Learning about Languagewith Normalizing Flows
Graham Neubig
Language Technologies Institute, Carnegie Mellon University
Chunting Zhou

Junxian He

Di Wang, Xuezhe Ma, Daniel Spokoyny, Taylor Berg-KirkpatrickLearning about Language?

2

Learning about Language?
• Syntactic structure2

Learning about Language?
• Syntactic structure

The cat sat on

a green wall2

Learning about Language?
• Syntactic structure

The cat sat on

a green wall

Parts-of-speech: DT NN VBD IN DT JJ

NN

2

Learning about Language?
• Syntactic structure

The cat sat on

a green wall

Parts-of-speech: DT NN VBD IN DT JJ

NN

Dependency:2

Learning about Language?
• Syntactic structure

The cat sat on

a green wall

Parts-of-speech: DT NN VBD IN DT JJ

NN

Dependency:• Cross-lingual correspondences

2

Learning about Language?
• Syntactic structure

The cat sat on

a green wall

Parts-of-speech: DT NN VBD IN DT JJ

NN

Dependency:• Cross-lingual correspondencesacatgreen onsat

the wallの は 上 壁 猫 緑 座った2

Learning about Language?
• Syntactic structure

The cat sat on

a green wall

Parts-of-speech: DT NN VBD IN DT JJ

NN

Dependency:• Cross-lingual correspondencesacatgreen onsat

the wallの は 上 壁 猫 緑 座った2

Supervised Approaches

X

Y
3Supervised Approaches

Supervised Learning

X

✓

Y

X

Y
4

Supervised Approaches

Supervised Learning

X

✓

Y

X

Y
5

Unsupervised ApproachesX

6Unsupervised Approaches
X• Learning language models P(X)6

Unsupervised Approaches
X• Learning language models P(X)
• Learning continuous features from language models(e.g. word2vec, skipthought, BERT)

6

Unsupervised Approaches
X• Learning language models P(X)
• Learning continuous features from language models(e.g. word2vec, skipthought, BERT)• But how do we turn this into interpretable structure?

6

Unsupervised Approaches
X• Learning language models P(X)
• Learning continuous features from language models(e.g. word2vec, skipthought, BERT)• But how do we turn this into interpretable structure?
• How do we do it while taking advantage ofcontinuous features?
6

Latent Variable Approaches

Unsupervised

X

✓

Y

?
?
?

Y

?

X
7

Latent Variable Approaches

Unsupervised

X

✓

Y

?
?
?

Y

?

X
8

Density Matching for
Bilingual Word Embedding
Chunting Zhou, Xuezhe Ma, Di Wang, Graham Neubig(NAACL 2019)9

Bilingual Word Embedding

10

Bilingual Word Embedding
pear
professor
schoolapple

caninedog

catpiazzaplanetearthbasketball10

Bilingual Word Embedding
pear
professor
schoolapple

canine

梨梨
りんご

教授
学校
ピアッツ

piazza
⽝犬

dog

cat

planetearthbasketball

猫

星
地球

バスケット

10

Bilingual Word Embedding
pear
professor
schoolapple

canine

梨梨
りんご

教授
学校
ピアッツ

piazza
⽝犬

dog

cat

planetearthbasketball

猫

星
地球

バスケット

•Map word embeddings from different
languages into a single vector space
10Bilingual Word Embedding
pear
professor
schoolapple

canine

梨梨
りんご

教授
学校
ピアッツ

piazza
⽝犬

dog

cat

planetearthbasketball

猫

星
地球

バスケット

•Map word embeddings from different
languages into a single vector space-Cross-lingual transfer
10Bilingual Word Embedding
pear
professor
schoolapple

canine

梨梨
りんご

教授
学校
ピアッツ

piazza
⽝犬

dog

cat

planetearthbasketball

猫

星
地球

バスケット

•Map word embeddings from different
languages into a single vector space-Cross-lingual transfer-Cross-lingual NLP tasks

10

Previous Work on Unsupervised BWE

Previous Work on Unsupervised BWE• Unsupervised methods of minimization some form of
distance between distributions of discrete vector sets:

Previous Work on Unsupervised BWE• Unsupervised methods of minimization some form of
distance between distributions of discrete vector sets:

Previous Work on Unsupervised BWE• Unsupervised methods of minimization some form of
distance between distributions of discrete vector sets:• No direct probabilistic interpretation, not a"typical" unsupervised generative modelDensity Mapping for Bilingual Word
Embedding (DeMa-BWE)

12

Density Mapping for Bilingual Word
Embedding (DeMa-BWE)
Japanese Space

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

12

Density Mapping for Bilingual Word
Embedding (DeMa-BWE)
Japanese Space

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

• Mapping function is learned with normalizingflow
12Normalizing Flows13

Normalizing FlowsX ⇠ P (X)<latexit sha1_base64="x6E2kl/41zLzrUTZRP09f040wUY=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkpSBT0WvXisYGsgDWWz3bRLd7NhdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHXS0zRWiHSC6VH2FNOUtoxzDDqZ8qikXE6WM0vp35j09UaSaTBzNJaSjwMGExI9hYKfBRTzOB2nX/vF+tuQ13DrRKvILUoEC7X/3qDSTJBE0M4VjrwHNTE+ZYGUY4nVZ6maYpJmM8pIGlCRZUh/n85Ck6s8oAxVLZSgyaq78nciy0nojIdgpsRnrZm4n/eUFm4uswZ0maGZqQxaI448hINPsfDZiixPCJJZgoZm9FZIQVJsamVLEheMsvr5Jus+FdNJr3l7XWTRFHGU7gFOrgwRW04A7a0AECEp7hFd4c47w4787HorXkFDPH8AfO5w+SX5Ai</latexit>

13

Normalizing FlowsX ⇠ P (X)<latexit sha1_base64="x6E2kl/41zLzrUTZRP09f040wUY=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkpSBT0WvXisYGsgDWWz3bRLd7NhdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHXS0zRWiHSC6VH2FNOUtoxzDDqZ8qikXE6WM0vp35j09UaSaTBzNJaSjwMGExI9hYKfBRTzOB2nX/vF+tuQ13DrRKvILUoEC7X/3qDSTJBE0M4VjrwHNTE+ZYGUY4nVZ6maYpJmM8pIGlCRZUh/n85Ck6s8oAxVLZSgyaq78nciy0nojIdgpsRnrZm4n/eUFm4uswZ0maGZqQxaI448hINPsfDZiixPCJJZgoZm9FZIQVJsamVLEheMsvr5Jus+FdNJr3l7XWTRFHGU7gFOrgwRW04A7a0AECEp7hFd4c47w4787HorXkFDPH8AfO5w+SX5Ai</latexit>

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13

Normalizing Flows

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Z = f✓ (X)<latexit sha1_base64="IDq8cdH7NY313/u9S3rZypQ+wFI=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSxCvZSkCnoRil48VrAf2Iay2W7apZtN2J0oJfanePGgiFd/iTf/jds2B219MPB4b4aZeX4suAbH+bZyK6tr6xv5zcLW9s7unl3cb+ooUZQ1aCQi1faJZoJL1gAOgrVjxUjoC9byR9dTv/XAlOaRvINxzLyQDCQPOCVgpJ5dvMeXOOilXRgyIJNy+6Rnl5yKMwNeJm5GSihDvWd/dfsRTUImgQqidcd1YvBSooBTwSaFbqJZTOiIDFjHUElCpr10dvoEHxulj4NImZKAZ+rviZSEWo9D33SGBIZ60ZuK/3mdBIILL+UyToBJOl8UJAJDhKc54D5XjIIYG0Ko4uZWTIdEEQomrYIJwV18eZk0qxX3tFK9PSvVrrI48ugQHaEyctE5qqEbVEcNRNEjekav6M16sl6sd+tj3pqzspkD9AfW5w/CxZMI</latexit>

X ⇠ P (X)<latexit sha1_base64="x6E2kl/41zLzrUTZRP09f040wUY=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkpSBT0WvXisYGsgDWWz3bRLd7NhdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHXS0zRWiHSC6VH2FNOUtoxzDDqZ8qikXE6WM0vp35j09UaSaTBzNJaSjwMGExI9hYKfBRTzOB2nX/vF+tuQ13DrRKvILUoEC7X/3qDSTJBE0M4VjrwHNTE+ZYGUY4nVZ6maYpJmM8pIGlCRZUh/n85Ck6s8oAxVLZSgyaq78nciy0nojIdgpsRnrZm4n/eUFm4uswZ0maGZqQxaI448hINPsfDZiixPCJJZgoZm9FZIQVJsamVLEheMsvr5Jus+FdNJr3l7XWTRFHGU7gFOrgwRW04A7a0AECEp7hFd4c47w4787HorXkFDPH8AfO5w+SX5Ai</latexit>

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13

Normalizing Flows

X = f✓ 1 (Z)<latexit sha1_base64="/fc4xHZsNC/YMH6XXAT25PjHrcA=">AAAB/3icbVDLSgNBEJyNrxhfq4IXL4NBiAfDbhT0IgS9eIxgHpisYXYymwyZfTDTK4R1D/6KFw+KePU3vPk3TpI9aGJBQ1HVTXeXGwmuwLK+jdzC4tLySn61sLa+sbllbu80VBhLyuo0FKFsuUQxwQNWBw6CtSLJiO8K1nSHV2O/+cCk4mFwC6OIOT7pB9zjlICWuuZeC19g7z45ttNu0oEBA5KW7o66ZtEqWxPgeWJnpIgy1LrmV6cX0thnAVBBlGrbVgROQiRwKlha6MSKRYQOSZ+1NQ2Iz5STTO5P8aFWetgLpa4A8ET9PZEQX6mR7+pOn8BAzXpj8T+vHYN37iQ8iGJgAZ0u8mKBIcTjMHCPS0ZBjDQhVHJ9K6YDIgkFHVlBh2DPvjxPGpWyfVKu3JwWq5dZHHm0jw5QCdnoDFXRNaqhOqLoET2jV/RmPBkvxrvxMW3NGdnMLvoD4/MHL4GU7g==</latexit>

Z = f✓ (X)<latexit sha1_base64="IDq8cdH7NY313/u9S3rZypQ+wFI=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSxCvZSkCnoRil48VrAf2Iay2W7apZtN2J0oJfanePGgiFd/iTf/jds2B219MPB4b4aZeX4suAbH+bZyK6tr6xv5zcLW9s7unl3cb+ooUZQ1aCQi1faJZoJL1gAOgrVjxUjoC9byR9dTv/XAlOaRvINxzLyQDCQPOCVgpJ5dvMeXOOilXRgyIJNy+6Rnl5yKMwNeJm5GSihDvWd/dfsRTUImgQqidcd1YvBSooBTwSaFbqJZTOiIDFjHUElCpr10dvoEHxulj4NImZKAZ+rviZSEWo9D33SGBIZ60ZuK/3mdBIILL+UyToBJOl8UJAJDhKc54D5XjIIYG0Ko4uZWTIdEEQomrYIJwV18eZk0qxX3tFK9PSvVrrI48ugQHaEyctE5qqEbVEcNRNEjekav6M16sl6sd+tj3pqzspkD9AfW5w/CxZMI</latexit>

X ⇠ P (X)<latexit sha1_base64="x6E2kl/41zLzrUTZRP09f040wUY=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkpSBT0WvXisYGsgDWWz3bRLd7NhdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHXS0zRWiHSC6VH2FNOUtoxzDDqZ8qikXE6WM0vp35j09UaSaTBzNJaSjwMGExI9hYKfBRTzOB2nX/vF+tuQ13DrRKvILUoEC7X/3qDSTJBE0M4VjrwHNTE+ZYGUY4nVZ6maYpJmM8pIGlCRZUh/n85Ck6s8oAxVLZSgyaq78nciy0nojIdgpsRnrZm4n/eUFm4uswZ0maGZqQxaI448hINPsfDZiixPCJJZgoZm9FZIQVJsamVLEheMsvr5Jus+FdNJr3l7XWTRFHGU7gFOrgwRW04A7a0AECEp7hFd4c47w4787HorXkFDPH8AfO5w+SX5Ai</latexit>

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Change of variable formula:@f✓ (x)p✓ (x) = pZ (f✓ (x)) det()@x
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13

Normalizing Flows

X = f✓ 1 (Z)<latexit sha1_base64="/fc4xHZsNC/YMH6XXAT25PjHrcA=">AAAB/3icbVDLSgNBEJyNrxhfq4IXL4NBiAfDbhT0IgS9eIxgHpisYXYymwyZfTDTK4R1D/6KFw+KePU3vPk3TpI9aGJBQ1HVTXeXGwmuwLK+jdzC4tLySn61sLa+sbllbu80VBhLyuo0FKFsuUQxwQNWBw6CtSLJiO8K1nSHV2O/+cCk4mFwC6OIOT7pB9zjlICWuuZeC19g7z45ttNu0oEBA5KW7o66ZtEqWxPgeWJnpIgy1LrmV6cX0thnAVBBlGrbVgROQiRwKlha6MSKRYQOSZ+1NQ2Iz5STTO5P8aFWetgLpa4A8ET9PZEQX6mR7+pOn8BAzXpj8T+vHYN37iQ8iGJgAZ0u8mKBIcTjMHCPS0ZBjDQhVHJ9K6YDIgkFHVlBh2DPvjxPGpWyfVKu3JwWq5dZHHm0jw5QCdnoDFXRNaqhOqLoET2jV/RmPBkvxrvxMW3NGdnMLvoD4/MHL4GU7g==</latexit>

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Z ⇠ N (0, I)<latexit sha1_base64="Z/gR+uIC+bj4zzXnm516ybItP9Q=">AAAB9XicbVBNSwMxEJ31s9avqkcvwSJUkLJbBT0WvehFKtgPbNeSTdM2NMkuSVYpS/+HFw+KePW/ePPfmLZ70NYHA4/3ZpiZF0ScaeO6387C4tLyympmLbu+sbm1ndvZrekwVoRWSchD1QiwppxJWjXMcNqIFMUi4LQeDC7Hfv2RKs1CeWeGEfUF7knWZQQbKz3co5ZmAt0U3GN0fdTO5d2iOwGaJ15K8pCi0s59tTohiQWVhnCsddNzI+MnWBlGOB1lW7GmESYD3KNNSyUWVPvJ5OoROrRKB3VDZUsaNFF/TyRYaD0Uge0U2PT1rDcW//Oaseme+wmTUWyoJNNF3ZgjE6JxBKjDFCWGDy3BRDF7KyJ9rDAxNqisDcGbfXme1EpF76RYuj3Nly/SODKwDwdQAA/OoAxXUIEqEFDwDK/w5jw5L8678zFtXXDSmT34A+fzB7AykK0=</latexit>

Change of variable formula:@f✓ (x)p✓ (x) = pZ (f✓ (x)) det()@x
<latexit sha1_base64="st0Jwp7pYeca65L8lAKjgjQIlE4=">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</latexit>

Intuitively, prevents degenerative mapping of everything to zero vector

13Normalizing Flows

X = f✓ 1 (Z)<latexit sha1_base64="/fc4xHZsNC/YMH6XXAT25PjHrcA=">AAAB/3icbVDLSgNBEJyNrxhfq4IXL4NBiAfDbhT0IgS9eIxgHpisYXYymwyZfTDTK4R1D/6KFw+KePU3vPk3TpI9aGJBQ1HVTXeXGwmuwLK+jdzC4tLySn61sLa+sbllbu80VBhLyuo0FKFsuUQxwQNWBw6CtSLJiO8K1nSHV2O/+cCk4mFwC6OIOT7pB9zjlICWuuZeC19g7z45ttNu0oEBA5KW7o66ZtEqWxPgeWJnpIgy1LrmV6cX0thnAVBBlGrbVgROQiRwKlha6MSKRYQOSZ+1NQ2Iz5STTO5P8aFWetgLpa4A8ET9PZEQX6mR7+pOn8BAzXpj8T+vHYN37iQ8iGJgAZ0u8mKBIcTjMHCPS0ZBjDQhVHJ9K6YDIgkFHVlBh2DPvjxPGpWyfVKu3JwWq5dZHHm0jw5QCdnoDFXRNaqhOqLoET2jV/RmPBkvxrvxMW3NGdnMLvoD4/MHL4GU7g==</latexit>

