a, P> 1 = b
max
hC, Pi + hf, anmf2R ,g2R
P2Rn⇥m+P1i + hg, b

P> 1i

Discrete Dual Problem
Wc (a, b) = min

P2Rn⇥m+= min<latexit sha1_base64="byX1JLQtzlA4jBFjTdVykmGvJhM=">AABFxXictVzvchu3EYfSf7HSNk77sV/OUdxxGtmV3HTaaSYzsSVZVizbtEXJTkxLwyOP9NlHHs0jadkMp4/Rx+lz9A3aT32CznSxCxxwJO4Wp7q+kYTD4be72AMWuwucw1ESZ5OtrX+sffCjH//kpz/78NL6Rz//xS8/vvzJr06ydDruRMedNEnHT8N2FiXxMDqexJMkejoaR+1BmERPwlc78vmTWTTO4nTYnLwdRc8H7f4w7sWd9gSqzi7/7eugdaV1JWhNB/Fw3moErXgYtB4/PvvidA6FSTyIsmCwWATr0KJ9Pm/1dIvT4Warn98MFgESWm9108lo3tpZALHF+heBuu/BfTu4HkgO6TDKrEd9eBTio9PWJB3R47PLG1s3tvBfsFrYVoUNof410k8+/Ui0RFekoiOmYiAiMRQTKCeiLTK4noltsSVGUPdczKFuDKUYn0diIdYBO4VWEbRoQ+0r+N2Hu2eqdgj3km
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sha1_base64="ArAP4pFqCC7ehj4ZFosmugbsUgQ=">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</latexit>

hC, Pi : P1 = a, P> 1 = b
max
hC, Pi + hf, anmf2R ,g2R
P2Rn⇥m+= max
hf, ai + hg, bi + minnm
f2R ,g2R<latexit sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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</latexit><latexit
sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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</latexit>

P2Rn⇥m+P1i + hg, b
hC, PiP> 1i

hf, P1i

hg, P> 1i

Discrete Dual Problem
Wc (a, b) = min

P2Rn⇥m+= min<latexit sha1_base64="byX1JLQtzlA4jBFjTdVykmGvJhM=">AABFxXictVzvchu3EYfSf7HSNk77sV/OUdxxGtmV3HTaaSYzsSVZVizbtEXJTkxLwyOP9NlHHs0jadkMp4/Rx+lz9A3aT32CznSxCxxwJO4Wp7q+kYTD4be72AMWuwucw1ESZ5OtrX+sffCjH//kpz/78NL6Rz//xS8/vvzJr06ydDruRMedNEnHT8N2FiXxMDqexJMkejoaR+1BmERPwlc78vmTWTTO4nTYnLwdRc8H7f4w7sWd9gSqzi7/7eugdaV1JWhNB/Fw3moErXgYtB4/PvvidA6FSTyIsmCwWATr0KJ9Pm/1dIvT4Warn98MFgESWm9108lo3tpZALHF+heBuu/BfTu4HkgO6TDKrEd9eBTio9PWJB3R47PLG1s3tvBfsFrYVoUNof410k8+/Ui0RFekoiOmYiAiMRQTKCeiLTK4noltsSVGUPdczKFuDKUYn0diIdYBO4VWEbRoQ+0r+N2Hu2eqdgj3km
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sha1_base64="ArAP4pFqCC7ehj4ZFosmugbsUgQ=">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</latexit>

hC, Pi : P1 = a, P> 1 = b
max
hC, Pi + hf, anmf2R ,g2R
P2Rn⇥m+= max
hf, ai + hg, bi + minnm
f2R ,g2R<latexit sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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</latexit><latexit
sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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</latexit>

P2Rn⇥m+= max
hf, ai + hg, bi + min hCnm
f2R ,g2R

P>0

P1i + hg, b
hC, PiP> 1i

hf, P1i(f1> + 1g> ), Pi

hg, P> 1i

Discrete Dual Problem
Wc (a, b) = min

P2Rn⇥m+= min<latexit sha1_base64="byX1JLQtzlA4jBFjTdVykmGvJhM=">AABFxXictVzvchu3EYfSf7HSNk77sV/OUdxxGtmV3HTaaSYzsSVZVizbtEXJTkxLwyOP9NlHHs0jadkMp4/Rx+lz9A3aT32CznSxCxxwJO4Wp7q+kYTD4be72AMWuwucw1ESZ5OtrX+sffCjH//kpz/78NL6Rz//xS8/vvzJr06ydDruRMedNEnHT8N2FiXxMDqexJMkejoaR+1BmERPwlc78vmTWTTO4nTYnLwdRc8H7f4w7sWd9gSqzi7/7eugdaV1JWhNB/Fw3moErXgYtB4/PvvidA6FSTyIsmCwWATr0KJ9Pm/1dIvT4Warn98MFgESWm9108lo3tpZALHF+heBuu/BfTu4HkgO6TDKrEd9eBTio9PWJB3R47PLG1s3tvBfsFrYVoUNof410k8+/Ui0RFekoiOmYiAiMRQTKCeiLTK4noltsSVGUPdczKFuDKUYn0diIdYBO4VWEbRoQ+0r+N2Hu2eqdgj3km
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sha1_base64="ArAP4pFqCC7ehj4ZFosmugbsUgQ=">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</latexit>

hC, Pi : P1 = a, P> 1 = b
max
hC, Pi + hf, anmf2R ,g2R
P2Rn⇥m+= max
hf, ai + hg, bi + minnm

P2Rn⇥m+f2R ,g2R<latexit sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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</latexit><latexit
sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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</latexit>= max
hf, ai + hg, bi + min hC
nm
P>0f2R ,g2R

min hQ, Pi =P>0⇢

P1i + hg, b
hC, Pihf, P1i(f1> + 1g> ), Pi

0 if Q > 0,1 otherwise.

Theorem:Wc (a, b) = max{hf, ai + hg, bi : fnmf2R ,g2R

P> 1i

g 6 C}

hg, P> 1i

Discrete Dual Problem
Wc (a, b) = min

P2Rn⇥m+= min

hC, Pi : P1 = a, P> 1 = b
max
hC, Pi + hf, anmf2R ,g2R
P2Rn⇥m+= max
hf, ai + hg, bi + minnm

P2Rn⇥m+f2R ,g2R<latexit sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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
sha1_base64="xD9q06rTZjW94zgCM8TF494RYlA=">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</latexit>= max
hf, ai + hg, bi + min hC
nm
P>0f2R ,g2R

min hQ, Pi =P>0⇢

P1i + hg, b
hC, PiP> 1i

hf, P1i

hg, P> 1i(f1> + 1g> ), Pi

0 if Q > 0,1 otherwise.