Z = f✓ (X)<latexit sha1_base64="IDq8cdH7NY313/u9S3rZypQ+wFI=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSxCvZSkCnoRil48VrAf2Iay2W7apZtN2J0oJfanePGgiFd/iTf/jds2B219MPB4b4aZeX4suAbH+bZyK6tr6xv5zcLW9s7unl3cb+ooUZQ1aCQi1faJZoJL1gAOgrVjxUjoC9byR9dTv/XAlOaRvINxzLyQDCQPOCVgpJ5dvMeXOOilXRgyIJNy+6Rnl5yKMwNeJm5GSihDvWd/dfsRTUImgQqidcd1YvBSooBTwSaFbqJZTOiIDFjHUElCpr10dvoEHxulj4NImZKAZ+rviZSEWo9D33SGBIZ60ZuK/3mdBIILL+UyToBJOl8UJAJDhKc54D5XjIIYG0Ko4uZWTIdEEQomrYIJwV18eZk0qxX3tFK9PSvVrrI48ugQHaEyctE5qqEbVEcNRNEjekav6M16sl6sd+tj3pqzspkD9AfW5w/CxZMI</latexit>

X ⇠ P (X)<latexit sha1_base64="x6E2kl/41zLzrUTZRP09f040wUY=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBahXkpSBT0WvXisYGsgDWWz3bRLd7NhdyOU0J/hxYMiXv013vw3btsctPXBwOO9GWbmRSln2rjut1NaW9/Y3CpvV3Z29/YPqodHXS0zRWiHSC6VH2FNOUtoxzDDqZ8qikXE6WM0vp35j09UaSaTBzNJaSjwMGExI9hYKfBRTzOB2nX/vF+tuQ13DrRKvILUoEC7X/3qDSTJBE0M4VjrwHNTE+ZYGUY4nVZ6maYpJmM8pIGlCRZUh/n85Ck6s8oAxVLZSgyaq78nciy0nojIdgpsRnrZm4n/eUFm4uswZ0maGZqQxaI448hINPsfDZiixPCJJZgoZm9FZIQVJsamVLEheMsvr5Jus+FdNJr3l7XWTRFHGU7gFOrgwRW04A7a0AECEp7hFd4c47w4787HorXkFDPH8AfO5w+SX5Ai</latexit>

Z ⇠ N (0, I)<latexit sha1_base64="Z/gR+uIC+bj4zzXnm516ybItP9Q=">AAAB9XicbVBNSwMxEJ31s9avqkcvwSJUkLJbBT0WvehFKtgPbNeSTdM2NMkuSVYpS/+HFw+KePW/ePPfmLZ70NYHA4/3ZpiZF0ScaeO6387C4tLyympmLbu+sbm1ndvZrekwVoRWSchD1QiwppxJWjXMcNqIFMUi4LQeDC7Hfv2RKs1CeWeGEfUF7knWZQQbKz3co5ZmAt0U3GN0fdTO5d2iOwGaJ15K8pCi0s59tTohiQWVhnCsddNzI+MnWBlGOB1lW7GmESYD3KNNSyUWVPvJ5OoROrRKB3VDZUsaNFF/TyRYaD0Uge0U2PT1rDcW//Oaseme+wmTUWyoJNNF3ZgjE6JxBKjDFCWGDy3BRDF7KyJ9rDAxNqisDcGbfXme1EpF76RYuj3Nly/SODKwDwdQAA/OoAxXUIEqEFDwDK/w5jw5L8678zFtXXDSmT34A+fzB7AykK0=</latexit>

Change of variable formula:@f✓ (x)p✓ (x) = pZ (f✓ (x)) det()@x
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Intuitively, prevents degenerative mapping of everything to zero vector
Normalizing Flow: A series of such invertible transformations f<latexit sha1_base64="aj1VrWgqSrqkgqJ/bLgtsTmRK/w=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtgvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPzD+M7g==</latexit>

13

DeMa-BWE: Preliminaries
Japanese Space

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

14

DeMa-BWE: Preliminaries
Japanese Space

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

Notations:14

DeMa-BWE: Preliminaries
Japanese Space

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

Notations:dy2R: denote vectors in the src and tgt embedding space
x2R ,
d

14

DeMa-BWE: Preliminaries
Japanese Space

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

Notations:dy2R: denote vectors in the src and tgt embedding spacex2R ,dxi , yj : denote an actual word in src and tgt vocabularies

14

DeMa-BWE: Preliminaries
Japanese Space

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

Notations:dy2R: denote vectors in the src and tgt embedding spacex2R ,dxi , yj : denote an actual word in src and tgt vocabularies

fxy , fyx : denote src->tgt, and tgt-src mapping functions
14

Prior Distribution• Assumption on the monolingual word embedding
space: Gaussian mixture model

⿃鳥

猫

⽝犬
15

Prior Distribution• Assumption on the monolingual word embedding
space: Gaussian mixture model

⿃鳥

猫

⽝犬
15

Prior Distribution• Assumption on the monolingual word embedding
space: Gaussian mixture model

⿃鳥

猫

⽝犬
15

DeMa-BWE: Density Matching

16

DeMa-BWE: Density Matching• Sampling a continuous vector from the GMM

16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )⽝犬16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )⽝犬• Apply the mapping function fxy to obtain the transformed
vector in the target space.

16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )• Apply the mapping function fxy to obtain the transformed
vector in the target space.

fxy (·) = Wxy ·⽝犬dogcanine

cat

bird

16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )• Apply the mapping function fxy to obtain the transformed
vector in the target space.

fxy (·) = Wxy ·⽝犬dogcaninecatbird• Computing the density of x in the mapped target space

16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )• Apply the mapping function fxy to obtain the transformed
vector in the target space.

fxy (·) = Wxy ·⽝犬dogcaninecatbird• Computing the density of x in the mapped target space

16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )• Apply the mapping function fxy to obtain the transformed
vector in the target space.

fxy (·) = Wxy ·⽝犬dogcaninecatbird• Computing the density of x in the mapped target space

16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )• Apply the mapping function fxy to obtain the transformed
vector in the target space.

fxy (·) = Wxy ·⽝犬dogcaninecatbird• Computing the density of x in the mapped target space• Objective:

16

DeMa-BWE: Density Matching
⿃鳥猫• Sampling a continuous vector from the GMM
xi ⇠ ⇡(xi )x ⇠ p̃(x|xi )• Apply the mapping function fxy to obtain the transformed
vector in the target space.

fxy (·) = Wxy ·⽝犬dogcaninecatbird• Computing the density of x in the mapped target space• Objective:

16

Method Details

17Method Details• Weak Orthogonality Constraint: Try to make surethat the transformation is close to orthogonal

17

Method Details• Weak Orthogonality Constraint: Try to make surethat the transformation is close to orthogonal

17

Method Details• Weak Orthogonality Constraint: Try to make surethat the transformation is close to orthogonal• Weak Supervision w/ Identical Strings: Take
advantage of the fact that identical strings areusually the same word in both languages

17

Method Details• Weak Orthogonality Constraint: Try to make surethat the transformation is close to orthogonal• Weak Supervision w/ Identical Strings: Take
advantage of the fact that identical strings areusually the same word in both languages

17

Method Details• Weak Orthogonality Constraint: Try to make surethat the transformation is close to orthogonal• Weak Supervision w/ Identical Strings: Take
advantage of the fact that identical strings areusually the same word in both languages• Alignment Selection Methods: Use cross-domain
similarity local scaling (CSLS)

17

Method Details• Weak Orthogonality Constraint: Try to make surethat the transformation is close to orthogonal• Weak Supervision w/ Identical Strings: Take
advantage of the fact that identical strings areusually the same word in both languages• Alignment Selection Methods: Use cross-domain
similarity local scaling (CSLS)

17

Experiments• Dataset and Tasks• Bilingual Lexicon Induction Task: MUSE dataset(Conneau el al., 2017)• Cross-lingual Word Similarity Task: SemEval 2017
• Languages• Baseline languages: en - es, de, fr, ru, zh, ja• Morphologically rich languages: en - et, fi, el, hu,pl, tr18

Main Results on BLI (close languages)
Procrustes(R)MUSE (U+R)

SL-unsup-ID

de-en

en-es

DeMa-BWE

85

81.25

77.5

73.75

70
en-de

es-en
19

Main Results on BLI (distant languages)
Procrustes(R)MUSE (U+R)

DeMa-BWE

65

48.75

32.5

16.25

0
en-et

et-en

en-el

el-en

en-ja

ja-en

20

Unsupervised Learning of
Syntactic Structure w/
Invertible Neural Projections
Junxian He, Graham Neubig, Taylor Berg-Kirkpatrick(EMNLP 2018)21

HMM for Part-of-Speech Induction

z1

z2

z3

x1

x2

x3

The

cat

sat

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22Gaussian HMM for POS Induction

z1

z2

z3

x1

x2

x3

The

cat

sat

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Gaussian HMM for POS Induction

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cat

sat

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Latent Embeddings w/ Neural Projection

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24Latent Embeddings w/ Neural Projectionzi ⇠<latexit sha1_base64="A0vAn9BPw9kDXHS9rkuB77rpTAE=">AAACDnicZVDLTgIxFO3gC/GFunTTSExcEDJjTNQd0Y1LTEQwMCGd0oGGdjpp75jghI8wbvU7XBm3/oKf4R/YgVkI3KS5p+ee25yeIBbcgOv+OIWV1bX1jeJmaWt7Z3evvH/wYFSiKWtSJZRuB8QwwSPWBA6CtWPNiAwEawWjm2zeemLacBXdwzhmviSDiIecErDU43OP467hEvfKFbfmTgsvAy8HFZRXo1f+7fYVTSSLgApiTMdzY/BTooFTwSalbmJYTOiIDFjHwohIZvx0aniCTyzTx6HS9kSAp+z/jZRIIwkMrTJrZm6WMaCUMFWrgqHMWvbM9G7GMqgGspqJtAnNghEIL/2UR3ECLKIzH2EiMCichYP7XDMKYmwBoZrbr2A6JJpQsBGWbEbeYiLLoHlWu6q5d+eV+nUeVhEdoWN0ijx0geroFjVQE1Ek0St6Q+/Oi/PhfDpfM2nByXcO0Vw5339MaJ0J</latexit>

Markov
Structure

z1

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e2

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24Latent Embeddings w/ Neural Projectionzi ⇠<latexit sha1_base64="A0vAn9BPw9kDXHS9rkuB77rpTAE=">AAACDnicZVDLTgIxFO3gC/GFunTTSExcEDJjTNQd0Y1LTEQwMCGd0oGGdjpp75jghI8wbvU7XBm3/oKf4R/YgVkI3KS5p+ee25yeIBbcgOv+OIWV1bX1jeJmaWt7Z3evvH/wYFSiKWtSJZRuB8QwwSPWBA6CtWPNiAwEawWjm2zeemLacBXdwzhmviSDiIecErDU43OP467hEvfKFbfmTgsvAy8HFZRXo1f+7fYVTSSLgApiTMdzY/BTooFTwSalbmJYTOiIDFjHwohIZvx0aniCTyzTx6HS9kSAp+z/jZRIIwkMrTJrZm6WMaCUMFWrgqHMWvbM9G7GMqgGspqJtAnNghEIL/2UR3ECLKIzH2EiMCichYP7XDMKYmwBoZrbr2A6JJpQsBGWbEbeYiLLoHlWu6q5d+eV+nUeVhEdoWN0ijx0geroFjVQE1Ek0St6Q+/Oi/PhfDpfM2nByXcO0Vw5339MaJ0J</latexit>

Markov
Structureei ⇠ N (µzi , ⌃zi )

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24Latent Embeddings w/ Neural Projection

Markov
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Neuralxi = f (ei )Projector<latexit sha1_base64="RMNkU799kfoHwXHS6vi9XiRQ2tw=">AAACDXicZVDLSgMxFM3UV62vquDGzWARXJQy1YW6K3XjsgXHFtqhZNJMG5pMhuSOUIb+g7jVX3DrSlwJfoMb/8E/MDPtwrYXwj0599xwcvyIMw2O823lVlbX1jfym4Wt7Z3dveL+wb2WsSLUJZJL1faxppyF1AUGnLYjRbHwOW35o5t03nqgSjMZ3sE4op7Ag5AFjGAwVLvri4ROehe9YsmpOFnZy6A6A6XaUfOHvdY/G73ib7cvSSxoCIRjrTtVJwIvwQoY4XRS6MaaRpiM8IB2DAyxoNpLMr8T+9QwfTuQypwQ7Iz9v5FgoQWGoVGmTc/NUgak5LpsVDAUaUufye56LPyyL8qpSOlALxiB4MpLWBjFQEMy9RHE3AZpp9nYfaYoAT42ABPFzFdsMsQKEzAJFkxG1cVEloF7XrmuOE2TVR1NK4+O0Qk6Q1V0iWroFjWQiwji6Ak9oxfr0Xqz3q2PqTRnzXYO0VxZX39bgqCP</latexit>
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x1<latexit sha1_base64="wPlw8y/0IqfHJrazcSQaHuwQ07k=">AAACDXicZVDLSgMxFM34rPVVdekmWAQXpcyIoO6KblxWcGyhLSWTZtrQZDIkd8Qy9B/ErX6HK3HrN/gZ/oGZ6SxseyHck3PPDScniAU34Lo/zsrq2vrGZmmrvL2zu7dfOTh8NCrRlPlUCaXbATFM8Ij5wEGwdqwZkYFgrWB8m81bT0wbrqIHmMSsJ8kw4iGnBCzV7gYyfZ72vX6l6tbdvPAy8ApQRUU1+5Xf7kDRRLIIqCDGdDw3hl5KNHAq2LTcTQyLCR2TIetYGBHJTC/N/U7xqWUGOFTanghwzv7fSIk0ksDIKrNm5mYZA0oJU7MqGMmsZc/kdzORQS2QtUykTWgWjEB41Ut5FCfAIjrzESYCg8JZNnjANaMgJhYQqrn9CqYjogkFm2DZZuQtJrIM/PP6dd29v6g2boqwSugYnaAz5KFL1EB3qIl8RJFAr+gNvTsvzofz6XzNpCtOsXOE5sr5/gMr950D</latexit>

x2<latexit sha1_base64="OUe9NQ3ny9IilmHPU3ONQ4TtfRo=">AAACDXicZVDLSgMxFM3UV62vqks3g0VwUcpMEdRd0Y3LCo4ttEPJpJk2NJkMyR2xDP0Hcavf4Urc+g1+hn9gZjoL214I9+Tcc8PJCWLONDjOj1VaW9/Y3CpvV3Z29/YPqodHj1omilCPSC5VN8CachZRDxhw2o0VxSLgtBNMbrN554kqzWT0ANOY+gKPIhYygsFQ3X4g0ufZoDmo1pyGk5e9CtwC1FBR7UH1tz+UJBE0AsKx1j3XicFPsQJGOJ1V+ommMSYTPKI9AyMsqPbT3O/MPjPM0A6lMicCO2f/b6RYaIFhbJRZ0wuzjAEpua4bFYxF1rJn8rueiqAeiHomUjrUS0YgvPJTFsUJ0IjMfYQJt0HaWTb2kClKgE8NwEQx8xWbjLHCBEyCFZORu5zIKvCajeuGc39Ra90UYZXRCTpF58hFl6iF7lAbeYggjl7RG3q3XqwP69P6mktLVrFzjBbK+v4DLZmdBA==</latexit>

x3<latexit sha1_base64="r7RTsVKDbYXIRCyPj9szuInKMVA=">AAACDXicZVDLSgMxFM3UV62vqks3g0VwUcqMCuqu6MZlBccW2qFk0kwbmkyG5I5Yhv6DuNXvcCVu/QY/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DRy0TRahHJJeqE2BNOYuoBww47cSKYhFw2g7Gt9m8/USVZjJ6gElMfYGHEQsZwWCoTi8Q6fO0f96v1pyGk5e9DNwC1FBRrX71tzeQJBE0AsKx1l3XicFPsQJGOJ1WeommMSZjPKRdAyMsqPbT3O/UPjHMwA6lMicCO2f/b6RYaIFhZJRZ03OzjAEpua4bFYxE1rJn8rueiKAeiHomUjrUC0YgvPJTFsUJ0IjMfIQJt0HaWTb2gClKgE8MwEQx8xWbjLDCBEyCFZORu5jIMvDOGtcN5/6i1rwpwiqjI3SMTpGLLlET3aEW8hBBHL2iN/RuvVgf1qf1NZOWrGLnEM2V9f0HLzudBQ==</latexit>

xi ⇠Point mass atf (ei )<latexit sha1_base64="4XmkaafD4EWeU6wrjePkY8qE6No=">AAACE3icZVDLTgIxFO3gC/GFunTTSExcEDIYE3VHdOMSE0dIGCSd0oGGdjpp7xjJhN8wbvU7XBm3foCf4R/YGVgI3KS5p+ee25yeIBbcgOv+OIWV1bX1jeJmaWt7Z3evvH/wYFSiKfOoEkq3A2KY4BHzgINg7VgzIgPBWsHoJpu3npg2XEX3MI5ZV5JBxENOCVjq0Q9k+jzpcewbLnGvXHFrbl54GdRnoIJm1eyVf/2+oolkEVBBjOnU3Ri6KdHAqWCTkp8YFhM6IgPWsTAikplumrue4BPL9HGotD0R4Jz9v5ESaSSBoVVmzczNMgaUEqZqVTCUWcueye9mLINqIKuZSJvQLBiB8LKb8ihOgEV06iNMBAaFs4Rwn2tGQYwtIFRz+xVMh0QTCjbHks2ovpjIMvDOalc19+680riehVVER+gYnaI6ukANdIuayEMUafSK3tC78+J8OJ/O11RacGY7h2iunO8/lwWfXA==</latexit><latexit sha1_base64="ISFMuiBWVv5rfvJnqq9uM8Osfvo=">AAACHnicZVBNS8NAEN34WetX1YMHL8EiVCglEUG9Fb14rGBsoS1hs920S3ezYXcilJAfI171d3gSr/oz/Adu0h5sO7DM2zdvhpkXxJxpcJwfa2V1bX1js7RV3t7Z3duvHBw+aZkoQj0iuVSdAGvKWUQ9YMBpJ1YUi4DTdjC+y+vtZ6o0k9EjTGLaF3gYsZARDIbyK8ehn/YCkfbiEcuyWg5p5rNzv1J1Gk4R9jJwZ6CKZtHyK7+9gSSJoBEQjrXuuk4M/RQrYITTrNxLNI0xGeMh7RoYYUF1Py0OyOwzwwzsUCrzIrAL9n9HioUWGEZGmSc9V8sZkJLrulHBSOQpH1P89UQE9UDUc5HSoV5YBMLrfsqiOAEakekeYcJtkHZulj1gihLgEwMwUcycYpMRVpiAsbRsPHIXHVkG3kXjpuE8XFabtzOzSugEnaIactEVaqJ71EIeIihDr+gNvVsv1of1aX1NpSvWrOcIzYX1/QfnyqO9</latexit>

24Latent Embeddings w/ Neural Projection

Markov
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xi ⇠Point mass atf (ei )<latexit sha1_base64="4XmkaafD4EWeU6wrjePkY8qE6No=">AAACE3icZVDLTgIxFO3gC/GFunTTSExcEDIYE3VHdOMSE0dIGCSd0oGGdjpp7xjJhN8wbvU7XBm3foCf4R/YGVgI3KS5p+ee25yeIBbcgOv+OIWV1bX1jeJmaWt7Z3evvH/wYFSiKfOoEkq3A2KY4BHzgINg7VgzIgPBWsHoJpu3npg2XEX3MI5ZV5JBxENOCVjq0Q9k+jzpcewbLnGvXHFrbl54GdRnoIJm1eyVf/2+oolkEVBBjOnU3Ri6KdHAqWCTkp8YFhM6IgPWsTAikplumrue4BPL9HGotD0R4Jz9v5ESaSSBoVVmzczNMgaUEqZqVTCUWcueye9mLINqIKuZSJvQLBiB8LKb8ihOgEV06iNMBAaFs4Rwn2tGQYwtIFRz+xVMh0QTCjbHks2ovpjIMvDOalc19+680riehVVER+gYnaI6ukANdIuayEMUafSK3tC78+J8OJ/O11RacGY7h2iunO8/lwWfXA==</latexit><latexit sha1_base64="ISFMuiBWVv5rfvJnqq9uM8Osfvo=">AAACHnicZVBNS8NAEN34WetX1YMHL8EiVCglEUG9Fb14rGBsoS1hs920S3ezYXcilJAfI171d3gSr/oz/Adu0h5sO7DM2zdvhpkXxJxpcJwfa2V1bX1js7RV3t7Z3duvHBw+aZkoQj0iuVSdAGvKWUQ9YMBpJ1YUi4DTdjC+y+vtZ6o0k9EjTGLaF3gYsZARDIbyK8ehn/YCkfbiEcuyWg5p5rNzv1J1Gk4R9jJwZ6CKZtHyK7+9gSSJoBEQjrXuuk4M/RQrYITTrNxLNI0xGeMh7RoYYUF1Py0OyOwzwwzsUCrzIrAL9n9HioUWGEZGmSc9V8sZkJLrulHBSOQpH1P89UQE9UDUc5HSoV5YBMLrfsqiOAEakekeYcJtkHZulj1gihLgEwMwUcycYpMRVpiAsbRsPHIXHVkG3kXjpuE8XFabtzOzSugEnaIactEVaqJ71EIeIihDr+gNvVsv1of1aX1NpSvWrOcIzYX1/QfnyqO9</latexit>

24Dependency Model with Valence

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VBD

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dog in Paris[Klein and Manning 2004]25

Dependency Model with Valence

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VBD

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[Klein and Manning 2004]
26

Dependency Parse Induction
from POS

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27

Grammar Induction from Raw Text?
z1
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28Grammar Induction from Raw Text?
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dog in Paris
29

Latent Embeddings w/ Neural Projectione1

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x2<latexit sha1_base64="OUe9NQ3ny9IilmHPU3ONQ4TtfRo=">AAACDXicZVDLSgMxFM3UV62vqks3g0VwUcpMEdRd0Y3LCo4ttEPJpJk2NJkMyR2xDP0Hcavf4Urc+g1+hn9gZjoL214I9+Tcc8PJCWLONDjOj1VaW9/Y3CpvV3Z29/YPqodHj1omilCPSC5VN8CachZRDxhw2o0VxSLgtBNMbrN554kqzWT0ANOY+gKPIhYygsFQ3X4g0ufZoDmo1pyGk5e9CtwC1FBR7UH1tz+UJBE0AsKx1j3XicFPsQJGOJ1V+ommMSYTPKI9AyMsqPbT3O/MPjPM0A6lMicCO2f/b6RYaIFhbJRZ0wuzjAEpua4bFYxF1rJn8rueiqAeiHomUjrUS0YgvPJTFsUJ0IjMfYQJt0HaWTb2kClKgE8NwEQx8xWbjLHCBEyCFZORu5zIKvCajeuGc39Ra90UYZXRCTpF58hFl6iF7lAbeYggjl7RG3q3XqwP69P6mktLVrFzjBbK+v4DLZmdBA==</latexit>

x3<latexit sha1_base64="r7RTsVKDbYXIRCyPj9szuInKMVA=">AAACDXicZVDLSgMxFM3UV62vqks3g0VwUcqMCuqu6MZlBccW2qFk0kwbmkyG5I5Yhv6DuNXvcCVu/QY/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DRy0TRahHJJeqE2BNOYuoBww47cSKYhFw2g7Gt9m8/USVZjJ6gElMfYGHEQsZwWCoTi8Q6fO0f96v1pyGk5e9DNwC1FBRrX71tzeQJBE0AsKx1l3XicFPsQJGOJ1WeommMSZjPKRdAyMsqPbT3O/UPjHMwA6lMicCO2f/b6RYaIFhZJRZ03OzjAEpua4bFYxE1rJn8rueiKAeiHomUjrUC0YgvPJTFsUJ0IjMfIQJt0HaWTb2gClKgE8MwEQx8xWbjLDCBEyCFZORu5jIMvDOGtcN5/6i1rwpwiqjI3SMTpGLLlET3aEW8hBBHL2iN/RuvVgf1qf1NZOWrGLnEM2V9f0HLzudBQ==</latexit>30Latent Embeddings w/ Neural Projectionzi ⇠<latexit sha1_base64="A0vAn9BPw9kDXHS9rkuB77rpTAE=">AAACDnicZVDLTgIxFO3gC/GFunTTSExcEDJjTNQd0Y1LTEQwMCGd0oGGdjpp75jghI8wbvU7XBm3/oKf4R/YgVkI3KS5p+ee25yeIBbcgOv+OIWV1bX1jeJmaWt7Z3evvH/wYFSiKWtSJZRuB8QwwSPWBA6CtWPNiAwEawWjm2zeemLacBXdwzhmviSDiIecErDU43OP467hEvfKFbfmTgsvAy8HFZRXo1f+7fYVTSSLgApiTMdzY/BTooFTwSalbmJYTOiIDFjHwohIZvx0aniCTyzTx6HS9kSAp+z/jZRIIwkMrTJrZm6WMaCUMFWrgqHMWvbM9G7GMqgGspqJtAnNghEIL/2UR3ECLKIzH2EiMCichYP7XDMKYmwBoZrbr2A6JJpQsBGWbEbeYiLLoHlWu6q5d+eV+nUeVhEdoWN0ijx0geroFjVQE1Ek0St6Q+/Oi/PhfDpfM2nByXcO0Vw5339MaJ0J</latexit>

Syntax
Model

z1

z2

z3

e1

e2

e3

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Neural xi = f (ei )Projectorf (e)<latexit sha1_base64="PuNsspGjpcc2SeVmkDWfr5ab2W0=">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</latexit>
sha1_base64="270VJLD7gt0vH7AtNf4cVJpk6VA=">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</latexit>

xi ⇠Point mass atf (ei )<latexit sha1_base64="RMNkU799kfoHwXHS6vi9XiRQ2tw=">AAACDXicZVDLSgMxFM3UV62vquDGzWARXJQy1YW6K3XjsgXHFtqhZNJMG5pMhuSOUIb+g7jVX3DrSlwJfoMb/8E/MDPtwrYXwj0599xwcvyIMw2O823lVlbX1jfym4Wt7Z3dveL+wb2WsSLUJZJL1faxppyF1AUGnLYjRbHwOW35o5t03nqgSjMZ3sE4op7Ag5AFjGAwVLvri4ROehe9YsmpOFnZy6A6A6XaUfOHvdY/G73ib7cvSSxoCIRjrTtVJwIvwQoY4XRS6MaaRpiM8IB2DAyxoNpLMr8T+9QwfTuQypwQ7Iz9v5FgoQWGoVGmTc/NUgak5LpsVDAUaUufye56LPyyL8qpSOlALxiB4MpLWBjFQEMy9RHE3AZpp9nYfaYoAT42ABPFzFdsMsQKEzAJFkxG1cVEloF7XrmuOE2TVR1NK4+O0Qk6Q1V0iWroFjWQiwji6Ak9oxfr0Xqz3q2PqTRnzXYO0VxZX39bgqCP</latexit>
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sha1_base64="8oxgnabmEamXCM6ELWdhWTrrt8c=">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</latexit>

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31

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sha1_base64="Yd99wiwenul3NLx8OscdFfA1yQk=">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</latexit>
sha1_base64="UtagCrqX87KPW4tGKv92duQD8gc=">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</latexit>⌘

p(xi |zi ; ⌘, )Gaussian embedding parameters<latexit sha1_base64="vUGVam4zX8tReom4E38JGVl+oAE=">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</latexit>

ei<latexit sha1_base64="86pgXmDcE2XVlZKOeiFEsb42X04=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="znhKAUtEwogmaPWollZUEdE8utY=">AAACQHicZVE9SwNBEJ3z2/gZW5vDIFiEcLFRO8HGMoJRwQthbzNnFvfj2J1Tw3F/wVZ/lb3/w0os3EtSGDOwzNs3M8vbN0kmhaMo+gwWFpeWV1bX1msbm1vbO7v1zRtncsuxy4009i5hDqXQ2CVBEu8yi0wlEm+Tx4uqfvuE1gmjr2mUYU+xBy1SwRlVVIx90d9tRK1oHOE8aE9BA6bR6deD43hgeK5QE5fMuft2lFGvYJYEl1jW4txhxvgje8B7DzVT6HrFWGwZHnpmEKbG+qMpHLN/JwqmnGI09J1VcjO1iiFjpGv6LhqqKlXPjO9upJJmoppVk3Wp+yeE0tNeIXSWE2o+0ZHmMiQTVsaEA2GRkxx5wLgV/ishHzLLOHn7ZkQkqqzVYo3P3CjF9KCIX8oiTlTxUpazPE549Lx3uf3f03nQPW6dtaKrCNZgHw7gCNpwAudwCR3oAochvMIbvAcfwVfwPVnHQjDdyx7MRPDzCypZtBY=</latexit>
sha1_base64="ViRgb/9Jv45ewVl+QYmhz6smaZo=">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</latexit>xf (e; )
<latexit sha1_base64="en/AeGOCu073Nk2yvXZpUPZ0fRU=">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</latexit>

xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>

32

Learning and Inferencezi<latexit sha1_base64="4V3LEopIC1ACF3cZlm/0mgv/HeE=">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</latexit><latexit sha1_base64="4gmLgsVwRggpXTovfy6uYJcrnUQ=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="Yd99wiwenul3NLx8OscdFfA1yQk=">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</latexit>
sha1_base64="UtagCrqX87KPW4tGKv92duQD8gc=">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</latexit>⌘

p(xi |zi ; ⌘, )Projection parameters<latexit sha1_base64="vUGVam4zX8tReom4E38JGVl+oAE=">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</latexit>

ei<latexit sha1_base64="86pgXmDcE2XVlZKOeiFEsb42X04=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="znhKAUtEwogmaPWollZUEdE8utY=">AAACQHicZVE9SwNBEJ3z2/gZW5vDIFiEcLFRO8HGMoJRwQthbzNnFvfj2J1Tw3F/wVZ/lb3/w0os3EtSGDOwzNs3M8vbN0kmhaMo+gwWFpeWV1bX1msbm1vbO7v1zRtncsuxy4009i5hDqXQ2CVBEu8yi0wlEm+Tx4uqfvuE1gmjr2mUYU+xBy1SwRlVVIx90d9tRK1oHOE8aE9BA6bR6deD43hgeK5QE5fMuft2lFGvYJYEl1jW4txhxvgje8B7DzVT6HrFWGwZHnpmEKbG+qMpHLN/JwqmnGI09J1VcjO1iiFjpGv6LhqqKlXPjO9upJJmoppVk3Wp+yeE0tNeIXSWE2o+0ZHmMiQTVsaEA2GRkxx5wLgV/ishHzLLOHn7ZkQkqqzVYo3P3CjF9KCIX8oiTlTxUpazPE549Lx3uf3f03nQPW6dtaKrCNZgHw7gCNpwAudwCR3oAochvMIbvAcfwVfwPVnHQjDdyx7MRPDzCypZtBY=</latexit>
sha1_base64="ViRgb/9Jv45ewVl+QYmhz6smaZo=">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</latexit>xf (e; )
<latexit sha1_base64="en/AeGOCu073Nk2yvXZpUPZ0fRU=">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</latexit><latexit sha1_base64="YFYVkXezeKmEAHc8b85dRLg5uLo=">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</latexit>

xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">AAACS3icZVDLSgMxFM3UV63P6tLNYBFclDIVQd2JblxWsLbQKSWT3rHBPIbkjrYM8wtu9avc+x+uxIWZtgurF8I9OffccHKiRHCLQfDhlZaWV1bXyuuVjc2t7Z3d6t691alh0GZaaNONqAXBFbSRo4BuYoDKSEAnerwu5p0nMJZrdYeTBPqSPigec0axoMLxgA92a0EjmJb/HzTnoEbm1RpUvZNwqFkqQSET1NpeM0iwn1GDnAnIK2FqIaHskT5Az0FFJdh+NjWb+0eOGfqxNu4o9Kfs742MSispjpyyaHZhVjCotbB1p8KRLFrxzPRuJzKqR7JeiIyN7R8jGJ/3M66SFEGxmY84FT5qvwjGH3IDDMXEAcoMd1/x2YgaytDFt2AiknmlEip4ZlpKqoZZOM6zMJLZOM8XeZjx4HiXcvNvpv9B+6Rx0QhuT2uXV/O4y+SAHJJj0iRn5JLckBZpE0ZG5IW8kjfv3fv0vrzvmbTkzXf2yUKVVn4AHsW04A==</latexit>

33

Learning and Inferencedim(x) = dim(e) and f is invertible<latexit sha1_base64="YXQGWOg/cF5S2BxOODZtuZLXmIc=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="LMxtcCziVcl5koy83KCXGg87iyE=">AAACPXicZVHLSgNBEOz1GePbq5dFETyEsPGi3gQvHhWMEdwQZie9ZnAey0yvJiz7BV71q/wCP8OTeHM2ycGYhqFrqruHmuokk8JRFH0GC4tLyyurtbX6+sbm1vbO7sadM7nl2OZGGnufMIdSaGyTIIn3mUWmEomd5Omyqnee0Tph9C2NMuwq9qhFKjgjT92kvZ3DqBmNI5wHrSk4hGlc93aDk7hveK5QE5fMuYdWlFG3YJYEl1jW49xhxvgTe8QHDzVT6LrFWGkZHnmmH6bG+qMpHLN/JwqmnGI08J1VcjO1iiFjpGv4LhqoKlXPjO9upJJGohpVk3Wp+yeE0rNuIXSWE2o+0ZHmMiQTVq6EfWGRkxx5wLgV/ishHzDLOHnvZkQkqqzXY40v3CjFdL+Ih2URJ6oYluUsjxMePe9dbv33dB60T5rnzegmghrswwEcQwtO4QKu4BrawAHhFd7gPfgIvoLvyToWgule9mAmgp9fnzKy2A==</latexit>
sha1_base64="UnN/qVlR88PXzV/bcsyXQUpjxXo=">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</latexit>

zi<latexit sha1_base64="4gmLgsVwRggpXTovfy6uYJcrnUQ=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="Yd99wiwenul3NLx8OscdFfA1yQk=">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</latexit>
sha1_base64="UtagCrqX87KPW4tGKv92duQD8gc=">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</latexit><latexit sha1_base64="4V3LEopIC1ACF3cZlm/0mgv/HeE=">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</latexit>⌘

p(xi |zi ; ⌘, )<latexit sha1_base64="vUGVam4zX8tReom4E38JGVl+oAE=">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</latexit>

ei<latexit sha1_base64="86pgXmDcE2XVlZKOeiFEsb42X04=">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</latexit>
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x
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1