Theorem:Wc (a, b) = max{hf, ai + hg, bi : fnmf2R ,g2R

g 6 C}

Primal-dual relations: {(i, j) : Pi,j 6= 0} ⇢ {(i, j) : fi + gj = Ci,j }<latexit sha1_base64="mXT9YhX8L7lf5BdFBO1oNpSQe30=">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</latexit><latexit
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Continuous Dual Problem
Wc (↵, ) =min⇡2M+ (X ⇥Y)def.

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def.

hf, ai =P

i fi aiContinuous Dual Problem
Wc (↵, ) ={hc, ⇡i : ⇡ 1 = ↵, ⇡ 2 = }min⇡2M+ (X ⇥Y)def.

hf , ↵i =<latexit sha1_base64="se5IX5+6njWKy85BP15PzazVNiY=">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</latexit><latexit
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def.

hf, ai =P

i fi ai

Theorem:Wc (↵, ) =max

f 2C(X ),g2C(Y)max

f 2C(X ),g2C(Y)⇢Z

f (x)d↵(x) +XZY{hf , ↵i + hg, i : f

g 6 c}

g(y)d (y) : 8(x, y), f (x) + g(y) 6 c(x, y)Continuous Dual Problem
Wc (↵, ) ={hc, ⇡i : ⇡ 1 = ↵, ⇡ 2 = }min⇡2M+ (X ⇥Y)def.

hf , ↵i =<latexit sha1_base64="se5IX5+6njWKy85BP15PzazVNiY=">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</latexit><latexit
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def.

hf, ai =P

i fi ai

Theorem:Wc (↵, ) =max

f 2C(X ),g2C(Y)max

f 2C(X ),g2C(Y)⇢Z

f (x)d↵(x) +XZY{hf , ↵i + hg, i : f

g 6 c}

g(y)d (y) : 8(x, y), f (x) + g(y) 6 c(x, y)Primal-dual relations: Supp(⇡) ⇢ {(x, y) : f (x) + g(y) = c(x, y)}C-transforms
Fixing f , solve the dual with respect to g:⇢Zmaxg(y)d (y) : 8(x, y), f (x) + g(y) 6 c(x, y)
g2C(Y)⇢Z Y
max
g(y)d (y) : 8y, g(y) 6 min c(x, y) f (x)<latexit sha1_base64="izjhNSpYSgi9QeTHRhVJz7BqTyg=">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</latexit><latexit
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C-transforms
Fixing f , solve the dual with respect to g:⇢Zmaxg(y)d (y) : 8(x, y), f (x) + g(y) 6 c(x, y)
g2C(Y)⇢Z Y
max
g(y)d (y) : 8y, g(y) 6 min c(x, y) f (x)<latexit sha1_base64="izjhNSpYSgi9QeTHRhVJz7BqTyg=">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</latexit><latexit
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x

Y

def.

c-transforms:

f c (y) = min c(x, y)
x2X

def.

g c (x) = min c(x, y)
y2Y

f (x)
g(y)

C-transforms
Fixing f , solve the dual with respect to g:⇢Zmaxg(y)d (y) : 8(x, y), f (x) + g(y) 6 c(x, y)
g2C(Y)⇢Z Y
max
g(y)d (y) : 8y, g(y) 6 min c(x, y) f (x)<latexit sha1_base64="izjhNSpYSgi9QeTHRhVJz7BqTyg=">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</latexit><latexit
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x

Y

def.

c-transforms:

f c (y) = min c(x, y)
x2X

def.

g c (x) = min c(x, y)
y2Y

f (x)
g(y)

Proposition: The dual solution on g is equal ↵-a.e to f c .

C-transforms
Fixing f , solve the dual with respect to g:⇢Zmaxg(y)d (y) : 8(x, y), f (x) + g(y) 6 c(x, y)
g2C(Y)⇢Z Y
max
g(y)d (y) : 8y, g(y) 6 min c(x, y) f (x)<latexit sha1_base64="izjhNSpYSgi9QeTHRhVJz7BqTyg=">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</latexit><latexit
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x

Y

def.

c-transforms:

f c (y) = min c(x, y)
x2X

def.

g c (x) = min c(x, y)
y2Y

f (x)
g(y)

Proposition: The dual solution on g is equal ↵-a.e to f c .
constant
Cost c(x, y) = hx, yi:
R
R
R
R
2
2
||x y|| d⇡(x, y) = 2 hx, yid⇡(x, y) + ||x|| d↵(x) + ||y||2 d (y)
c

f =

( f)

⇤

⇤

def.

where h (y) = suphx, yi
x

f (x)

Overview• Dual Problem• W1• Semi-discrete Problem• Optimal Quantization

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Alternate Minimization and W1(f , g) 7 ! (f , f c ) 7 ! (f cc , f c ) 7 ! (f cc , f ccc )<latexit sha1_base64="2SxZEYL5yk2jBr7j7X7cufh6P9A=">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</latexit><latexit
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Alternate Minimization and W1(f , g) 7 ! (f , f c ) 7 ! (f cc , f c ) 7 ! (f cc , f ccc ) = (f cc , f c )<latexit sha1_base64="5f50aKJRcgV4L/4RPQpP5LnOUlY=">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Proposition: f ccc = f c .

Alternate Minimization and W1(f , g) ! (f , f c ) ! (f cc , f c ) ! (f cc , f ccc ) = (f cc , f c )Proposition: f ccc = f c .

W1 case: c(x, y) = d(x, y)<latexit sha1_base64="3pLz5EsNOhKHqOGj1+2u5vKJpHg=">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</latexit><latexit
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Proposition:<latexit sha1_base64="/g1rrPKuJaU0YV3ASj/5Gw4Vt2Y=">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</latexit><latexit
sha1_base64="/g1rrPKuJaU0YV3ASj/5Gw4Vt2Y=">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</latexit>

9f s.t. g = f c<latexit sha1_base64="c75+5GfYei81PRvPg7r39TJHnEs=">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
sha1_base64="c75+5GfYei81PRvPg7r39TJHnEs=">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</latexit>()f cc =

fc

g is 1-Lipschitz.

Alternate Minimization and W1(f , g) ! (f , f c ) ! (f cc , f c ) ! (f cc , f ccc ) = (f cc , f c )Proposition: f ccc = f c .