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xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>

34

Learning and Inferencedim(x) = dim(e) and f is invertible<latexit sha1_base64="YXQGWOg/cF5S2BxOODZtuZLXmIc=">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</latexit>
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p(xi |zi ; ⌘, )1@f
= p(f 1 (xi )|zi ; ⌘) det@xi⌘<latexit sha1_base64="vUGVam4zX8tReom4E38JGVl+oAE=">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</latexit><latexit sha1_base64="hDcZgH5iScA9U7++4o6kHOLgWK0=">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</latexit>

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x
f

1

(e; )

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34

Learning and Inferencedim(x) = dim(e) and f is invertible<latexit sha1_base64="YXQGWOg/cF5S2BxOODZtuZLXmIc=">AAACSHicZVDLSgMxFM3UV61vXboZLIKLUqYiqDvRjcsKVgudUjLpnTY0jyG5o5ZhvsCtfpVf4Ge4Endm2i5svRDuybnnhpMTJYJbDIJPr7S0vLK6Vl6vbGxube/s7u0/WJ0aBi2mhTbtiFoQXEELOQpoJwaojAQ8RqObYv74BMZyre5xnEBX0oHiMWcUHXUX93arQT2YlP8fNGagSmbV7O15p2Ffs1SCQiaotZ1GkGA3owY5E5BXwtRCQtmIDqDjoKISbDebOM39Y8f0/VgbdxT6E/bvRkallRSHTlk0OzcrGNRa2JpT4VAWrXhmcrdjGdUiWStExsZ2wQjGF92MqyRFUGzqI06Fj9ovUvH73ABDMXaAMsPdV3w2pIYydNnNmYhkXqmECp6ZlpKqfha+5FkYyewlz+d5mPLgeJdyYzHT/6B1Wr+sB3dn1avrWdxlckiOyAlpkHNyRW5Jk7QII0BeyRt59z68L+/b+5lKS95s54DMVan0C2JLs4w=</latexit>
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p(xi |zi ; ⌘, )1@f
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x
f

1

(e; )

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Determinant of Jacobian matrix<latexit sha1_base64="YFYVkXezeKmEAHc8b85dRLg5uLo=">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</latexit>

xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">AAACS3icZVDLSgMxFM3UV63P6tLNYBFclDIVQd2JblxWsLbQKSWT3rHBPIbkjrYM8wtu9avc+x+uxIWZtgurF8I9OffccHKiRHCLQfDhlZaWV1bXyuuVjc2t7Z3d6t691alh0GZaaNONqAXBFbSRo4BuYoDKSEAnerwu5p0nMJZrdYeTBPqSPigec0axoMLxgA92a0EjmJb/HzTnoEbm1RpUvZNwqFkqQSET1NpeM0iwn1GDnAnIK2FqIaHskT5Az0FFJdh+NjWb+0eOGfqxNu4o9Kfs742MSispjpyyaHZhVjCotbB1p8KRLFrxzPRuJzKqR7JeiIyN7R8jGJ/3M66SFEGxmY84FT5qvwjGH3IDDMXEAcoMd1/x2YgaytDFt2AiknmlEip4ZlpKqoZZOM6zMJLZOM8XeZjx4HiXcvNvpv9B+6Rx0QhuT2uXV/O4y+SAHJJj0iRn5JLckBZpE0ZG5IW8kjfv3fv0vrzvmbTkzXf2yUKVVn4AHsW04A==</latexit>

35

Learning and Inferencedim(x) = dim(e) and f is invertible<latexit sha1_base64="YXQGWOg/cF5S2BxOODZtuZLXmIc=">AAACSHicZVDLSgMxFM3UV61vXboZLIKLUqYiqDvRjcsKVgudUjLpnTY0jyG5o5ZhvsCtfpVf4Ge4Endm2i5svRDuybnnhpMTJYJbDIJPr7S0vLK6Vl6vbGxube/s7u0/WJ0aBi2mhTbtiFoQXEELOQpoJwaojAQ8RqObYv74BMZyre5xnEBX0oHiMWcUHXUX93arQT2YlP8fNGagSmbV7O15p2Ffs1SCQiaotZ1GkGA3owY5E5BXwtRCQtmIDqDjoKISbDebOM39Y8f0/VgbdxT6E/bvRkallRSHTlk0OzcrGNRa2JpT4VAWrXhmcrdjGdUiWStExsZ2wQjGF92MqyRFUGzqI06Fj9ovUvH73ABDMXaAMsPdV3w2pIYydNnNmYhkXqmECp6ZlpKqfha+5FkYyewlz+d5mPLgeJdyYzHT/6B1Wr+sB3dn1avrWdxlckiOyAlpkHNyRW5Jk7QII0BeyRt59z68L+/b+5lKS95s54DMVan0C2JLs4w=</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
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p(xi |zi ; ⌘, )1@f
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1

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Gaussian distribution
Determinant of Jacobian matrix<latexit sha1_base64="YFYVkXezeKmEAHc8b85dRLg5uLo=">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</latexit>

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36

Learning and Inference

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37

Learning and Inference

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z3<latexit sha1_base64="VwYPdoeVjXkUU912kwmjdymaS+E=">AAACCHicZVBLTsMwFHT4lvIrsGQTUSGxqKoEkIBdBRuWRRBaqY0qx3Vaq3Yc2S9IJeoJEFs4ByvElltwDG6Ak2ZB2ydZbzxvnjWeIOZMg+P8WEvLK6tr66WN8ubW9s5uZW//UctEEeoRyaVqB1hTziLqAQNO27GiWASctoLRTTZvPVGlmYweYBxTX+BBxEJGMBjq/rl31qtUnbqTl70I3AJUUVHNXuW325ckETQCwrHWHdeJwU+xAkY4nZS7iaYxJiM8oB0DIyyo9tPc6sQ+NkzfDqUyJwI7Z/9vpFhogWFolFnTM7OMASm5rhkVDEXWsmfyux6LoBaIWiZSOtRzRiC89FMWxQnQiEx9hAm3QdpZLHafKUqAjw3ARDHzFZsMscIETHhlk5E7n8gi8E7rV3Xn7rzauC7CKqFDdIROkIsuUAPdoibyEEED9Ire0Lv1Yn1Yn9bXVLpkFTsHaKas7z/xkZqy</latexit>1@fp(f 1 (xi )|zi ; ⌘) det@xi

37Learning and Inference

Example of Markov priorz1<latexit sha1_base64="jbtr+W9vpRfg2saI7YImOblkNik=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcqMCOqu6MZlRccW2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DRy0TRahHJJeqE2BNOYuoBww47cSKYhFw2g7GN9m8/USVZjJ6gElMfYGHEQsZwWCo++e+26/WnIaTl70M3ALUUFGtfvW3N5AkETQCwrHWXdeJwU+xAkY4nVZ6iaYxJmM8pF0DIyyo9tPc6tQ+MczADqUyJwI7Z/9vpFhogWFklFnTc7OMASm5rhsVjETWsmfyu56IoB6IeiZSOtQLRiC89FMWxQnQiMx8hAm3QdpZLPaAKUqATwzARDHzFZuMsMIETHgVk5G7mMgy8M4aVw3n7rzWvC7CKqMjdIxOkYsuUBPdohbyEEFD9Ire0Lv1Yn1Yn9bXTFqyip1DNFfW9x/uTZqw</latexit><latexit sha1_base64="NOdjYPNrY5zneVpTueePUz9eQrc=">AAAConicZVHLbhMxFHWGVwmvFJZsLKJKqRSimQoJEJuqbJDYtILQojqMPJ47iVU/RvYdSJjOt/EdfABb+AU8k0iQ9kqWj4/Pubo+zkolPcbxz1504+at23d27vbv3X/w8NFg9/EnbysnYCqssu4s4x6UNDBFiQrOSgdcZwpOs4u37f3pV3BeWvMRVyXMNJ8bWUjBMVDp4HM5Kr7Uz5MmrVm5kM2ILVO5f/k9lW8oA+T77EjOLxnCEuscsGGF4yJIuUPJFV17m39Ea286TzoYxpO4K3odJBswJJs6Tnd7Byy3otJgUCju/XkSlzir285CQdNnlYeSiws+h/MADdfgZ3WXQUP3ApPTwrqwDNKO/d9Rc+01x0VQtpvfumsZtFb5cVDhQrdb26Y7+5XOxpketyLnC39lECxezWppygrBiPUcRaUoWtrmTXPpQKBaBcCFk+EpVCx4CBHDr2wNkemm32cGvgmrNTd5zZYh2EzXy6bZ5mHNQ+BDysnVTK+D6cHk9SQ+eTE8PNrEvUOekmdkRBLykhySd+SYTIkgP8gv8pv8ifai99FJ9GEtjXobzxOyVRH7C6CC1NA=</latexit>

z2<latexit sha1_base64="ZNKuRe23lzCJMNesh6cRgweKXJI=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcpMEdRd0Y3Lio4W2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DBy0TRahHJJeqE2BNOYuoBww47cSKYhFw+hiMr7P54xNVmsnoHiYx9QUeRixkBIOh7p77zX615jScvOxl4Baghopq96u/vYEkiaAREI617rpODH6KFTDC6bTSSzSNMRnjIe0aGGFBtZ/mVqf2iWEGdiiVORHYOft/I8VCCwwjo8yanptlDEjJdd2oYCSylj2T3/VEBPVA1DOR0qFeMALhhZ+yKE6ARmTmI0y4DdLOYrEHTFECfGIAJoqZr9hkhBUmYMKrmIzcxUSWgddsXDac27Na66oIq4yO0DE6RS46Ry10g9rIQwQN0St6Q+/Wi/VhfVpfM2nJKnYO0VxZ33/v75qx</latexit>z3

log p(x) = log pGHMM (f

1

(x))

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sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="Ku8E5xMhx2yjUIS3NBNvbWk1Bus=">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</latexit>
sha1_base64="xChZ9DUVMtsn1KFXp2HSXCyWRic=">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</latexit><latexit sha1_base64="VwYPdoeVjXkUU912kwmjdymaS+E=">AAACCHicZVBLTsMwFHT4lvIrsGQTUSGxqKoEkIBdBRuWRRBaqY0qx3Vaq3Yc2S9IJeoJEFs4ByvElltwDG6Ak2ZB2ydZbzxvnjWeIOZMg+P8WEvLK6tr66WN8ubW9s5uZW//UctEEeoRyaVqB1hTziLqAQNO27GiWASctoLRTTZvPVGlmYweYBxTX+BBxEJGMBjq/rl31qtUnbqTl70I3AJUUVHNXuW325ckETQCwrHWHdeJwU+xAkY4nZS7iaYxJiM8oB0DIyyo9tPc6sQ+NkzfDqUyJwI7Z/9vpFhogWFolFnTM7OMASm5rhkVDEXWsmfyux6LoBaIWiZSOtRzRiC89FMWxQnQiEx9hAm3QdpZLHafKUqAjw3ARDHzFZsMscIETHhlk5E7n8gi8E7rV3Xn7rzauC7CKqFDdIROkIsuUAPdoibyEEED9Ire0Lv1Yn1Yn9bXVLpkFTsHaKas7z/xkZqy</latexit>1@fp(f 1 (xi )|zi ; ⌘) det@xi

37Learning and Inference

Example of Markov priorz1<latexit sha1_base64="jbtr+W9vpRfg2saI7YImOblkNik=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcqMCOqu6MZlRccW2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DRy0TRahHJJeqE2BNOYuoBww47cSKYhFw2g7GN9m8/USVZjJ6gElMfYGHEQsZwWCo++e+26/WnIaTl70M3ALUUFGtfvW3N5AkETQCwrHWXdeJwU+xAkY4nVZ6iaYxJmM8pF0DIyyo9tPc6tQ+MczADqUyJwI7Z/9vpFhogWFklFnTc7OMASm5rhsVjETWsmfyu56IoB6IeiZSOtQLRiC89FMWxQnQiMx8hAm3QdpZLPaAKUqATwzARDHzFZuMsMIETHgVk5G7mMgy8M4aVw3n7rzWvC7CKqMjdIxOkYsuUBPdohbyEEFD9Ire0Lv1Yn1Yn9bXTFqyip1DNFfW9x/uTZqw</latexit>

z2<latexit sha1_base64="ZNKuRe23lzCJMNesh6cRgweKXJI=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcpMEdRd0Y3Lio4W2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DBy0TRahHJJeqE2BNOYuoBww47cSKYhFw+hiMr7P54xNVmsnoHiYx9QUeRixkBIOh7p77zX615jScvOxl4Baghopq96u/vYEkiaAREI617rpODH6KFTDC6bTSSzSNMRnjIe0aGGFBtZ/mVqf2iWEGdiiVORHYOft/I8VCCwwjo8yanptlDEjJdd2oYCSylj2T3/VEBPVA1DOR0qFeMALhhZ+yKE6ARmTmI0y4DdLOYrEHTFECfGIAJoqZr9hkhBUmYMKrmIzcxUSWgddsXDac27Na66oIq4yO0DE6RS46Ry10g9rIQwQN0St6Q+/Wi/VhfVpfM2nJKnYO0VxZ33/v75qx</latexit>

z3<latexit sha1_base64="VwYPdoeVjXkUU912kwmjdymaS+E=">AAACCHicZVBLTsMwFHT4lvIrsGQTUSGxqKoEkIBdBRuWRRBaqY0qx3Vaq3Yc2S9IJeoJEFs4ByvElltwDG6Ak2ZB2ydZbzxvnjWeIOZMg+P8WEvLK6tr66WN8ubW9s5uZW//UctEEeoRyaVqB1hTziLqAQNO27GiWASctoLRTTZvPVGlmYweYBxTX+BBxEJGMBjq/rl31qtUnbqTl70I3AJUUVHNXuW325ckETQCwrHWHdeJwU+xAkY4nZS7iaYxJiM8oB0DIyyo9tPc6sQ+NkzfDqUyJwI7Z/9vpFhogWFolFnTM7OMASm5rhkVDEXWsmfyux6LoBaIWiZSOtRzRiC89FMWxQnQiEx9hAm3QdpZLHafKUqAjw3ARDHzFZsMscIETHhlk5E7n8gi8E7rV3Xn7rzauC7CKqFDdIROkIsuUAPdoibyEEED9Ire0Lv1Yn1Yn9bXVLpkFTsHaKas7z/xkZqy</latexit>

log p(x) = log pGHMM (f<latexit sha1_base64="oq/g0iP5izTeEta3oHw43r5mrTc=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="Ku8E5xMhx2yjUIS3NBNvbWk1Bus=">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</latexit>
sha1_base64="xChZ9DUVMtsn1KFXp2HSXCyWRic=">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</latexit>

1

(x))

x
f

1

(e; )