W1 case: c(x, y) = d(x, y)<latexit sha1_base64="3pLz5EsNOhKHqOGj1+2u5vKJpHg=">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</latexit><latexit
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W1 (↵, ) = max {hf , ↵i + hg, i : f
f ,g= max hf , ↵i + hf c , i
ff cc =

Proposition:fc

g is 1-Lipschitz.

g 6 c}

Alternate Minimization and W1(f , g) ! (f , f c ) ! (f cc , f c ) ! (f cc , f ccc ) = (f cc , f c )Proposition: f ccc = f c .

W1 case: c(x, y) = d(x, y)<latexit sha1_base64="3pLz5EsNOhKHqOGj1+2u5vKJpHg=">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</latexit><latexit
sha1_base64="3pLz5EsNOhKHqOGj1+2u5vKJpHg=">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</latexit>f cc =

Proposition:<latexit sha1_base64="/g1rrPKuJaU0YV3ASj/5Gw4Vt2Y=">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</latexit><latexit
sha1_base64="/g1rrPKuJaU0YV3ASj/5Gw4Vt2Y=">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</latexit>

9f s.t. g = f c<latexit sha1_base64="c75+5GfYei81PRvPg7r39TJHnEs=">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
sha1_base64="c75+5GfYei81PRvPg7r39TJHnEs=">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</latexit>()

g is 1-Lipschitz.

g 6 c}

W1 (↵, ) = max {hf , ↵i + hg, i : f
f ,g= max hf , ↵i + hf c , i = max hf c ,
f

fc

f

↵i

Alternate Minimization and W1(f , g) ! (f , f c ) ! (f cc , f c ) ! (f cc , f ccc ) = (f cc , f c )Proposition: f ccc = f c .

W1 case: c(x, y) = d(x, y)<latexit sha1_base64="3pLz5EsNOhKHqOGj1+2u5vKJpHg=">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</latexit><latexit
sha1_base64="3pLz5EsNOhKHqOGj1+2u5vKJpHg=">AABFmnictVzbchu5EYU3t7Vz8yaPycNkJafslKKVlE0ltVtbtdbFslayLZuU7LVpq4bkiKZNcWjeLJurH8jX5DV5yXfkD5Kn/EL6AgwwJGYaoziakogBcbobPUCju4FRc9DrjsZra/+88tH3vv+DH/7o46vXfvyTn/7s59c/+cXxKJ0MW8lRK+2lwyfNeJT0uv3kaNwd95Ing2ESnzV7yePm6y38/vE0GY66ab8+fjdInp/FnX73tNuKx1B1cn15+fHJ+nJjnJwDeBa1gNQXF9Fy6+b5yrtbX7XpY/nk+tLa6hr9RIuFdV1YUvrnMP3k089UQ7VVqlpqos5UovpqDOWeitUIrmdqXa2pAdQ9VzOoG0KpS98n6kJdA+wEWiXQIoba1/C3A3fPdG0f7pHmiNAt4NKD3yEgI3UDMCm0G0IZuUX0/YQoY20R7RnRRNnewWdT0zqD2rF6CbUSzrQMxWFfxupU/Zn60IU+DagGe9fSVCakFZQ8cno1BgoDqMNyG74fQrlFSKPniDAj6jvqNqbv/0UtsRbvW7rtRP2bpLwBV6RquvdpRiFWU6If0dOcwHcsTw84d4BCovuIpbek6zPqfR/az6D+PlwXVDI6acI1o9qLUuQWXD7klojchcuH3BWRB3D5kAci8hAuH/JQIxE7JJ378TW4fPiayPkhXD7kQxH5CC4f8pGIPIbLhzwWkU/h8iGfisg7cPmQd0TkPlw+5L6IrMPlQ9ZF5BFcPuSRiNyBy4fc0cjimTqEKyU6XWFW3oZyngdaih7U3Bbl2yTr6MNuBszpVgFWntXb8OnHbgfoNCnA7gSMu9MCrDzydsFG+rGyLbpLq4kPe1fE7sEI8GP3ROw36lUB9puAmfa6ACvPtQNo58fK1vce3Pmx90TsfSj5sfIa9QBq/NgHASvGoAB7KGIfqjcF2BCrPyzAyna/BnbFj5XXqTq092NDrOmkACvb02PwYPxYebV6DLV+7GMR+0SdExbL1kedkReAlmyR5hOiWU71W7D7Pqrol03Iyi3S/TaA7lP1vqCnZmW+RutOh+RPYJ6XUYv1XC7uO7YYAJdYlKyXrVU98rWbUB8LmE6G6RDmTETsZojdQMRBhjgIlmuU2eUR+c8yl1qGqAUimnqtKxsh2GYcoPt2tm5iqacx5au0QWzPIcq8XxwJpp9T8mRMjYQcZ6sklpIACdNsrcBSosdKuZU3iAc5BM+IlzRfVigyw2gNNVVG7WXmTzAyovsyxFuKFE0vDQ8ZN9YWKI86F1FND6opot55UO9E1MSDmoioqQc1FVHWKri4RsAIsPrHZzGjOx4B7I8XXxF4ILdhhbsL8zeC8XMIHucjqnkAnzWK86WrTDLMHOCajBmV5zn7PYTSTC1BvY1AtymW79EMS0AybvlA5xPwjm0Fzzm20ReZ1xBl2ZlwOl2Sp5PRQc80ovlUjc4+1VyQJ8mlavi72bw3pWr4HdI44k2pGn6spR9fQva6xtYvga3BbBpo7dtyVRqc62EapnyNVmS0uPhUz/SYQXrnFenv6Sezd4nnskUl1o8tV6Mxcvo3yvWvCg2r55Gj52pU0OdiD9uUoso96esY25arypDSKtrXcti7qk8G27T1kzHlajQOwRvbovh+5pSrjt5B1htbrkbjWHGO9YKiBlOuRqND96wPW65GAzM7sc4p2HJVy44a4Djdlqta9T5lnDHfxGOea6xXNCQ/aaKpdck/KM8MuRHB4jqG+aEXykQW5ZSsd1tMp5mtZeUSGX8hAas2rigH+hcTxwfL05ipDdH/ZhnGufV9kY5d41HzB6DFCGY/7zdI+XmMCUz+A613DyiuixFZvmcGtyHicJSczqEaunYseouW76sswrF1J1QrxWy2t1aPDbLXIxp7A/IJD0izkh4OCp9wEUVJQwc5Dcn0qujuvZ6vee2vibjBHGKQjbQW7T7xrl15DOvTes3R8Q29ozSGi/eX7PjFzPaptjYY86Rki1CWMp5uO5OzcutwXV1RNp/O30X0RNFeTclqdGn3ayRGoSYzzd74jO4t7SPa/0MeTKMFzzHSVAaKd+gwY4+5+4gsqmtvJd6oL5MN5PKIrK6xx+XojoPueNDVY5wtWDHuQ6kOMcMR3NUDopxrma5S0vhQ/T7biU3pCZZH9L2chTQ02N4kOQtZFmW/zFF5C2gcDRylh9OYp2PwjQVKctTvk8fGrnnLf4N2ic1eekxjvHg0F2di2sR1g7hGNGt4B5nv5jmwBDPvNxvkv5b3EvlV4Yg2VOL6wuHMeunT6YKEItgBecY9mm3S7Mi3dvNT898YTofK7NPjznlKFjIi+xfB+pTSmIzo1z2nYHbr2SL0yEaG2J1u5t34fJ2uOMasH9dVfILCjreEbNmE+Bu67uwa0VjkiIHXgYu5sW10ckC+YEJch9q627ldvvog0p7JcEcJU7Rj5Sbxv0V/za8ZJ0sLIwI1jE9gpG2d73mkFLOgjmJa5cttkGnrSrmcyfBCS23XPyvTck6ybYq4UB5crdvAuUX3zAtHyZDkHi204XW0LJuLlAdzesTenlIUz3a/o1dglHuFVsklmnMNGiUdGAXjLIowbaUs8jzfcl556mG0R/8X6lbXea0hxUjZDC5rSMr9JxStuVL2YFTz+H1Ns8mv9eFcq3I+fRqLZ85c/g5qfwN/jdzmPoxOM2cVNmkMMAV7ZzXCNdFCizBemzleZmQaWvbe8rNj0rRyay4TX7N1szH2tDKVQxo15zprYcqXofHKofEqUId12te0WjT1xhKdiLFFXe+MhvKrwq1egfJEpCx7ZAbVDZDSjaXCqLZFqnKMb1DvRVprIq0YZqu7G+DO+RCkf67Pz+7vstU9UnfIt2mRB8bxS5tmaZd8LlNbHqkxBeT8ubav7uxvUA1yb5IFRcp8ZhRnDO86tei6yCT9rV7ZUrLz1iKYM1JvdRtjYxtU/sMC8ozmxIjmpUF8Ti0SLb8rRzRnkVYdnyOizH9MPhX7HeUxs9vaPpMo50/YeJNnleXFkUKf9C9l3vYWotc9J36NKCacaO+6CbSqP2GkwBiTSfB7liN6QrjK8U4Ce7RNsp+Ldop38fqORKsk9Ux9FWBjOOq1Y90dW6bHpm+/g5aodfvUfS1kfr1gjhK/y+zoxbSqnWkfdTZ3fzlasV7l8vdlepjM8bX6mFAbN7KwUV4e01BfBnNhiapxYUwIl2q9qCJ/NcmryMy7U6GUTWtDOZ9pYBvzkuIl6cwpInze3U2vN3dL6EdzgV6TsC41rpEoYTYu1fkB19JiVurqwjrEtVdLV6OesxIVrRSGurtaWPvNFjIh69dTUs6GW7uyN3JRipyFYQotxaeHi+JDl+aX6gv6GylfdGg4huQOa+Df3lZbaucDnIZ4o8uc0YyoBm1Bey72jnU/8y3KdfTGoe7SD+EQzqMLupak79JKWlV2pixL7lIPp/+WrMBQJaL0tmX1Prhc5J4scqrSny5ZNrk3XWXe+6naF8MhpCd5LuF8eF9D6sWpMu9PVeuDoS73IM+hCg9zjiHsmdvW1Xm5nMr1tcgllAevAmbHxeBw5684VrHtQizU0HkiH54DWofTEupmtfhf+2H4WE7VeYVyG9F7ba8Cnjq3S3RGFv3h6nPGcgsZzcUcw3mmWe+st+Tnx35fVOlJpU5vPjx99EftGDC8ZorzoLJ0jHdHkZU3lAruC/hkSNV/1D+uyG8+vMloFMlRhZLZpyimZlrI1Mzbnb7eme9CZLJ0imTKU7NxRI1OxG6pPXUHfrcyD7Dq6VB+b5M/Eet/V7cNtadkPUwWnTMHDapLKPthd9HadG/PzxZJjGd5+WxvHWpwL/yAavGc731qj2d967m+Fb+twnP9nkpVOxeRzO/u2XnVhB7kd944B2TeKY7oLD1nsfjk2VnA3iKfn+IIyfcGx7ykMxUrm3cpo9z0UnbfT1ik3XT6Vp5NLJPanNwuH9EdQb4ZtbCabtF8G+jzBrj7gW8JxFmOK1KfUV2s1yr0GyQdHXpkeEb5iSaNhjWIF/+oVuBzRZf9ejtckHREIyIv0bnzXfl5tAvvKLXvb96gbJzJF051u5RyC3YPs/wJbhdy4adXju+U4DuOlDV6Wq8p+h+q8gzmpITmRMvk7ir3lcm+sh4wto6z8VEezU9LeE0D+r9fiN53JN0FWZqU849oV3FI9HpaNzskPZ/uLM8e3y2R1rynyjTt+U47DsxJzfKdiZ4ed8W2iE9jShmjpIDO/JtR8i6LLI8sTYgsEpWJKMkk4O3oaYAs0wA6p4I0pyKFjiiJnsEn15fW5///yGLheGN1fW11/eHG0teb+n+TfKx+pT5VN2Gt/JP6GkbooToCTn9Rf1V/U3/f+fXO5s7ezj43/eiKxvxS5X526v8FRLZsBQ==</latexit>f cc =

Proposition:<latexit sha1_base64="/g1rrPKuJaU0YV3ASj/5Gw4Vt2Y=">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</latexit><latexit
sha1_base64="/g1rrPKuJaU0YV3ASj/5Gw4Vt2Y=">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</latexit>

9f s.t. g = f c<latexit sha1_base64="c75+5GfYei81PRvPg7r39TJHnEs=">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
sha1_base64="c75+5GfYei81PRvPg7r39TJHnEs=">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</latexit>()

g is 1-Lipschitz.

g 6 c}

W1 (↵, ) = max {hf , ↵i + hg, i : f
f ,g= max hf , ↵i + hf c , i = max hf c ,ff= max hf , ↵Lip(f )61

i = ||↵

fc

||W1

↵i

Euclidean and Graphs W1
Case d(x, y) = ||xy||:<latexit sha1_base64="jFQHQr6/IwFJ5kA2P7JfL9IvbFs=">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</latexit><latexit
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max||rf (x)||1 61

hf , ↵i=min

div(u)=↵Z||u(x)||dx

Euclidean and Graphs W1
Case d(x, y) = ||xy||:<latexit sha1_base64="jFQHQr6/IwFJ5kA2P7JfL9IvbFs=">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</latexit><latexit
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max||rf (x)||1 61

hf , ↵

i=b||W1 = max|ri,j f|61hf, adiv(u)=↵||u(x)||dx

ri,j f = wi,j1 (fi fj )P
mini,j wi,j |ui,j |def.