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37Learning and Inference

Example of Markov priorz1<latexit sha1_base64="jbtr+W9vpRfg2saI7YImOblkNik=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcqMCOqu6MZlRccW2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DRy0TRahHJJeqE2BNOYuoBww47cSKYhFw2g7GN9m8/USVZjJ6gElMfYGHEQsZwWCo++e+26/WnIaTl70M3ALUUFGtfvW3N5AkETQCwrHWXdeJwU+xAkY4nVZ6iaYxJmM8pF0DIyyo9tPc6tQ+MczADqUyJwI7Z/9vpFhogWFklFnTc7OMASm5rhsVjETWsmfyu56IoB6IeiZSOtQLRiC89FMWxQnQiMx8hAm3QdpZLPaAKUqATwzARDHzFZuMsMIETHgVk5G7mMgy8M4aVw3n7rzWvC7CKqMjdIxOkYsuUBPdohbyEEFD9Ire0Lv1Yn1Yn9bXTFqyip1DNFfW9x/uTZqw</latexit>

z2<latexit sha1_base64="ZNKuRe23lzCJMNesh6cRgweKXJI=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcpMEdRd0Y3Lio4W2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DBy0TRahHJJeqE2BNOYuoBww47cSKYhFw+hiMr7P54xNVmsnoHiYx9QUeRixkBIOh7p77zX615jScvOxl4Baghopq96u/vYEkiaAREI617rpODH6KFTDC6bTSSzSNMRnjIe0aGGFBtZ/mVqf2iWEGdiiVORHYOft/I8VCCwwjo8yanptlDEjJdd2oYCSylj2T3/VEBPVA1DOR0qFeMALhhZ+yKE6ARmTmI0y4DdLOYrEHTFECfGIAJoqZr9hkhBUmYMKrmIzcxUSWgddsXDac27Na66oIq4yO0DE6RS46Ry10g9rIQwQN0St6Q+/Wi/VhfVpfM2nJKnYO0VxZ33/v75qx</latexit>z3

log p(x) = log pGHMM (f(x))<latexit sha1_base64="oq/g0iP5izTeEta3oHw43r5mrTc=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="Ku8E5xMhx2yjUIS3NBNvbWk1Bus=">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</latexit>
sha1_base64="xChZ9DUVMtsn1KFXp2HSXCyWRic=">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</latexit><latexit sha1_base64="VwYPdoeVjXkUU912kwmjdymaS+E=">AAACCHicZVBLTsMwFHT4lvIrsGQTUSGxqKoEkIBdBRuWRRBaqY0qx3Vaq3Yc2S9IJeoJEFs4ByvElltwDG6Ak2ZB2ydZbzxvnjWeIOZMg+P8WEvLK6tr66WN8ubW9s5uZW//UctEEeoRyaVqB1hTziLqAQNO27GiWASctoLRTTZvPVGlmYweYBxTX+BBxEJGMBjq/rl31qtUnbqTl70I3AJUUVHNXuW325ckETQCwrHWHdeJwU+xAkY4nZS7iaYxJiM8oB0DIyyo9tPc6sQ+NkzfDqUyJwI7Z/9vpFhogWFolFnTM7OMASm5rhkVDEXWsmfyux6LoBaIWiZSOtRzRiC89FMWxQnQiEx9hAm3QdpZLHafKUqAjw3ARDHzFZsMscIETHhlk5E7n8gi8E7rV3Xn7rzauC7CKqFDdIROkIsuUAPdoibyEEED9Ire0Lv1Yn1Yn9bXVLpkFTsHaKas7z/xkZqy</latexit>+<latexit sha1_base64="SkZYIwUJm36ifl+bt7GkjNGFpcg=">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</latexit><latexit sha1_base64="NOdjYPNrY5zneVpTueePUz9eQrc=">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</latexit>

1X

log det

@f

1

x
f

1

(e; )

<latexit sha1_base64="0TvUX8MzRC5izm1cmvW9PX4J8iQ=">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</latexit>

@xi

1
@f
p(f 1 (xi )|zi ; ⌘) det@xi

37Learning and Inference

Example of Markov priorz1<latexit sha1_base64="jbtr+W9vpRfg2saI7YImOblkNik=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcqMCOqu6MZlRccW2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DRy0TRahHJJeqE2BNOYuoBww47cSKYhFw2g7GN9m8/USVZjJ6gElMfYGHEQsZwWCo++e+26/WnIaTl70M3ALUUFGtfvW3N5AkETQCwrHWXdeJwU+xAkY4nVZ6iaYxJmM8pF0DIyyo9tPc6tQ+MczADqUyJwI7Z/9vpFhogWFklFnTc7OMASm5rhsVjETWsmfyu56IoB6IeiZSOtQLRiC89FMWxQnQiMx8hAm3QdpZLPaAKUqATwzARDHzFZuMsMIETHgVk5G7mMgy8M4aVw3n7rzWvC7CKqMjdIxOkYsuUBPdohbyEEFD9Ire0Lv1Yn1Yn9bXTFqyip1DNFfW9x/uTZqw</latexit>

z2<latexit sha1_base64="ZNKuRe23lzCJMNesh6cRgweKXJI=">AAACCHicZVDLSgMxFM3UV62vqks3g0VwUcpMEdRd0Y3Lio4W2qFk0kwbmkyG5I5Qh36BuNXvcCVu/Qs/wz8wM52FbS+Ee3LuueHkBDFnGhznxyqtrK6tb5Q3K1vbO7t71f2DBy0TRahHJJeqE2BNOYuoBww47cSKYhFw+hiMr7P54xNVmsnoHiYx9QUeRixkBIOh7p77zX615jScvOxl4Baghopq96u/vYEkiaAREI617rpODH6KFTDC6bTSSzSNMRnjIe0aGGFBtZ/mVqf2iWEGdiiVORHYOft/I8VCCwwjo8yanptlDEjJdd2oYCSylj2T3/VEBPVA1DOR0qFeMALhhZ+yKE6ARmTmI0y4DdLOYrEHTFECfGIAJoqZr9hkhBUmYMKrmIzcxUSWgddsXDac27Na66oIq4yO0DE6RS46Ry10g9rIQwQN0St6Q+/Wi/VhfVpfM2nJKnYO0VxZ33/v75qx</latexit>z3

log p(x) = log pGHMM (f

1

(x))

<latexit sha1_base64="oq/g0iP5izTeEta3oHw43r5mrTc=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="Ku8E5xMhx2yjUIS3NBNvbWk1Bus=">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</latexit>
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1

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1

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37Learning with Inverse Projectionei<latexit sha1_base64="n3cjhTJw9tlLdu5m9eJ8NNiBIqk=">AAACDnicZVDLTgIxFO3gC/GFmrhxM5GYuCBkcKPuCG5cQuIIBiakUzrQ0E4n7R0TMuEjjFv9BNeujBsX/oIb/8E/sDOwELhJc0/PPbc5PX7EmQbH+bZyK6tr6xv5zcLW9s7uXnH/4E7LWBHqEsmlavtYU85C6gIDTtuRolj4nLb80XU6bz1QpZkMb2EcUU/gQcgCRjAY6r7ri4ROeqzQK5acipOVvQyqM1CqHTV/2Gv9s9Er/nb7ksSChkA41rpTdSLwEqyAEU4nhW6saYTJCA9ox8AQC6q9JDM8sU8N07cDqcwJwc7Y/xsJFlpgGBpl2vTcLGVASq7LRgVDkbb0meyux8Iv+6KcipQO9IIRCC69hIVRDDQkUx9BzG2QdhqO3WeKEuBjAzBRzHzFJkOsMAETYZpRdTGRZeCeV64qTtNkVUfTyqNjdILOUBVdoBq6QQ3kIoIEekLP6MV6tN6sd+tjKs1Zs51DNFfW1x/sYqDZ</latexit>
sha1_base64="MrqiPJqbEeBmKU3LaA4LyQylndY=">AAACDnicZVDLTgIxFL2DL8QXauLGzURi4oKQwY26I7pxCYkIBgh2Sgca2umkvWNCJnyEcavf4cq49Rfc+A/+gR1gIXCT5p6ee25zevxIcIOe9+1kVlbX1jeym7mt7Z3dvfz+wb1RsaasTpVQuukTwwQPWR05CtaMNCPSF6zhD2/SeeOJacNVeIejiHUk6Yc84JSgpR7avkzYuMtz3XzBK3mTcpdBeQYKlaPazyMAVLv533ZP0ViyEKkgxrTKXoSdhGjkVLBxrh0bFhE6JH3WsjAkkplOMjE8dk8t03MDpe0J0Z2w/zcSIo0kOLDKtJm5WcqgUsIUrQoHMm3pM5O7GUm/6MtiKtImMAtGMLjsJDyMYmQhnfoIYuGictNw3B7XjKIYWUCo5vYrLh0QTSjaCNOMyouJLIP6eemq5NVsVtcwrSwcwwmcQRkuoAK3UIU6UJDwAq/w5jw7786H8zmVZpzZziHMlfP1B2cZnx0=</latexit>

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Inversef<latexit sha1_base64="NQDPiKGUHm9/+KiRn5wz6HvhrUk=">AAACBnicZVDLTgIxFO3gC/GFutSYicTEBSEzbtQd0Y1LSBwhAUI6pQMN7XTS3jEhE5YujFv9DldGlv6G3+DKP7AzsBC4SXNPzz23OT1+xJkGx/m2ciura+sb+c3C1vbO7l5x/+BBy1gR6hHJpWr6WFPOQuoBA06bkaJY+Jw2/OFtOm88UqWZDO9hFNGOwP2QBYxgMFQ96BZLTsXJyl4G7gyUqseT+s/TyaTWLf62e5LEgoZAONa65ToRdBKsgBFOx4V2rGmEyRD3acvAEAuqO0lmdGyfGaZnB1KZE4Kdsf83Eiy0wDAwyrTpuVnKgJRcl40KBiJt6TPZXY+EX/ZFORUpHegFIxBcdRIWRjHQkEx9BDG3QdppKHaPKUqAjwzARDHzFZsMsMIETHQFk5G7mMgy8C4q1xWnbrK6QdPKoyN0is6Riy5RFd2hGvIQQRS9oFf0Zj1b79aH9TmV5qzZziGaK+vrD/ImnZ8=</latexit>
sha1_base64="wkf9utTHcQFbfvS3Vq9hZY9DMZU=">AAACBnicZVDLTgIxFL3jE/GFutTFRGLigpDBjbojunEJiSMkQEindKChnU7aOyZkwgcYt/odrowu/Q13/oJ/YAdYCNykuafnntucniAW3KDnfTsrq2vrG5u5rfz2zu7efuHg8MGoRFPmUyWUbgbEMMEj5iNHwZqxZkQGgjWC4W02bzwybbiK7nEUs44k/YiHnBK0VD3sFope2ZuUuwwqM1CsnnzWfwCg1i38tnuKJpJFSAUxplXxYuykRCOngo3z7cSwmNAh6bOWhRGRzHTSidGxe2aZnhsqbU+E7oT9v5ESaSTBgVVmzczNMgaVEqZkVTiQWcuemdzNSAalQJYykTahWTCC4VUn5VGcIIvo1EeYCBeVm4Xi9rhmFMXIAkI1t19x6YBoQtFGl7cZVRYTWQb+Rfm67NVtVjcwrRwcwymcQwUuoQp3UAMfKDB4hhd4dZ6cN+fd+ZhKV5zZzhHMlfP1B+ntnDk=</latexit>

f

1

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<latexit sha1_base64="fjhIxwws58GsuvPWxj5BWS9qhG8=">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</latexit>
sha1_base64="T5O2W5fdBe9YVbRkg7tBseU/cmg=">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</latexit>

xi<latexit sha1_base64="YTThJcjf7ThL3C1arIc7iYN4iXU=">AAACDXicZVDLSgMxFM34rPVVdekmWAQXpUxFUHdFNy4rOLbQDiWTZtrQZDIkd8Qy9B/ErX6HK3HrN/gZ/oGZ6SxseyHck3PPDScniAU34Lo/zsrq2vrGZmmrvL2zu7dfOTh8NCrRlHlUCaU7ATFM8Ih5wEGwTqwZkYFg7WB8m83bT0wbrqIHmMTMl2QY8ZBTApbq9AKZPk/7vF+punU3L7wMGgWooqJa/cpvb6BoIlkEVBBjug03Bj8lGjgVbFruJYbFhI7JkHUtjIhkxk9zv1N8apkBDpW2JwKcs/83UiKNJDCyyqyZuVnGgFLC1KwKRjJr2TP53UxkUAtkLRNpE5oFIxBe+SmP4gRYRGc+wkRgUDjLBg+4ZhTExAJCNbdfwXRENKFgEyzbjBqLiSwD77x+XXfvL6rNmyKsEjpGJ+gMNdAlaqI71EIeokigV/SG3p0X58P5dL5m0hWn2DlCc+V8/wGHZ507</latexit>

xi<latexit sha1_base64="qlOuSkQawl0c7uB9u7+xlyfG6YA=">AAACDXicZVDLTgIxFO3gC/GFmrhxM5GYuCBkcKPuCG5cQuIICUxIp3SgoZ1O2jtGMuEfjFv9BbeujCsTv8GN/+Af2BlYCNykuafnntucHj/iTIPjfFu5ldW19Y38ZmFre2d3r7h/cKdlrAh1ieRStX2sKWchdYEBp+1IUSx8Tlv+6Dqdt+6p0kyGtzCOqCfwIGQBIxgM1e76InmY9FivWHIqTlb2MqjOQKl21Pxhr/XPRq/42+1LEgsaAuFY607VicBLsAJGOJ0UurGmESYjPKAdA0MsqPaSzO/EPjVM3w6kMicEO2P/byRYaIFhaJRp03OzlAEpuS4bFQxF2tJnsrseC7/si3IqUjrQC0YguPQSFkYx0JBMfQQxt0HaaTZ2nylKgI8NwEQx8xWbDLHCBEyCBZNRdTGRZeCeV64qTtNkVUfTyqNjdILOUBVdoBq6QQ3kIoI4ekLP6MV6tN6sd+tjKs1Zs51DNFfW1x/S7aDY</latexit>
sha1_base64="cABDScOf/SzJSeJxwEj+ZulgKro=">AAACDXicZVDLTgIxFL2DL8QXauLGzURi4oKQwY26I7pxCYkICRDslA40tNNJe8dIJvyDcavf4cq49Rvc+A/+gR1gIXCT5p6ee25zevxIcIOe9+1kVlbX1jeym7mt7Z3dvfz+wb1RsaasTpVQuukTwwQPWR05CtaMNCPSF6zhD2/SeeORacNVeIejiHUk6Yc84JSgpZptXyZP4y7v5gteyZuUuwzKM1CoHNV+HgCg2s3/tnuKxpKFSAUxplX2IuwkRCOngo1z7diwiNAh6bOWhSGRzHSSid+xe2qZnhsobU+I7oT9v5EQaSTBgVWmzczNUgaVEqZoVTiQaUufmdzNSPpFXxZTkTaBWTCCwWUn4WEUIwvp1EcQCxeVm2bj9rhmFMXIAkI1t19x6YBoQtEmmLMZlRcTWQb189JVyavZrK5hWlk4hhM4gzJcQAVuoQp1oCDgBV7hzXl23p0P53MqzTiznUOYK+frD02knxw=</latexit>

38

Learning with Inverse Projectionei<latexit sha1_base64="n3cjhTJw9tlLdu5m9eJ8NNiBIqk=">AAACDnicZVDLTgIxFO3gC/GFmrhxM5GYuCBkcKPuCG5cQuIIBiakUzrQ0E4n7R0TMuEjjFv9BNeujBsX/oIb/8E/sDOwELhJc0/PPbc5PX7EmQbH+bZyK6tr6xv5zcLW9s7uXnH/4E7LWBHqEsmlavtYU85C6gIDTtuRolj4nLb80XU6bz1QpZkMb2EcUU/gQcgCRjAY6r7ri4ROeqzQK5acipOVvQyqM1CqHTV/2Gv9s9Er/nb7ksSChkA41rpTdSLwEqyAEU4nhW6saYTJCA9ox8AQC6q9JDM8sU8N07cDqcwJwc7Y/xsJFlpgGBpl2vTcLGVASq7LRgVDkbb0meyux8Iv+6KcipQO9IIRCC69hIVRDDQkUx9BzG2QdhqO3WeKEuBjAzBRzHzFJkOsMAETYZpRdTGRZeCeV64qTtNkVUfTyqNjdILOUBVdoBq6QQ3kIoIEekLP6MV6tN6sd+tjKs1Zs51DNFfW1x/sYqDZ</latexit>
sha1_base64="MrqiPJqbEeBmKU3LaA4LyQylndY=">AAACDnicZVDLTgIxFL2DL8QXauLGzURi4oKQwY26I7pxCYkIBgh2Sgca2umkvWNCJnyEcavf4cq49Rfc+A/+gR1gIXCT5p6ee25zevxIcIOe9+1kVlbX1jeym7mt7Z3dvfz+wb1RsaasTpVQuukTwwQPWR05CtaMNCPSF6zhD2/SeeOJacNVeIejiHUk6Yc84JSgpR7avkzYuMtz3XzBK3mTcpdBeQYKlaPazyMAVLv533ZP0ViyEKkgxrTKXoSdhGjkVLBxrh0bFhE6JH3WsjAkkplOMjE8dk8t03MDpe0J0Z2w/zcSIo0kOLDKtJm5WcqgUsIUrQoHMm3pM5O7GUm/6MtiKtImMAtGMLjsJDyMYmQhnfoIYuGictNw3B7XjKIYWUCo5vYrLh0QTSjaCNOMyouJLIP6eemq5NVsVtcwrSwcwwmcQRkuoAK3UIU6UJDwAq/w5jw7786H8zmVZpzZziHMlfP1B2cZnx0=</latexit>(1)hi,l = xi,l<latexit sha1_base64="ecOX4VbbnENfsay0Sy5ZWNwjfdg=">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</latexit>