On graph: d(x, y) = Geodw (x, y)
||a

min

Z

bi =

div(u)=a b

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f
small

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Dual Norms
||↵

def.

||B = supB = {f : ||f ||1 6 1}

f 2B

R

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B = {f : ||rf ||1 6 1}

f (x)d(↵)(x)

Total variation
W1

B = {f : ||rf ||1 + ||f ||1 6 1}

Flat norm (unbalanced)

B = f : ||r(k) f ||2 6 1

Sobolev Ḣ k

B = {neural networks}?? (Non-convex)<latexit sha1_base64="0GX0Nk7SG8gAydNAoKuOXKqCjbQ=">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</latexit><latexit
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||↵

def.

||B = supB = {f : ||f ||1 6 1}

f 2B

R

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B = {f : ||rf ||1 6 1}

f (x)d(↵)(x)

Total variation
W1

B = {f : ||rf ||1 + ||f ||1 6 1}

Flat norm (unbalanced)

B = f : ||r(k) f ||2 6 1

Sobolev Ḣ k

B = {neural networks}?? (Non-convex)<latexit sha1_base64="0GX0Nk7SG8gAydNAoKuOXKqCjbQ=">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</latexit><latexit
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If C(X ) ⇢ Span(B), then || · ||B is stronger than weak convergence.

If B ⇢ C(X ) is compact, then || · ||B is weaker than weak convergence.

|| · ||B metrizes weak convergence.

Wasserstein GANs
Generative model fitting:<latexit sha1_base64="IQy9YZb9VPHZFEiCSHnt16Y7tG4=">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z

g✓

x

Wasserstein GANs
Generative model fitting: min ||↵✓✓
RP
= min sup f d↵✓
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f 2B

||B

⇣
g✓

↵✓
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z

g✓

x

Wasserstein GANs||B
Generative model fitting: min ||↵✓✓
RP
= min sup f d↵✓
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✓

f 2B

⇣
g✓

↵✓
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z

g✓

x

Wasserstein GANs||B
Generative model fitting: min ||↵✓✓
RP
= min sup f d↵✓
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✓

g✓

f 2B

{f⇠ }⇠ ⇢ B
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2R

z

g✓

x

x

f⇠

Wasserstein GANs||B
Generative model fitting: min ||↵✓✓
RP
= min sup f d↵✓
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✓

g✓

f 2B

{f⇠ }⇠ ⇢ B
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Wasserstein GANs,

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[Arjovsky, Chintala,Bottou, 2017]:

Overview• Dual Problem• W1• Semi-discrete Problem• Optimal Quantization

Semi-discrete Problem
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L3

j=1 bj yj

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Optimal Monge map:
T : L j ! yjSemi-discrete Problem
L1

0.6
Pm
=0.4j=1 bj yj

y0.2
1

0.6

Optimal Monge map:
1.5
T : L j ! yj

0.4
0.2

1

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L4
L2

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L3

y4

0

y2
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0 y3

2

c-transform:
def.
c
) gj
-0.2 g (x) = min c(x, yj0.5
0

0 0.5

16j6m

0.5 1

1

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p = 1/21
p p==1/2c(x, y)==3/2||x||x y||2pp=2
1 y|| c(x,py)p===3/2

Figure 5.1: Top: Figure
examples5.1:of semi-discretec̄-transforms
gc̄ in 1-D,
for ground
Top: examplesof semi-discretec̄-transformsgc̄
p
cost c(x, y) = |x ≠ y|costforc(x,varyingp (seeThep red
are atlocationsy) = |x≠ y|pcolorbar).
for varying(seepoints
colorbar).
The
red poic̄(yj , ≠gj )j . Bottom: (yexamplessemi-discretec̄-transforms
g in 2-D,
for ground
Bottom:examplesof semi-discretec̄-transformsgc̄
j , ≠gj )j . ofpcost c(x, y) = Îx ≠ yÎ
forvaryingp.≠Thelocationsyj œare
R2 at
, and
cost
c(x,
y) = Îx
yÎpredfor points
varyingare
p. at
The
red points
loc
their size is proportional
regions delimited
byregionsbold black
curvesbyare
theb
their tosizegjis. The
proportional
to gj . The
delimitedbold
Laguerre cells (Lg (yjLaguerre))j associatedtog (ythese(yj )jto. these points (yj )j .
cells (Lassociatedj ))j points

Semi-discrete Problem
0.6Pm=0.4j=1 bj yj

L1

y0.2
1

0.6

Optimal Monge map:
1.5
T : L j ! yj

0.4
0.2

1

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0.5 1

1

<latexit sha1_base64="yvH7FkZz/xWKXSCwMkfXV0U26WU=">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Semi-discrete OT:
def.

max E(g) =g2Rm

c-transform:
def.
c
) gj
-0.2 g (x) = min c(x, yj0.5
0

y2
-0.2

↵

L2

y4

2

R

c

g (x)(x)d↵(x) +p = 1/2

P

j gj bj1
p p==1/2c(x, y)==3/2||x||x y||2pp=2
1 y|| c(x,py)p===3/2

Figure 5.1: Top: Figure
examples5.1:of semi-discretec̄-transforms
gc̄ in 1-D,
for ground
Top: examplesof semi-discretec̄-transformsgc̄
p
cost c(x, y) = |x ≠ y|costforc(x,varyingp (seeThep red
are atlocationsy) = |x≠ y|pcolorbar).
for varying(seepoints
colorbar).
The
red poic̄(yj , ≠gj )j . Bottom: (yexamplessemi-discretec̄-transforms
g in 2-D,
for ground
Bottom:examplesof semi-discretec̄-transformsgc̄
j , ≠gj )j . ofpcost c(x, y) = Îx ≠ yÎ
forvaryingp.≠Thelocationsyj œare
R2 at
, and
cost
c(x,
y) = Îx
yÎpredfor points
varyingare
p. at
The
red points
loc
their size is proportional
regions delimited
byregionsbold black
curvesbyare
theb
their tosizegjis. The
proportional
to gj . The
delimitedbold
Laguerre cells (Lg (yjLaguerre))j associatedtog (ythese(yj )jto. these points (yj )j .
cells (Lassociatedj ))j points

Semi-discrete Problem
0.6Pm=0.4j=1 bj yj

L1

y0.2
1

0.6

Optimal Monge map:
1.5
T : L j ! yj

0.4
0.2

1

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0 y3

L3

def.

max E(g) =R

Laguerre cells:def.

Lj (g) =

0 0.5

16j6m

0.5 1

1

<latexit sha1_base64="yvH7FkZz/xWKXSCwMkfXV0U26WU=">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Semi-discrete OT:
g2Rm

c-transform:
def.
c
) gj
-0.2 g (x) = min c(x, yj0.5
0

y2
-0.2

↵

L2

y4

2

c

g (x)(x)d↵(x) +p = 1/2

P

j gj bj1
p p==1/2c(x, y)==3/2||x||x y||2pp=2
1 y|| c(x,py)p===3/2

Figure 5.1: Top: Figure
examples5.1:of semi-discretec̄-transforms
gc̄ in 1-D,
for ground
Top: examplesof semi-discretec̄-transforms
gc̄
pp
cost c(x, y) =2 |x ≠ y|costforc(x,varyingp (seeThep red
are atlocationsy) = |x≠
for varying(seepoints
colorbar).
The
red poi
2 y| colorbar).
c̄
x ; 8`,(y||x
y
||
g
6
||x
y
||
g
`
semi-discrete
g in 2-D,
for ground
Bottom:examplesof semi-discretec̄-transformsgc̄
jexamples` c̄-transforms
j , ≠gj )j .j Bottom: (yj , ≠gj )j . ofpcost c(x, y) = Îx ≠ yÎ
forvaryingp.≠Thelocationsyj œare
R2 at
, and
cost
c(x,
y) = Îx
yÎpredfor points
varyingare
p. at
The
red points
loc
their size is proportional
regions delimited
byregionsbold black
curvesbyare
theb
their tosizegjis. The
proportional
to gj . The
delimitedbold
Laguerre cells (Lg (yjLaguerre))j associatedtog (ythese(yj )jto. these points (yj )j .
cells (Lassociatedj ))j points

Semi-discrete Problem
0.6Pm=0.4j=1 bj yj

L1

y0.2
1

0.6

Optimal Monge map:
1.5
T : L j ! yj

0.4
0.2

1

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0 y3

L3

0 0.5

def.

max E(g) =Rcg (x)(x)d↵(x) +p = 1/2

P

def.

0.5 1

1

j gj bjc(x, y)==3/2||x||x y||2pp=2
1 y|| c(x,py)p===3/2

1p p==1/2

Laguerre cells:
Lj (g) =16j6m<latexit sha1_base64="yvH7FkZz/xWKXSCwMkfXV0U26WU=">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
sha1_base64="yvH7FkZz/xWKXSCwMkfXV0U26WU=">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</latexit>

Semi-discrete OT:
g2Rm

c-transform:
def.
c
) gj
-0.2 g (x) = min c(x, yj0.5
0

y2
-0.2

↵

L2

y4

0

2

Figure 5.1: Top: Figure
examples5.1:of semi-discretec̄-transforms
gc̄ in 1-D,
for ground
Top: examplesof semi-discretec̄-transforms
gc̄
pp
cost c(x, y) =2 |x ≠ y|costforc(x,varyingp (seeThep red
are atlocationsy) = |x≠
for varying(seepoints
colorbar).
The
red poi
2 y| colorbar).
c̄
x ; 8`,(y||x
y
||
g
6
||x
y
||
g
`
semi-discrete
g in 2-D,
for ground
Bottom:examplesof semi-discretec̄-transformsgc̄
jexamples` c̄-transforms
j , ≠gj )j .j Bottom: (yj , ≠gj )j . ofpcost c(x, y) = Îx ≠ yÎ
forvaryingp.≠Thelocationsyj œare
R2 at
, and
cost
c(x,
y) = Îx
yÎpredfor points
varyingare
p. at
The
red points
loc
their size is proportional
regions delimited
byregionsbold black
curvesbyare
theb
their tosizegjis. The
proportional
to gj . The
delimitedbold
Laguerre cells (Lg (yjLaguerre))j associatedtog (ythese(yj )jto. these points (yj )j .
cells (Lassociatedj ))j points

Proposition:rE(g)j =R

Lj (g)d↵bj

Semi-discrete Optimization
Gradient descent:R(1 ⌧ )g + ⌧ Lj (g) d↵<latexit sha1_base64="CsN7aT6sxVORH6Ub5GXs8/VrFDQ=">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</latexit><latexit
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Franz
Aurenhammer

VladimirOlikerQuentinMérigot

Bruno
Levy

Semi-discrete Optimization
Gradient descent:R(1 ⌧ )g + ⌧ Lj (g) d↵<latexit sha1_base64="CsN7aT6sxVORH6Ub5GXs8/VrFDQ=">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</latexit><latexit
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Franz
Aurenhammer

VladimirOlikerQuentinMérigot

Bruno
Levy

Semi-discrete Optimization

Semi-discrete Optimization

Overview• Dual Problem• W1• Semi-discrete Problem• Optimal Quantization

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Qm (↵) = min Wp (↵,<latexit sha1_base64="GOtAFge1SPYVu3blkpa6HEvWdms=">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
sha1_base64="GOtAFge1SPYVu3blkpa6HEvWdms=">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</latexit>

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Qm (↵) = min Wp (↵,b, Yconvex<latexit sha1_base64="HL+CWgVEA+ho9MS6+Yq/V1Xlz0w=">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Optimal Quantization↵=<latexit sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit><latexit
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quantizationPj b j yjdef.

Qm (↵) = min Wp (↵,b, Yconvex<latexit sha1_base64="HL+CWgVEA+ho9MS6+Yq/V1Xlz0w=">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b j yj )non-convex<latexit 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sha1_base64="Pv8Yn3B5GzNCGZ4ksVEmu3Zl+xk=">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</latexit>

In general: Qm (↵) ⇠ 1/m1/d .

Optimal Quantization↵=<latexit sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit><latexit
sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit>Pj b j yjdef.

Qm (↵) = min Wp (↵,b, Yj b j yj )non-convex

convexquantizationP<latexit sha1_base64="HL+CWgVEA+ho9MS6+Yq/V1Xlz0w=">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sha1_base64="Pv8Yn3B5GzNCGZ4ksVEmu3Zl+xk=">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</latexit>

In general: Qm (↵) ⇠ 1/m1/d .
Qm (↵) = min max
m
Y

g2R

R

c

g (x)(x)d↵(x) + minbP

j gj b j

Optimal Quantization↵=<latexit sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit><latexit
sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit>Pj b j yjdef.