(1)

ei

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Inversef(1)<latexit sha1_base64="NQDPiKGUHm9/+KiRn5wz6HvhrUk=">AAACBnicZVDLTgIxFO3gC/GFutSYicTEBSEzbtQd0Y1LSBwhAUI6pQMN7XTS3jEhE5YujFv9DldGlv6G3+DKP7AzsBC4SXNPzz23OT1+xJkGx/m2ciura+sb+c3C1vbO7l5x/+BBy1gR6hHJpWr6WFPOQuoBA06bkaJY+Jw2/OFtOm88UqWZDO9hFNGOwP2QBYxgMFQ96BZLTsXJyl4G7gyUqseT+s/TyaTWLf62e5LEgoZAONa65ToRdBKsgBFOx4V2rGmEyRD3acvAEAuqO0lmdGyfGaZnB1KZE4Kdsf83Eiy0wDAwyrTpuVnKgJRcl40KBiJt6TPZXY+EX/ZFORUpHegFIxBcdRIWRjHQkEx9BDG3QdppKHaPKUqAjwzARDHzFZsMsMIETHQFk5G7mMgy8C4q1xWnbrK6QdPKoyN0is6Riy5RFd2hGvIQQRS9oFf0Zj1b79aH9TmV5qzZziGaK+vrD/ImnZ8=</latexit>
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1

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(1)

hi,l

hi,r

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xi<latexit sha1_base64="jHj0NqdQ+87fbQS5prPS5RHLnOU=">AAACBnicZVDLSgMxFM34rPVVdekmWAQXpUxFUHdFNy5bcGyhHUomzbShyWRI7ghl6AeIW/0OV+LW3/Az/AMz01nY9kK4J+eeG05OEAtuwHV/nLX1jc2t7dJOeXdv/+CwcnT8ZFSiKfOoEkp3A2KY4BHzgINg3VgzIgPBOsHkPpt3npk2XEWPMI2ZL8ko4iGnBCzVHg0qVbfu5oVXQaMAVVRUa1D57Q8VTSSLgApiTK/hxuCnRAOngs3K/cSwmNAJGbGehRGRzPhpbnSGzy0zxKHS9kSAc/b/RkqkkQTGVpk1szDLGFBKmJpVwVhmLXsmv5upDGqBrGUibUKzZATCGz/lUZwAi+jcR5gIDApnoeAh14yCmFpAqOb2K5iOiSYUbHRlm1FjOZFV4F3Wb+tu+6ravCvCKqFTdIYuUANdoyZ6QC3kIYoYekVv6N15cT6cT+drLl1zip0TtFDO9x+Yv5n5</latexit>g<latexit sha1_base64="YTThJcjf7ThL3C1arIc7iYN4iXU=">AAACDXicZVDLSgMxFM34rPVVdekmWAQXpUxFUHdFNy4rOLbQDiWTZtrQZDIkd8Qy9B/ErX6HK3HrN/gZ/oGZ6SxseyHck3PPDScniAU34Lo/zsrq2vrGZmmrvL2zu7dfOTh8NCrRlHlUCaU7ATFM8Ih5wEGwTqwZkYFg7WB8m83bT0wbrqIHmMTMl2QY8ZBTApbq9AKZPk/7vF+punU3L7wMGgWooqJa/cpvb6BoIlkEVBBjug03Bj8lGjgVbFruJYbFhI7JkHUtjIhkxk9zv1N8apkBDpW2JwKcs/83UiKNJDCyyqyZuVnGgFLC1KwKRjJr2TP53UxkUAtkLRNpE5oFIxBe+SmP4gRYRGc+wkRgUDjLBg+4ZhTExAJCNbdfwXRENKFgEyzbjBqLiSwD77x+XXfvL6rNmyKsEjpGJ+gMNdAlaqI71EIeokigV/SG3p0X58P5dL5m0hWn2DlCc+V8/wGHZ507</latexit>

xi,l<latexit sha1_base64="vwq6y+PoHkbNqh+fpEPZilaEdao=">AAACEnicZVDLTgIxFO3gC/GFunTTSExcTMhgTNQd0Y1LTERIYEI6pQMN7XRs7xjJZD7DuNXvcGXc+gN+hn9gB1gI3KS5p+ee25yeIBbcgOf9OIWV1bX1jeJmaWt7Z3evvH/wYFSiKWtSJZRuB8QwwSPWBA6CtWPNiAwEawWjm3zeemLacBXdwzhmviSDiIecErCU3w1k+pz1Uu5ikfXKFa/qTQovg9oMVNCsGr3yb7evaCJZBFQQYzo1LwY/JRo4FSwrdRPDYkJHZMA6FkZEMuOnE9MZPrFMH4dK2xMBnrD/N1IijSQwtMq8mblZzoBSwrhWBUOZt/yZyd2MZeAG0s1F2oRmwQiEl37KozgBFtGpjzARGBTOA8J9rhkFMbaAUM3tVzAdEk0o2BhLNqPaYiLLoHlWvap6d+eV+vUsrCI6QsfoFNXQBaqjW9RATUTRI3pFb+jdeXE+nE/nayotOLOdQzRXzvcfFoSfHQ==</latexit>

xi,r<latexit sha1_base64="nykeAuIuyivai+ip12Gc3TWLdk8=">AAACEnicZVDLTgIxFO3gC/GFunTTSExcTMhgTNQd0Y1LTERIYEI6pQMN7XRs7xjJZD7DuNXvcGXc+gN+hn9gB1gI3KS5p+ee25yeIBbcgOf9OIWV1bX1jeJmaWt7Z3evvH/wYFSiKWtSJZRuB8QwwSPWBA6CtWPNiAwEawWjm3zeemLacBXdwzhmviSDiIecErCU3w1k+pz1Uu5infXKFa/qTQovg9oMVNCsGr3yb7evaCJZBFQQYzo1LwY/JRo4FSwrdRPDYkJHZMA6FkZEMuOnE9MZPrFMH4dK2xMBnrD/N1IijSQwtMq8mblZzoBSwrhWBUOZt/yZyd2MZeAG0s1F2oRmwQiEl37KozgBFtGpjzARGBTOA8J9rhkFMbaAUM3tVzAdEk0o2BhLNqPaYiLLoHlWvap6d+eV+vUsrCI6QsfoFNXQBaqjW9RATUTRI3pFb+jdeXE+nE/nayotOLOdQzRXzvcfIFafIw==</latexit>

xi<latexit sha1_base64="qlOuSkQawl0c7uB9u7+xlyfG6YA=">AAACDXicZVDLTgIxFO3gC/GFmrhxM5GYuCBkcKPuCG5cQuIICUxIp3SgoZ1O2jtGMuEfjFv9BbeujCsTv8GN/+Af2BlYCNykuafnntucHj/iTIPjfFu5ldW19Y38ZmFre2d3r7h/cKdlrAh1ieRStX2sKWchdYEBp+1IUSx8Tlv+6Dqdt+6p0kyGtzCOqCfwIGQBIxgM1e76InmY9FivWHIqTlb2MqjOQKl21Pxhr/XPRq/42+1LEgsaAuFY607VicBLsAJGOJ0UurGmESYjPKAdA0MsqPaSzO/EPjVM3w6kMicEO2P/byRYaIFhaJRp03OzlAEpuS4bFQxF2tJnsrseC7/si3IqUjrQC0YguPQSFkYx0JBMfQQxt0HaaTZ2nylKgI8NwEQx8xWbDLHCBEyCBZNRdTGRZeCeV64qTtNkVUfTyqNjdILOUBVdoBq6QQ3kIoI4ekLP6MV6tN6sd+tjKs1Zs51DNFfW1x/S7aDY</latexit>
sha1_base64="cABDScOf/SzJSeJxwEj+ZulgKro=">AAACDXicZVDLTgIxFL2DL8QXauLGzURi4oKQwY26I7pxCYkICRDslA40tNNJe8dIJvyDcavf4cq49Rvc+A/+gR1gIXCT5p6ee25zevxIcIOe9+1kVlbX1jeym7mt7Z3dvfz+wb1RsaasTpVQuukTwwQPWR05CtaMNCPSF6zhD2/SeeORacNVeIejiHUk6Yc84JSgpZptXyZP4y7v5gteyZuUuwzKM1CoHNV+HgCg2s3/tnuKxpKFSAUxplX2IuwkRCOngo1z7diwiNAh6bOWhSGRzHSSid+xe2qZnhsobU+I7oT9v5EQaSTBgVWmzczNUgaVEqZoVTiQaUufmdzNSPpFXxZTkTaBWTCCwWUn4WEUIwvp1EcQCxeVm2bj9rhmFMXIAkI1t19x6YBoQtEmmLMZlRcTWQb189JVyavZrK5hWlk4hhM4gzJcQAVuoQp1oCDgBV7hzXl23p0P53MqzTiznUOYK+frD02knxw=</latexit>[Dinh et al. 2014]39

Learning with Inverse Projectionei<latexit sha1_base64="n3cjhTJw9tlLdu5m9eJ8NNiBIqk=">AAACDnicZVDLTgIxFO3gC/GFmrhxM5GYuCBkcKPuCG5cQuIIBiakUzrQ0E4n7R0TMuEjjFv9BNeujBsX/oIb/8E/sDOwELhJc0/PPbc5PX7EmQbH+bZyK6tr6xv5zcLW9s7uXnH/4E7LWBHqEsmlavtYU85C6gIDTtuRolj4nLb80XU6bz1QpZkMb2EcUU/gQcgCRjAY6r7ri4ROeqzQK5acipOVvQyqM1CqHTV/2Gv9s9Er/nb7ksSChkA41rpTdSLwEqyAEU4nhW6saYTJCA9ox8AQC6q9JDM8sU8N07cDqcwJwc7Y/xsJFlpgGBpl2vTcLGVASq7LRgVDkbb0meyux8Iv+6KcipQO9IIRCC69hIVRDDQkUx9BzG2QdhqO3WeKEuBjAzBRzHzFJkOsMAETYZpRdTGRZeCeV64qTtNkVUfTyqNjdILOUBVdoBq6QQ3kIoIEekLP6MV6tN6sd+tjKs1Zs51DNFfW1x/sYqDZ</latexit>
sha1_base64="MrqiPJqbEeBmKU3LaA4LyQylndY=">AAACDnicZVDLTgIxFL2DL8QXauLGzURi4oKQwY26I7pxCYkIBgh2Sgca2umkvWNCJnyEcavf4cq49Rfc+A/+gR1gIXCT5p6ee25zevxIcIOe9+1kVlbX1jeym7mt7Z3dvfz+wb1RsaasTpVQuukTwwQPWR05CtaMNCPSF6zhD2/SeeOJacNVeIejiHUk6Yc84JSgpR7avkzYuMtz3XzBK3mTcpdBeQYKlaPazyMAVLv533ZP0ViyEKkgxrTKXoSdhGjkVLBxrh0bFhE6JH3WsjAkkplOMjE8dk8t03MDpe0J0Z2w/zcSIo0kOLDKtJm5WcqgUsIUrQoHMm3pM5O7GUm/6MtiKtImMAtGMLjsJDyMYmQhnfoIYuGictNw3B7XjKIYWUCo5vYrLh0QTSjaCNOMyouJLIP6eemq5NVsVtcwrSwcwwmcQRkuoAK3UIU6UJDwAq/w5jw7786H8zmVZpzZziHMlfP1B2cZnx0=</latexit>(1)hi,l = xi,l<latexit sha1_base64="ecOX4VbbnENfsay0Sy5ZWNwjfdg=">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</latexit>

(1)

ei

hi,r = xi,r + g(xi,l )<latexit sha1_base64="REkAECskl6vTPQb6KrE5yd3HUvc=">AAACDXicZVDLSgMxFM3UV62vqks3wSK4KGUqgrorunFZwbGFdiiZNNOGJpMhuSOUof8gbvU7XIlbv8HP8A/MTGdh2wvhnpx7bjg5QSy4Adf9cUpr6xubW+Xtys7u3v5B9fDoyahEU+ZRJZTuBsQwwSPmAQfBurFmRAaCdYLJXTbvPDNtuIoeYRozX5JRxENOCViq2w9kymYDPqjW3IabF14FzQLUUFHtQfW3P1Q0kSwCKogxvaYbg58SDZwKNqv0E8NiQidkxHoWRkQy46e53xk+s8wQh0rbEwHO2f8bKZFGEhhbZdbMwixjQClh6lYFY5m17Jn8bqYyqAeynom0Cc2SEQiv/ZRHcQIsonMfYSIwKJxlg4dcMwpiagGhmtuvYDommlCwCVZsRs3lRFaBd9G4abgPl7XWbRFWGZ2gU3SOmugKtdA9aiMPUSTQK3pD786L8+F8Ol9zackpdo7RQjnff2gonSg=</latexit><latexit sha1_base64="GqPvuAJfnZZO3NgGi6xcpTtqYcE=">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</latexit>

Inversef<latexit sha1_base64="NQDPiKGUHm9/+KiRn5wz6HvhrUk=">AAACBnicZVDLTgIxFO3gC/GFutSYicTEBSEzbtQd0Y1LSBwhAUI6pQMN7XTS3jEhE5YujFv9DldGlv6G3+DKP7AzsBC4SXNPzz23OT1+xJkGx/m2ciura+sb+c3C1vbO7l5x/+BBy1gR6hHJpWr6WFPOQuoBA06bkaJY+Jw2/OFtOm88UqWZDO9hFNGOwP2QBYxgMFQ96BZLTsXJyl4G7gyUqseT+s/TyaTWLf62e5LEgoZAONa65ToRdBKsgBFOx4V2rGmEyRD3acvAEAuqO0lmdGyfGaZnB1KZE4Kdsf83Eiy0wDAwyrTpuVnKgJRcl40KBiJt6TPZXY+EX/ZFORUpHegFIxBcdRIWRjHQkEx9BDG3QdppKHaPKUqAjwzARDHzFZsMsMIETHQFk5G7mMgy8C4q1xWnbrK6QdPKoyN0is6Riy5RFd2hGvIQQRS9oFf0Zj1b79aH9TmV5qzZziGaK+vrD/ImnZ8=</latexit>
sha1_base64="wkf9utTHcQFbfvS3Vq9hZY9DMZU=">AAACBnicZVDLTgIxFL3jE/GFutTFRGLigpDBjbojunEJiSMkQEindKChnU7aOyZkwgcYt/odrowu/Q13/oJ/YAdYCNykuafnntucniAW3KDnfTsrq2vrG5u5rfz2zu7efuHg8MGoRFPmUyWUbgbEMMEj5iNHwZqxZkQGgjWC4W02bzwybbiK7nEUs44k/YiHnBK0VD3sFope2ZuUuwwqM1CsnnzWfwCg1i38tnuKJpJFSAUxplXxYuykRCOngo3z7cSwmNAh6bOWhRGRzHTSidGxe2aZnhsqbU+E7oT9v5ESaSTBgVVmzczNMgaVEqZkVTiQWcuemdzNSAalQJYykTahWTCC4VUn5VGcIIvo1EeYCBeVm4Xi9rhmFMXIAkI1t19x6YBoQtFGl7cZVRYTWQb+Rfm67NVtVjcwrRwcwymcQwUuoQp3UAMfKDB4hhd4dZ6cN+fd+ZhKV5zZzhHMlfP1B+ntnDk=</latexit>f<latexit sha1_base64="fjhIxwws58GsuvPWxj5BWS9qhG8=">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</latexit>
sha1_base64="T5O2W5fdBe9YVbRkg7tBseU/cmg=">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</latexit>

1

(xi )