Qm (↵) = min Wp (↵,b, Yj b j yj )non-convex

convexquantizationP<latexit sha1_base64="HL+CWgVEA+ho9MS6+Yq/V1Xlz0w=">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In general: Qm (↵) ⇠ 1/m1/d .
Qm (↵) = min max
m
Y

g2R

R

c

g (x)(x)d↵(x) + min
b

def.

P

j gj b j

Voronoi cells: Lj (0) = Vj (Y ) = {x ; 8`, ||x=)

yj || 6 ||xg=0y` ||}

Optimal Quantization↵=<latexit sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit><latexit
sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">AABB7HictVzdchu3FYbTv9j9c9rL3mwru+NkHFdSM9POZDITWZJlxbRNh5TsJLQ1XHJFU15yaS4p/zB8h170rtPbPkJv09foG7RXfYWeH2CBJbELQHWNkYQF8Z1zcBY4OOcAdDxJh/lsc/Ofl9773vd/8MMfvX/5yo9/8tOf/fzqB784zrP5tJcc9bI0mz6Ju3mSDsfJ0Ww4S5Mnk2nSHcVp8jh+sYufPz5PpvkwG7dnbybJ01F3MB6eDnvdGTSdXP3oWoeILOJ0niyjTpxEn0WdfD46OYMH/NVPThZvTs6W106ubmze2qR/0XplS1Y2hPzXzD6IjkRH9EUmemIuRiIRYzGDeiq6IofyjdgSm2ICbU/FAtqmUBvS54lYiiuAnUOvBHp0ofUF/B7A0zeydQzPSDMndA+4pPAzBWQkrgMmg35TqCO3iD6fE2VsraK9IJoo2xv4G0taI2idiefQ6sKpnr44HMtMnIo/0hiGMKYJteDoepLKnLSCkkfGqGZAYQJtWO/D51Oo9wip9BwRJqexo2679Pm/qCe24nNP9p2Lf5OU16FEoiVHnxUUuuKc6Ef0NufwGcuTAucBUEjkGLH2inQ9otGPof8C2h9AWVJN6SSGsqDWZS1yF4oNuetEHkCxIQ+cyAYUG7LhRDah2JBNiUTslHRux7eg2PAtJ+dHUGzIR07kl1BsyC+dyGMoNuSxE/k1FBvyayfyDhQb8o4TeQ+KDXnPiWxDsSHbTuQRFBvyyInch2JD7ktk9UqdQsmIztCxKnegXuaBliKFlh2nfLfJOtqwtz3WdK8C617Ve/DXjt3z0GlSgd33mHenFVj3zDsAG2nHum3RXdpNbNi7TuwhzAA79tCJ/UKcVWC/8FhpLyqw7rXWgH52rNv63ocnO/a+E/sAanase496CC127EOPHWNSgW06sY/Eywqsj9WfVmDddr8FdsWOde9Tbehvx/pY03kF1m1Pj8GDsWPdu9VjaLVjHzuxT8TrCuwTJ/YrsO527FceO+zbCqzaY6/QDjIgfySBFVtHrVusSqxNgFrXwT8t9paUfOMY2l2YQYEZEGbkRBwUiANPRKNANLzlygs7mpO/6+bSKhAtT0Rc7E1Ymzn794v+WEs9EHsFYm8FUeeR4rtWYzkn70K1uJCzYufCms+YssJ+Yy2R86He8irEwxKC5/Zzmvk3KVrCCAo1VUftebHHMzKi5zrEK4re1CgVDzduVlgFE/XaiYotqNiJemNBvXGi5hbU3Ik6t6DOnSi98k1cx2MGaP3ju1jQE88A9pGrSwRewQ7sOndhjUYwf5rgBX5JLQ/hb4tib1epkwyjedwnMcvxtGSJp1BbiA1o11HhHsXXKa2wBCTjng9ljI9PmNtYyDXHVnhZ7ORRkTHxpzMkeQYFHfQWI1pPYXTuUcuSvDuuheHvFute1cLw+6TxJXnxXAvDz6T0swvI3pbY9gWwLVhNE6l9XQ+lwfkXpqHqV2jXRYuLb3Uk5wzSex1I/1C+mcMLvJddqrF+dD2MRm6MLy+NL4SG1nNu6DmMCnpP7PWqWhQ8krGMe3U9VIaMdtGxlEM/hb4Z7NOXb0bVw2g0wePapZh7YdRDZ++kGI2uh9E4Fpz3XJInr+phNAb0zPrQ9TAamG3pyjhf10MtO2qAY2ddD7XqY8oCYw6I5zy3aK9oSn7SXFIbkn9Qn60xff71fQxzNs+KGKGekvZtq+nExV5WL5HyFxKwarNAOdC/mBs+WJnGQmw74yuWYVba39fp6D0eNd8ALUaw+vkMwJUzT0FClZNA650CxS1n1FUemcJtO3E4S05XUB3ZOnN6i5ovZ43KbSfU6orL9Gi1Hjtkr3OaexPyCRukWZceGpVvuIqiS0ONkobc9EJ091au17L2N524yQpiUsy0Hp0I8UlafZxq03rL0PF1ecozg8JnPnr+Yrb5VFobjHkyskUoSx1Ps5/KI5ltuK/eFDrHzZ9F9EbRXp2T1RjSiVTujEJVtpi98QU9a9pHdCaHPJhGD95jJKlMBJ+aYRYd8+kRWVTT3rp4o75Uho7rOVldZY/r0QMDPbCgw2OcXdgxHkCtDTHDETy1PaKcK4WuMtL4VHxcnI5m9AbrI/q0ZCEVDbY3SclC1kXZz0tUXgEaZwNH6f40VukofGeNkjvqt8mjY9ey5b9OJ7fqfLtLc7x6NldnYvrEdZu4RrRq+FSXn1Y5sAQL6yfb5L/WjxL5hXBEG+ri+szgzHoZ04l/QhHshDzjlFaba3WUe5v5qdVPFKemUGfneJqdkYWMyP5FsD9lNCcj+jHvDqgTdLYIKdlIH7szLLwbm68zdM4x7ccNBd9q0PMtIVs2J/6Krrm6cpqLHDHwPrBcmdtKJw3yBRPiOpXWXa/t+t0HkfqehDlLmKKeKzeI/4f0W/2oebKxNiNQw/gGcmnrbO8jo5gFddSlXb7eBqm+ppTXChmeSan1/qdlulaSbI8iLpQHd+s+cO7RM/PCWTIlufO1PryP1mVzkfJkRY842lOK4tnuD+QOjHLfpF1yg9Zch2bJAGbBrIgiVF9XFnmVbz2vMnU/2vn/hbrWdVlrSDESOoPLGnLl9xOK1kwpU5jVPH9f0Gqya3260quez5jm4shYy99C66/ht5JbPfvRiUtW4TbNAaagn7RGuCVa6+HH63aJl5qZipZ+1vz0nFS9zJaLxNds3XSMfR5MpUmz5rXMWqj6RWicGTTOPHXYprNGrUXVrizRiTO2aMvTSl9+IdzaAZTnTspuj0yhhh5SmrGUH9W+k6o7xleot05am05aXVit5mmAueZ9kPa1vrq6vy1290jcId+mRx4Yxy99WqVD8rlUa32kxhSQ8yfSvpqrv0MtyD0mC4qU+R4nrhg+depRWRaS/lbubBnZeW0R1L2lV7KPsrEdqv9+DTmiNZHTulSIT6hHIuU35YhWLNItw+eIKPPfJZ+K/Y76mNnsrd9JVPIndLzJq0rz4khhTPp3Zd4O16LXQyN+jSgmnEvvOgZa4W8YKTBGZRLsnmVObwh3OT5JYI82Jvu5bqf4FG9sSHSLpF6IzzxsDEe9eq6bc0uNWI3tI+iJWtdv3dbDzS/15ujid5ETvS7taiPpoy5Wni9Gqyt3ufJznR7mK3y1PubUx4wsdJRXxnTEp95cWKIwLozx4RI2ihD5wyQPkZlPp3wpq96KcjnTwDbmOcVLrnugiLB5dzes3tyHjnHEa/RiwprUuMVFCbNxmcwPmJYWs1KX1/Yhbr1cuxulxk5UtVMo6uZuoe03W8iErF8qXDkb7m3K3ilFKe4sDFPoCb7RWxUfmjQ/hYK/I2GLDhVHn9xhC/zbHbEr9t/BbYiXss4ZzYha0Bb0V2LvrhxnuUe9jl4a1E36Phz8eQxB1y7ph7SThsrOlN2Sm9T96b8iKzAViVN63TN8DCYX90jWOYWMZ0iWzT2aoVDfxQkdi+LgM5IyF38+fK7hGsWpUN9pChuDou4eQZlDCA91j8Hvneve4bxMTvX6Wufiy4N3AXXionB48lcdq+h+PhZqaryRd88BrcNpDXW1W/yv41B8NKdwXr7ccvqu2ZnHW+d+iczIoj8cvmY0N5/ZXM3Rn2dWjE57S3Z+7PdFQW8qM0bz7umjP6rngOK1EJwHdUvHeHMWaXl9qeC5gE2GTPxHfHfJ/W2ElwWNKjlCKKlzimpqqoebmvrGpW106jMfmTSdKpnK1HQc0aIbsbviUNyBn93CAwy9HcrfpeS/iLV/f7YPradkPVQWnTMHHWpLKPuhT9H69Kzvz1ZJjHd5+W5vG1rwLLxBrXjP9wH1x7u+7dLYqr9Bwmv9vshEvxSRrJ7u6XUVwwjKJ2+cA1Lf843oLj1nsfjm2cjjbFHdn1qVaEGfuG8WxJX42JCyR3N1Is/q8eQAb9h3i/xQJH5HbV1p53HPdXFuVnJurnDOSTtlDq+Nz+rvZlVx2TW49Ivc2bnsl1Gcrc/z6nOje5Vc+A56PX5Qgx8YUrZI+y8oEp6K+mzevIbmXMpknrCOhcpEsh4wzuwW77s+sj2v4XXuMf57leh7hqQHIEtM+e+ITtimRC+Vutkn6fmmY30m9W6NtPJ7lCdXN7ZW/y+D9crx9q2tzVtbj7Y3Pr8t/5+D98WvxG/EDVjjfxCfA7WmOAIOfxJ/F9+Jf+yMd/6885edv3LX9y5JzC9F6d/O3/4LH5mxDw==</latexit>Pj b j yjdef.

Qm (↵) = min Wp (↵,b, Yj b j yj )non-convex

convexquantizationP<latexit sha1_base64="HL+CWgVEA+ho9MS6+Yq/V1Xlz0w=">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In general: Qm (↵) ⇠ 1/m1/d .
Qm (↵) = min max
m
Y

g2R

R

c

g (x)(x)d↵(x) + min
b

P

def.

j gj b j

Voronoi cells: Lj (0) = Vj (Y ) = {x ; 8`, ||xProposition:

Qm (↵) = min
Y

Z

=)

yj || 6 ||x

min c(x, yj )d↵(x)

16j6m

g=0
y` ||}

Optimal Quantization↵=<latexit sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit><latexit
sha1_base64="J/UT0rKb/egZhQXgBj4nhb0pjSg=">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</latexit>Pj b j yjdef.

Qm (↵) = min Wp (↵,b, Yj b j yj )non-convex

convexquantizationP<latexit sha1_base64="HL+CWgVEA+ho9MS6+Yq/V1Xlz0w=">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In general: Qm (↵) ⇠ 1/m1/d .
Qm (↵) = min max
m
Y

g2R

R

c

g (x)(x)d↵(x) + minbP

j gj b j

def.

Voronoi cells: Lj (0) = Vj (Y ) = {x ; 8`, ||xProposition:

Qm (↵) = minYZyj || 6 ||x

min c(x, yj )d↵(x)

16j6m

Local minimizers: yj = argmin
y

=)

Z

Vj (Y )c(x, y)d↵(x)g=0y` ||}

Lloyd’s Algorithm

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Z

y||2
Z
1
2
y|| d↵(x) =xd↵(x)|Vj | Vi

Cost c(x, y) = ||x<latexit sha1_base64="IV+9JenfsXWHPn/R5reZROyRHdk=">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</latexit><latexit
sha1_base64="IV+9JenfsXWHPn/R5reZROyRHdk=">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</latexit>

Vj (Y )||x

Lloyd’s Algorithm

argminy

Lloyd algorithm:

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Vj (Y )

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8j,

Vj

8j,

yj

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V1
y1

x3

{x ; 8` 6= j, ||x yj || 6 ||xZ
1
xd↵(x)|Vj | Vi

C3

y1

V1

y||2
Z
1
2
y|| d↵(x) =xd↵(x)|Vj | Vi

Cost c(x, y) = ||x

...

V4
y4

Iter #1

Iter #2

Iter #3` ||}Lloyd’s Algorithm

Lloyd’s Algorithm