(1)
hi,l

(1)

hi,r

<latexit sha1_base64="BSeNPzIye1eC45VcyN+Ot1HADN0=">AAACGnicZVDLSgMxFM3UV62vapdugkWoUMqMCOqu6MZlBWsLbS2ZNNOGJpMhuSOUYT5F3Op3uBK3bvwM/8BM24VtL4R7cu654eT4keAGXPfHya2tb2xu5bcLO7t7+wfFw6NHo2JNWZMqoXTbJ4YJHrImcBCsHWlGpC9Yyx/fZvPWM9OGq/ABJhHrSTIMecApAUv1i6WuL5NR2k94FYv0Kal4Z2m/WHZr7rTwKvDmoIzm1egXf7sDRWPJQqCCGNPx3Ah6CdHAqWBpoRsbFhE6JkPWsTAkkpleMjWf4lPLDHCgtD0h4Cn7fyMh0kgCI6vMmlmYZQwoJUzVqmAks5Y9M72bifSrvqxmIm0Cs2QEgqtewsMoBhbSmY8gFhgUzoLCA64ZBTGxgFDN7VcwHRFNKNg4CzYjbzmRVdA8r13X3PuLcv1mHlYeHaMTVEEeukR1dIcaqIkomqBX9IbenRfnw/l0vmbSnDPfKaGFcr7/AHw0oVI=</latexit>
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ei,l
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sha1_base64="ZndHDGIBuRmtjyQp3+WY268Rqpk=">AAACGnicZVBNTwIxEJ31E/AL5ehlIzHxQMjiRb0RvXjExBUSINgtXWhot5t21oRs+CEejFf9HZ6MVy/+DP+BXeAgMEkzr2/etDMviAU36Hk/ztr6xubWdi5f2Nnd2z8oHh49GJVoynyqhNKtgBgmeMR85ChYK9aMyECwZjC6yerNJ6YNV9E9jmPWlWQQ8ZBTgpbqFUudQIm+GUub0g5yycykVyx7VW8a7iqozUG5no+fHwGg0Sv+dvqKJpJFSAUxpl3zYuymRCOngk0KncSwmNARGbC2hRGxv3TT6fAT99QyfTdU2p4I3Sn7vyMl0kiCQ6vMklmoZQwqJUzFqnAos5Q9M71nW1UCWclE2oRmaRAML7spj+IEWURnc4SJcFG5mVFun2tGUYwtIFRzu4pLh0QTitbOgvWotuzIKvDPq1dV7856dQ2zyMExnMAZ1OAC6nALDfCBwhhe4Q3enRfnw/l0vmbSNWfeU4KFcL7/AOnOpC4=</latexit>(1)hi,l = xi,l<latexit sha1_base64="ecOX4VbbnENfsay0Sy5ZWNwjfdg=">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</latexit>+=<latexit sha1_base64="J710iawjlDt3Kcch2yC+lHTgmcU=">AAACE3icZVBNSwMxEM3Wr7p+VT16WSyCh1K2XtRDsejFYwVrC+1astlsG5pslmRWKEv/hnhV/4YnETx5Fn+G/8Bs24NtB8K8vHkzzDw/5kyD6/5YuaXlldW1/Lq9sbm1vVPY3bvTMlGENojkUrV8rClnEW0AA05bsaJY+Jw2/cFVVm8+UKWZjG5hGFNP4F7EQkYwGOq+40se6KEwKa2OuoWiW3bH4SyCyhQULz7tavz6bde7hd9OIEkiaASEY63bFTcGL8UKGOF0ZHcSTWNMBrhH2wZGWFDtpeOtR86RYQInlMq8CJwx+78jxUILDH2jzJKeqWUMSMl1yaigL7KUjRn/s4NKvihlIqVDPbcIhGdeyqI4ARqRyR5hwh2QTuaQEzBFCfChAZgoZk5xSB8rTMD4aBuPKvOOLILGSfm87N64xdolmkQeHaBDdIwq6BTV0DWqowYiSKEn9IxerEfrzXq3PibSnDXt2UczYX39ASoHoxk=</latexit><latexit
sha1_base64="tt7sHfQoCO2k4YfwNUEEi7Nc9MQ=">AAACE3icZVDNSgMxEJ6tf7X+VT16WSyCh1K2XtRDsejFYwVrC20t2TTbhiabJZkVylLwKcSrPocnETz5AD6Gb2C29WDbgTBfvvlmmPn8SHCDnvftZJaWV1bXsuu5jc2t7Z387t6dUbGmrE6VULrpE8MED1kdOQrWjDQj0hes4Q+v0nrjgWnDVXiLo4h1JOmHPOCUoKXu274SPTOSNiWVcTdf8EreJNxFUP4DhYvPXOURAGrd/E+7p2gsWYhUEGNaZS/CTkI0cirYONeODYsIHZI+a1kYEslMJ5lsPXaPLNNzA6XtC9GdsP87EiKNJDiwyjSZmVrKoFLCFK0KBzJN6ZjJPz2o6MtiKtImMHOLYHDWSXgYxchCOt0jiIWLyk0dcntcM4piZAGhmttTXDogmlC0PuasR+V5RxZB/aR0XvJuvEL1EqaRhQM4hGMowylU4RpqUAcKGp7hBV6dJ+fNeXc+ptKM89ezDzPhfP0CDOChpQ==</latexit><latexit sha1_base64="3duua+im21Knd6Mo/ulJRO+F6GY=">AAACC3icZVDLTgIxFO3gC8cX6tLNRGLigpDBjbowEt24xMQBEiCkUzpQaaeT9o4JmfALxq1u/QVXxq3xE4yf4R/YGVgI3KS5p+ee25weP+JMg+v+WLml5ZXVtfy6vbG5tb1T2N2raxkrQj0iuVRNH2vKWUg9YMBpM1IUC5/Thj+8TueNB6o0k+EdjCLaEbgfsoARDIaqt0lPgu4Wim7ZzcpZBJUpKF5+2RfR67dd6xZ+2z1JYkFDIBxr3aq4EXQSrIARTsd2O9Y0wmSI+7RlYIgF1Z0kczt2jgzTcwKpzAnBydj/GwkWWmAYGGXa9MwsZUBKrktGBQORtvSZ7K5Hwi/5opSKlA70nBEIzjoJC6MYaEgmPoKYOyCdNBmnxxQlwEcGYKKY+YpDBlhhAiY/22RUmU9kEXgn5fOye+sWq1doUnl0gA7RMaqgU1RFN6iGPETQPXpCz+jFerTerHfrYyLNWdOdfTRT1ucfeoefeg==</latexit>
sha1_base64="n3mWCM7/BmwuNjxBIg9VkyIruU0=">AAACC3icZVDLTgIxFL2DL8QX6tLNRGLigpDBjbowEt24xMQBEiCkUzpQaaeT9o4JISR+gXGr3+HKuDV+g5/hH9gBFwI3ae7puec2pyeIBTfoed9OZml5ZXUtu57b2Nza3snv7tWMSjRlPlVC6UZADBM8Yj5yFKwRa0ZkIFg9GFyn8/oD04ar6A6HMWtL0ot4yClBS9VatKvQdPIFr+RNyl0E5T9QuPzKXTwCQLWT/2l1FU0ki5AKYkyz7MXYHhGNnAo2zrUSw2JCB6THmhZGRDLTHk3cjt0jy3TdUGl7InQn7P+NEZFGEuxbZdrMzCxlUClhilaFfZm29JnJ3QxlUAxkMRVpE5o5IxietUc8ihNkEZ36CBPhonLTZNwu14yiGFpAqOb2Ky7tE00o2vxyNqPyfCKLwD8pnZe8W69QuYJpZeEADuEYynAKFbiBKvhA4R6e4QVenSfnzXl3PqbSjPO3sw8z5Xz+Al1gngY=</latexit>

(2)
hi,l

<latexit sha1_base64="3duua+im21Knd6Mo/ulJRO+F6GY=">AAACC3icZVDLTgIxFO3gC8cX6tLNRGLigpDBjbowEt24xMQBEiCkUzpQaaeT9o4JmfALxq1u/QVXxq3xE4yf4R/YGVgI3KS5p+ee25weP+JMg+v+WLml5ZXVtfy6vbG5tb1T2N2raxkrQj0iuVRNH2vKWUg9YMBpM1IUC5/Thj+8TueNB6o0k+EdjCLaEbgfsoARDIaqt0lPgu4Wim7ZzcpZBJUpKF5+2RfR67dd6xZ+2z1JYkFDIBxr3aq4EXQSrIARTsd2O9Y0wmSI+7RlYIgF1Z0kczt2jgzTcwKpzAnBydj/GwkWWmAYGGXa9MwsZUBKrktGBQORtvSZ7K5Hwi/5opSKlA70nBEIzjoJC6MYaEgmPoKYOyCdNBmnxxQlwEcGYKKY+YpDBlhhAiY/22RUmU9kEXgn5fOye+sWq1doUnl0gA7RMaqgU1RFN6iGPETQPXpCz+jFerTerHfrYyLNWdOdfTRT1ucfeoefeg==</latexit>
sha1_base64="n3mWCM7/BmwuNjxBIg9VkyIruU0=">AAACC3icZVDLTgIxFL2DL8QX6tLNRGLigpDBjbowEt24xMQBEiCkUzpQaaeT9o4JISR+gXGr3+HKuDV+g5/hH9gBFwI3ae7puec2pyeIBTfoed9OZml5ZXUtu57b2Nza3snv7tWMSjRlPlVC6UZADBM8Yj5yFKwRa0ZkIFg9GFyn8/oD04ar6A6HMWtL0ot4yClBS9VatKvQdPIFr+RNyl0E5T9QuPzKXTwCQLWT/2l1FU0ki5AKYkyz7MXYHhGNnAo2zrUSw2JCB6THmhZGRDLTHk3cjt0jy3TdUGl7InQn7P+NEZFGEuxbZdrMzCxlUClhilaFfZm29JnJ3QxlUAxkMRVpE5o5IxietUc8ihNkEZ36CBPhonLTZNwu14yiGFpAqOb2Ky7tE00o2vxyNqPyfCKLwD8pnZe8W69QuYJpZeEADuEYynAKFbiBKvhA4R6e4QVenSfnzXl3PqbSjPO3sw8z5Xz+Al1gngY=</latexit><latexit sha1_base64="REkAECskl6vTPQb6KrE5yd3HUvc=">AAACDXicZVDLSgMxFM3UV62vqks3wSK4KGUqgrorunFZwbGFdiiZNNOGJpMhuSOUof8gbvU7XIlbv8HP8A/MTGdh2wvhnpx7bjg5QSy4Adf9cUpr6xubW+Xtys7u3v5B9fDoyahEU+ZRJZTuBsQwwSPmAQfBurFmRAaCdYLJXTbvPDNtuIoeYRozX5JRxENOCViq2w9kymYDPqjW3IabF14FzQLUUFHtQfW3P1Q0kSwCKogxvaYbg58SDZwKNqv0E8NiQidkxHoWRkQy46e53xk+s8wQh0rbEwHO2f8bKZFGEhhbZdbMwixjQClh6lYFY5m17Jn8bqYyqAeynom0Cc2SEQiv/ZRHcQIsonMfYSIwKJxlg4dcMwpiagGhmtuvYDommlCwCVZsRs3lRFaBd9G4abgPl7XWbRFWGZ2gU3SOmugKtdA9aiMPUSTQK3pD786L8+F8Ol9zackpdo7RQjnff2gonSg=</latexit>

(1)

···

···

ei

<latexit sha1_base64="9nq30MFPFBgzWoiP8ja7QQU/IdQ=">AAACE3icZVDNSsNAEN7Uv1r/qh69BIsgWEoqgnorevFYwdhCG8tms2mX7mbD7kQooa8hXvU5PIlXH8DH8A3cpDnYdmCZb7/5Zpj5/JgzDY7zY5VWVtfWN8qbla3tnd296v7Bo5aJItQlkkvV9bGmnEXUBQacdmNFsfA57fjj26zeeaZKMxk9wCSmnsDDiIWMYDDUU9+XPNATYVJ6Nh1Ua07DycNeBs0C1FAR7UH1tx9IkggaAeFY617TicFLsQJGOJ1W+ommMSZjPKQ9AyMsqPbSfOupfWKYwA6lMi8CO2f/d6RYaIFhZJRZ0nO1jAEpua4bFYxElrIx+T87qO6LeiZSOtQLi0B45aUsihOgEZntESbcBmlnDtkBU5QAnxiAiWLmFJuMsMIEjI8V41Fz0ZFl4J43rhvO/UWtdVOYVUZH6Bidoia6RC10h9rIRQQp9Ire0Lv1Yn1Yn9bXTFqyip5DNBfW9x9LXZ/K</latexit><latexit<latexit sha1_base64="cJNeTq2Nhpm5E1hwBlQIZx9c5Vc=">AAACGnicZVDLSgMxFM34rPU12qWbYBEqlDItgrorunFZwdpCW0smzbShyWRI7ghlmE8Rt/odrsStGz/DPzDTdmHbC+GenHtuODl+JLgBz/tx1tY3Nre2czv53b39g0P36PjRqFhT1qRKKN32iWGCh6wJHARrR5oR6QvW8se32bz1zLThKnyAScR6kgxDHnBKwFJ9t9D1ZTJK+wkvY5E+JaXaedp3i17FmxZeBdU5KKJ5Nfrub3egaCxZCFQQYzpVL4JeQjRwKlia78aGRYSOyZB1LAyJZKaXTM2n+MwyAxwobU8IeMr+30iINJLAyCqzZhZmGQNKCVO2KhjJrGXPTO9mIv2yL8uZSJvALBmB4KqX8DCKgYV05iOIBQaFs6DwgGtGQUwsIFRz+xVMR0QTCjbOvM2oupzIKmjWKtcV7/6iWL+Zh5VDJ+gUlVAVXaI6ukMN1EQUTdArekPvzovz4Xw6XzPpmjPfKaCFcr7/AH3YoVM=</latexit>

hi,r = xi,r + g(xi,l )<latexit sha1_base64="GqPvuAJfnZZO3NgGi6xcpTtqYcE=">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</latexit>

(2)
hi,r
<latexit sha1_base64="PLD+o/nimrkoifAVqZrGlSo/hiw=">AAACGnicZVDLSgMxFM34rPU12qWbYBEqlDItgrorunFZwdpCW0smzbShyWRI7ghlmE8Rt/odrsStGz/DPzDTdmHbC+GenHtuODl+JLgBz/tx1tY3Nre2czv53b39g0P36PjRqFhT1qRKKN32iWGCh6wJHARrR5oR6QvW8se32bz1zLThKnyAScR6kgxDHnBKwFJ9t9D1ZTJK+wkvY50+JaXaedp3i17FmxZeBdU5KKJ5Nfrub3egaCxZCFQQYzpVL4JeQjRwKlia78aGRYSOyZB1LAyJZKaXTM2n+MwyAxwobU8IeMr+30iINJLAyCqzZhZmGQNKCVO2KhjJrGXPTO9mIv2yL8uZSJvALBmB4KqX8DCKgYV05iOIBQaFs6DwgGtGQUwsIFRz+xVMR0QTCjbOvM2oupzIKmjWKtcV7/6iWL+Zh5VDJ+gUlVAVXaI6ukMN1EQUTdArekPvzovz4Xw6XzPpmjPfKaCFcr7/AIfOoVk=</latexit>+

Inversef<latexit sha1_base64="NQDPiKGUHm9/+KiRn5wz6HvhrUk=">AAACBnicZVDLTgIxFO3gC/GFutSYicTEBSEzbtQd0Y1LSBwhAUI6pQMN7XTS3jEhE5YujFv9DldGlv6G3+DKP7AzsBC4SXNPzz23OT1+xJkGx/m2ciura+sb+c3C1vbO7l5x/+BBy1gR6hHJpWr6WFPOQuoBA06bkaJY+Jw2/OFtOm88UqWZDO9hFNGOwP2QBYxgMFQ96BZLTsXJyl4G7gyUqseT+s/TyaTWLf62e5LEgoZAONa65ToRdBKsgBFOx4V2rGmEyRD3acvAEAuqO0lmdGyfGaZnB1KZE4Kdsf83Eiy0wDAwyrTpuVnKgJRcl40KBiJt6TPZXY+EX/ZFORUpHegFIxBcdRIWRjHQkEx9BDG3QdppKHaPKUqAjwzARDHzFZsMsMIETHQFk5G7mMgy8C4q1xWnbrK6QdPKoyN0is6Riy5RFd2hGvIQQRS9oFf0Zj1b79aH9TmV5qzZziGaK+vrD/ImnZ8=</latexit>
sha1_base64="wkf9utTHcQFbfvS3Vq9hZY9DMZU=">AAACBnicZVDLTgIxFL3jE/GFutTFRGLigpDBjbojunEJiSMkQEindKChnU7aOyZkwgcYt/odrowu/Q13/oJ/YAdYCNykuafnntucniAW3KDnfTsrq2vrG5u5rfz2zu7efuHg8MGoRFPmUyWUbgbEMMEj5iNHwZqxZkQGgjWC4W02bzwybbiK7nEUs44k/YiHnBK0VD3sFope2ZuUuwwqM1CsnnzWfwCg1i38tnuKJpJFSAUxplXxYuykRCOngo3z7cSwmNAh6bOWhRGRzHTSidGxe2aZnhsqbU+E7oT9v5ESaSTBgVVmzczNMgaVEqZkVTiQWcuemdzNSAalQJYykTahWTCC4VUn5VGcIIvo1EeYCBeVm4Xi9rhmFMXIAkI1t19x6YBoQtFGl7cZVRYTWQb+Rfm67NVtVjcwrRwcwymcQwUuoQp3UAMfKDB4hhd4dZ6cN+fd+ZhKV5zZzhHMlfP1B+ntnDk=</latexit>f<latexit sha1_base64="fjhIxwws58GsuvPWxj5BWS9qhG8=">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</latexit>
sha1_base64="T5O2W5fdBe9YVbRkg7tBseU/cmg=">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</latexit>

1

(xi )

=
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sha1_base64="cABDScOf/SzJSeJxwEj+ZulgKro=">AAACDXicZVDLTgIxFL2DL8QXauLGzURi4oKQwY26I7pxCYkICRDslA40tNNJe8dIJvyDcavf4cq49Rvc+A/+gR1gIXCT5p6ee25zevxIcIOe9+1kVlbX1jeym7mt7Z3dvfz+wb1RsaasTpVQuukTwwQPWR05CtaMNCPSF6zhD2/SeeORacNVeIejiHUk6Yc84JSgpZptXyZP4y7v5gteyZuUuwzKM1CoHNV+HgCg2s3/tnuKxpKFSAUxplX2IuwkRCOngo1z7diwiNAh6bOWhSGRzHSSid+xe2qZnhsobU+I7oT9v5EQaSTBgVWmzczNUgaVEqZoVTiQaUufmdzNSPpFXxZTkTaBWTCCwWUn4WEUIwvp1EcQCxeVm2bj9rhmFMXIAkI1t19x6YBoQtEmmLMZlRcTWQb189JVyavZrK5hWlk4hhM4gzJcQAVuoQp1oCDgBV7hzXl23p0P53MqzTiznUOYK+frD02knxw=</latexit>[Dinh et al. 2014]40

Experiments• Dataset: English Penn Treebank• POS tagging
Trained and tested on whole PTB• Grammar induction
Trained on sentences of length <= 10 in section 2-21
Tested on sentences in section 23

41

Part-of-speech Induction

Many-to-1 %Feature-based 80.2

75

50

75.4

80.8

VM

100

100

75

Feature-based 72.1
68.5

62.7

HM

Gaussia

Neural

50

74.1

53.8

HM

Gaussia

Neural42

Part-of-speech Induction

Many-to-1 %Feature-based 80.2

75

50

75.4

80.8

VM

100

100

75

Feature-based 72.1
68.5

62.7

HM

Gaussia

Neural

50

74.1

53.8

HM

Gaussia

NeuralOutperform feature-based SOTA
42

75

75

Oracle POS
50

25

49.6

DM

55.4

60.2

Directed % on all≤

Directed % on len <= 10

Dependency Parse Induction≤

50

Oracle POS
35.8
25

Gaussi

Neural

43.1

47.9

DM

Gaussi

Neural43

Original Embedding Space

adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number

44

Projected Embedding Space w/ Markov Priorzi<latexit sha1_base64="4gmLgsVwRggpXTovfy6uYJcrnUQ=">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</latexit>
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sha1_base64="UtagCrqX87KPW4tGKv92duQD8gc=">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</latexit>

ei<latexit sha1_base64="86pgXmDcE2XVlZKOeiFEsb42X04=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="znhKAUtEwogmaPWollZUEdE8utY=">AAACQHicZVE9SwNBEJ3z2/gZW5vDIFiEcLFRO8HGMoJRwQthbzNnFvfj2J1Tw3F/wVZ/lb3/w0os3EtSGDOwzNs3M8vbN0kmhaMo+gwWFpeWV1bX1msbm1vbO7v1zRtncsuxy4009i5hDqXQ2CVBEu8yi0wlEm+Tx4uqfvuE1gmjr2mUYU+xBy1SwRlVVIx90d9tRK1oHOE8aE9BA6bR6deD43hgeK5QE5fMuft2lFGvYJYEl1jW4txhxvgje8B7DzVT6HrFWGwZHnpmEKbG+qMpHLN/JwqmnGI09J1VcjO1iiFjpGv6LhqqKlXPjO9upJJmoppVk3Wp+yeE0tNeIXSWE2o+0ZHmMiQTVsaEA2GRkxx5wLgV/ishHzLLOHn7ZkQkqqzVYo3P3CjF9KCIX8oiTlTxUpazPE549Lx3uf3f03nQPW6dtaKrCNZgHw7gCNpwAudwCR3oAochvMIbvAcfwVfwPVnHQjDdyx7MRPDzCypZtBY=</latexit>
sha1_base64="ViRgb/9Jv45ewVl+QYmhz6smaZo=">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</latexit>

x
f

1

(e; )

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xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>

45

Projected Embedding Space w/ Markov Priorzi<latexit sha1_base64="4gmLgsVwRggpXTovfy6uYJcrnUQ=">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</latexit>
sha1_base64="jBAzhNAJ26J1i9dTzZ4SO1PLZqY=">AAACOXicZVHLTsMwEHTKq4Ty6JlLBELiUFVpL8ANiQtHkAhUIhVynA216kdkb6BVlB/gylfxCXwFJ8QVp+2BlpWsHc/uWuPZJBfcYhh+eo219Y3Nrea2v9Pyd/f2D1r3VheGQcS00GaQUAuCK4iQo4BBboDKRMBDMr6q6w8vYCzX6g6nOQwlfVY844yio26eDo7DbjiL4D/oLcAxWcRT2+vHqWaFBIVMUGsfe2GOw5Ia5ExA5ceFhZyyMX2GRwcVlWCH5UxnFZw4Jg0ybdxRGMzYvxMllVZSHLnOOtmlWs2g1sJ2XBeOZJ3qZ2Z3O5VJJ5GdusnYzK4Iwex8WHKVFwiKzXVkhQhQB7UnQcoNMBRTBygz3H0lYCNqKEPn3JKIRFa+Hyt4ZVpKqtIynlRlnMhyUlXLPMx5cLwzubdq6X8Q9bsX3fA2JE1ySI7IKemRM3JJrskNiQgjKXkj796H9+V9z3fR8BZLaZOl8H5+AY4YsXk=</latexit>
sha1_base64="Yd99wiwenul3NLx8OscdFfA1yQk=">AAACP3icZVG7TsNAEFzzJrxpaSwQEkUUOTRAh0RDCYIAEo6i82WdnLiHdbeGBMufQAtfxQfwHVSIhnOSgsBKp52b3T3NzSaZFI6i6COYmZ2bX1hcWq6trK6tb2xurd44k1uOLW6ksXcJcyiFxhYJkniXWWQqkXibPJxV9dtHtE4YfU3DDNuK9bRIBWfkqavnjuhs7kWNaBThf9CcgD2YxEVnKziMu4bnCjVxyZy7b0YZtQtmSXCJZS3OHWaMP7Ae3nuomULXLkZay3DfM90wNdYfTeGI/T1RMOUUo77vrJKbqlUMGSNd3XdRX1WpemZ0d0OV1BNVr5qsS90fIZQetwuhs5xQ87GONJchmbDyJewKi5zk0APGrfBfCXmfWcbJuzclIlFlrRZrfOJGKaa7RTwoizhRxaAsp3kc8+h573Lzr6f/QeuwcdKILiNYgh3YhQNowhGcwjlcQAs49OAFXuEteA8+g6/xOmaCyV62YSqC7x+BKbPG</latexit>
sha1_base64="UtagCrqX87KPW4tGKv92duQD8gc=">AAACSnicZVC7TgMxEPSF8AqvACXNiQiJIoouNECHoKEMggASF0U+Zy+x4sfJ3gPC6T6BFr6KD+A7qBANviQFCStZO56dtcYTJYJbDIJPr7RQXlxaXlmtrK1vbG5Vt3durU4NgzbTQpv7iFoQXEEbOQq4TwxQGQm4i4YXxfzuEYzlWt3gKIGOpH3FY84oOur6pcu71VrQCMbl/wfNKaiRabW6295R2NMslaCQCWrtQzNIsJNRg5wJyCthaiGhbEj78OCgohJsJxt7zf0Dx/T8WBt3FPpj9u9GRqWVFAdOWTQ7MysY1FrYulPhQBateGZ8tyMZ1SNZL0TGxnbOCMYnnYyrJEVQbOIjToWP2i9y8XvcAEMxcoAyw91XfDaghjJ06c2YiGReqYQKnpiWkqpeFj7nWRjJ7DnPZ3mY8OB4l3JzPtP/oH3UOG0EV0Ht7Hwa9wrZI/vkkDTJMTkjl6RF2oSRPnklb+Td+/C+vG/vZyItedOdXTJTpfIvTPa0eA==</latexit>

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sha1_base64="ViRgb/9Jv45ewVl+QYmhz6smaZo=">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</latexit>

x
f

1

(e; )

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xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>

45

Projected Embedding Space w/ DMV Prior

adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number

46

Projected Embedding Space w/ DMV Prior

adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number

46

Projected Embedding Space w/ DMV Prior

adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number

46

Projected Embedding Space w/ DMV Prior

adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number

47

Projected Embedding Space w/ DMV Prior

adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number

48

Conclusion

49

Conclusion

50

Conclusion• Normalizing flows for unsupervised learning
X = f✓ 1 (Z)<latexit sha1_base64="/fc4xHZsNC/YMH6XXAT25PjHrcA=">AAAB/3icbVDLSgNBEJyNrxhfq4IXL4NBiAfDbhT0IgS9eIxgHpisYXYymwyZfTDTK4R1D/6KFw+KePU3vPk3TpI9aGJBQ1HVTXeXGwmuwLK+jdzC4tLySn61sLa+sbllbu80VBhLyuo0FKFsuUQxwQNWBw6CtSLJiO8K1nSHV2O/+cCk4mFwC6OIOT7pB9zjlICWuuZeC19g7z45ttNu0oEBA5KW7o66ZtEqWxPgeWJnpIgy1LrmV6cX0thnAVBBlGrbVgROQiRwKlha6MSKRYQOSZ+1NQ2Iz5STTO5P8aFWetgLpa4A8ET9PZEQX6mR7+pOn8BAzXpj8T+vHYN37iQ8iGJgAZ0u8mKBIcTjMHCPS0ZBjDQhVHJ9K6YDIgkFHVlBh2DPvjxPGpWyfVKu3JwWq5dZHHm0jw5QCdnoDFXRNaqhOqLoET2jV/RmPBkvxrvxMW3NGdnMLvoD4/MHL4GU7g==</latexit>

Z = f✓ (X)<latexit sha1_base64="IDq8cdH7NY313/u9S3rZypQ+wFI=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSxCvZSkCnoRil48VrAf2Iay2W7apZtN2J0oJfanePGgiFd/iTf/jds2B219MPB4b4aZeX4suAbH+bZyK6tr6xv5zcLW9s7unl3cb+ooUZQ1aCQi1faJZoJL1gAOgrVjxUjoC9byR9dTv/XAlOaRvINxzLyQDCQPOCVgpJ5dvMeXOOilXRgyIJNy+6Rnl5yKMwNeJm5GSihDvWd/dfsRTUImgQqidcd1YvBSooBTwSaFbqJZTOiIDFjHUElCpr10dvoEHxulj4NImZKAZ+rviZSEWo9D33SGBIZ60ZuK/3mdBIILL+UyToBJOl8UJAJDhKc54D5XjIIYG0Ko4uZWTIdEEQomrYIJwV18eZk0qxX3tFK9PSvVrrI48ugQHaEyctE5qqEbVEcNRNEjekav6M16sl6sd+tj3pqzspkD9AfW5w/CxZMI</latexit>

50

Conclusion• Normalizing flows for unsupervised learning
X = f✓ 1 (Z)<latexit sha1_base64="/fc4xHZsNC/YMH6XXAT25PjHrcA=">AAAB/3icbVDLSgNBEJyNrxhfq4IXL4NBiAfDbhT0IgS9eIxgHpisYXYymwyZfTDTK4R1D/6KFw+KePU3vPk3TpI9aGJBQ1HVTXeXGwmuwLK+jdzC4tLySn61sLa+sbllbu80VBhLyuo0FKFsuUQxwQNWBw6CtSLJiO8K1nSHV2O/+cCk4mFwC6OIOT7pB9zjlICWuuZeC19g7z45ttNu0oEBA5KW7o66ZtEqWxPgeWJnpIgy1LrmV6cX0thnAVBBlGrbVgROQiRwKlha6MSKRYQOSZ+1NQ2Iz5STTO5P8aFWetgLpa4A8ET9PZEQX6mR7+pOn8BAzXpj8T+vHYN37iQ8iGJgAZ0u8mKBIcTjMHCPS0ZBjDQhVHJ9K6YDIgkFHVlBh2DPvjxPGpWyfVKu3JwWq5dZHHm0jw5QCdnoDFXRNaqhOqLoET2jV/RmPBkvxrvxMW3NGdnMLvoD4/MHL4GU7g==</latexit>

Z = f✓ (X)<latexit sha1_base64="IDq8cdH7NY313/u9S3rZypQ+wFI=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSxCvZSkCnoRil48VrAf2Iay2W7apZtN2J0oJfanePGgiFd/iTf/jds2B219MPB4b4aZeX4suAbH+bZyK6tr6xv5zcLW9s7unl3cb+ooUZQ1aCQi1faJZoJL1gAOgrVjxUjoC9byR9dTv/XAlOaRvINxzLyQDCQPOCVgpJ5dvMeXOOilXRgyIJNy+6Rnl5yKMwNeJm5GSihDvWd/dfsRTUImgQqidcd1YvBSooBTwSaFbqJZTOiIDFjHUElCpr10dvoEHxulj4NImZKAZ+rviZSEWo9D33SGBIZ60ZuK/3mdBIILL+UyToBJOl8UJAJDhKc54D5XjIIYG0Ko4uZWTIdEEQomrYIJwV18eZk0qxX3tFK9PSvVrrI48ugQHaEyctE5qqEbVEcNRNEjekav6M16sl6sd+tj3pqzspkD9AfW5w/CxZMI</latexit>Space• Learning ofJapanesebilinguallexicons

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

50

Conclusion• Normalizing flows for unsupervised learning
X = f✓ 1 (Z)<latexit sha1_base64="/fc4xHZsNC/YMH6XXAT25PjHrcA=">AAAB/3icbVDLSgNBEJyNrxhfq4IXL4NBiAfDbhT0IgS9eIxgHpisYXYymwyZfTDTK4R1D/6KFw+KePU3vPk3TpI9aGJBQ1HVTXeXGwmuwLK+jdzC4tLySn61sLa+sbllbu80VBhLyuo0FKFsuUQxwQNWBw6CtSLJiO8K1nSHV2O/+cCk4mFwC6OIOT7pB9zjlICWuuZeC19g7z45ttNu0oEBA5KW7o66ZtEqWxPgeWJnpIgy1LrmV6cX0thnAVBBlGrbVgROQiRwKlha6MSKRYQOSZ+1NQ2Iz5STTO5P8aFWetgLpa4A8ET9PZEQX6mR7+pOn8BAzXpj8T+vHYN37iQ8iGJgAZ0u8mKBIcTjMHCPS0ZBjDQhVHJ9K6YDIgkFHVlBh2DPvjxPGpWyfVKu3JwWq5dZHHm0jw5QCdnoDFXRNaqhOqLoET2jV/RmPBkvxrvxMW3NGdnMLvoD4/MHL4GU7g==</latexit>

Z = f✓ (X)<latexit sha1_base64="IDq8cdH7NY313/u9S3rZypQ+wFI=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSxCvZSkCnoRil48VrAf2Iay2W7apZtN2J0oJfanePGgiFd/iTf/jds2B219MPB4b4aZeX4suAbH+bZyK6tr6xv5zcLW9s7unl3cb+ooUZQ1aCQi1faJZoJL1gAOgrVjxUjoC9byR9dTv/XAlOaRvINxzLyQDCQPOCVgpJ5dvMeXOOilXRgyIJNy+6Rnl5yKMwNeJm5GSihDvWd/dfsRTUImgQqidcd1YvBSooBTwSaFbqJZTOiIDFjHUElCpr10dvoEHxulj4NImZKAZ+rviZSEWo9D33SGBIZ60ZuK/3mdBIILL+UyToBJOl8UJAJDhKc54D5XjIIYG0Ko4uZWTIdEEQomrYIJwV18eZk0qxX3tFK9PSvVrrI48ugQHaEyctE5qqEbVEcNRNEjekav6M16sl6sd+tj3pqzspkD9AfW5w/CxZMI</latexit>Space• Learning ofJapanesebilinguallexicons

English Space
dog

澆

ሞ

caninemapping function

ᇦ

cat

bird

• Learning of syntactic structure

The cat sat on

a green wall50

Thank You! Questions?

https://github.com/violet-zct/DeMa-BWE

The cat sat on

a green wall

https://github.com/jxhe/struct-learning-with-flow51