Carnegie Mellon University

Learning with
Latent Linguistic Structure
Graham Neubig@ BlackBoxNLP 11/1/2018
C

with:

Junxian He

Pengcheng Yin

Chunting Zhou

Taylor Berg-Kirkpatrick

How to Achieve Interpretability in Neural Nets?

Black Box

Glass Box!
Analysis
SyntaxSemantics

SyntaxSemantics

Research Problems• Fundamentally highly interpretable models (e.g. discrete
HMMs) are not sufficiently powerful• How can we harness the power of neural networks, with
underlying interpretable representations?
• How can we learn them on unlabeled data?

Carnegie Mellon University

e.g. Syntactic Analysis
Dependency:
Parts-of-speech:

DT NN VBD IN DT JJ

The cat sat on!4NN

a green wallCarnegie Mellon University

Supervised Approach

X

Y
!5

Carnegie Mellon University

Supervised Approach
Supervised Learning

X

✓

X

Y

Y
!6

Carnegie Mellon University

Supervised Approach
Supervised Learning

X

✓

X

Y

Y
!7

Carnegie Mellon University

Latent Variable Approach
Unsupervised Learning

X

✓

Y

Y

?
?

X

?
!8

?

Carnegie Mellon University

Latent Variable Approach
Unsupervised Learning

X

✓

Y

Y

?
?

X

?
!9

?

Carnegie Mellon University

Multi-space Variational
Encoder-Decoders
C

Chunting Zhou and Graham Neubig(ACL 2017)10

Features of Words• Syntax:• What syntactic features does the word have?
• Closed-class, generally enumerable for a specific language.
• Meaning/Symbol:• What is the meaning of the word, how is it spelled/
pronounced?
• Open-class, complicated regularities and relationships.
• Can we create a model that elegantly models both?

Background: Variational Auto-encoder(Kingma et al., 2014, Bowman et al., 2016)
p

l

a

y

s




q (z|x): Recognition model/Encoderp✓ (x|z): Generation model/Decoder

p

l

a

y

s




Maximize the Variational lower bound:

log p✓ (x)Ez⇠q (z|x) [log p✓ (x|z)]KL(q (z|x)||p(z))

Proposed Model: Multi-space Variational Encoder-Decoders
Latent Continuous
Lemma
p

l

a

y

s 


Latent Discrete
+PP

p

l

a

y

e

d 


+PAST
+VERB

• Modeling complicated higher-level structure (e.g. meaning or
symbol of the word): incorporation of continuous latent variables• Modeling closed-class and interpretable features (e.g. syntax):
incorporation of discrete latent variables

plays, played, playing• How can we learn in a un- or semi-supervised way?

Variable Definitions
z :continuous latent variable(s)x : a source sequence
p

l

a

y

Lemma

(t)

x : a target sequences 


p

+PP

l

a

y

e

d 


+PAST
+VERB

y

(t)

(t) (t)(t)= [y1 , y2 , · · · yK ] :discrete labels for each target sequenceSupervised Learning:
Labeled Multi-space Variational Autoencoders
p

a

l

y

s 


Training Data:
surface form + labels
q (z|x) = N (z|µ (x), diag( 2 (x)))

Discrete Labels

Latent Continuous

+3

Lemma

+PRS

``play’’+VERB

Discriminative label classifier:Decoder: p✓ (x|y, z) = f (x|y, z; ✓)D(x, y) = E(x,y)⇠pl (x,y) [ log q (y|x)](with label attention)

p

l

a

y

s 


Maximize : U (x) = Variational Lower Bound of log p(x, y)Unsupervised Learning:
Unlabeled Multi-space Variational Auto-encoders
p

a

l

y

s 


Training Data:
surface form only
q (y|x) =q (z|x) = N (z|µ (x), diag( 2 (x)))``play’’YkLatent Continuous

+3

Lemma

+PRS

q (yk |x) =YkCat(yk |⇡ (x)Latent Discrete+VERB

Decoder: p✓ (x|y, z) = f (x|y, z; ✓)

p

l

a

y

s 


Maximize : U (x) = Variational Lower Bound of log p(x, y)Labeled Sequence-to-sequence Training:
Multi-space Variational Encoder-Decoders
p

a

l

y




Training Data:
two surface forms+ labelsq (z|x) = N (z|µ (x), diag( 2 (x)))

Latent Continuous

+3

Lemma

+PRS

``play’’Discrete Labels+VERB

Decoder: p✓ (x|y, z) = f (x|y, z; ✓)

p
(t)

(t)

Maximize : Ll (x , y |x

(s)

l

a

y

s 
(t)(t)) = Variational Lower Bound of log p(x , y |x(s))

Learning MSVED• Learning Continuous Latent Variables:
Reparameterization trick (Kingma et al., 2014):✏ ⇠ N (0, 1), z = µ (x) +(x) ✏• Learning Discrete Latent Variables:
Gumbel-Softmax (Maddison et al., 2017)

exp((log(⇡ij ) + gij )/⌧ )ŷij = PNi

k=1 exp((log(⇡ik ) + gik )/⌧• Training tricks (Bowman et al. 2016):•KL-divergence Annealing•Input dropout in the decoder

Experimental Setup
Task: Morphology re-inflection
Dataset: SIGMORPHON 2016 task 3
source word: communicatedtarget word: communicatestarget labels: V;3;SG;PRS

Language: Turkish, Arabic, Maltese, Finnish,
Spanish, German, Hungarian, Navajo,
Georgian, Russian

Results and Analysis

Helped bysemi-supervised training
Helped by
bi-directional training
Competitive w/
standard attention
Single-directional supervised model (SD-Sup);
Bidirectional supervised model (BD-Sup);
Semi-supervised model (Semi-Sup)Analysis on Tag Attention
a l - - i mā r ā t i y y ā t ucase=NOM
mood=None
pos=ADJ
per=None
poss=None
num=PLtense=None
aspect=Nonevoice=None
gen=FEM
def=DEF

Visualization of Latent Continuous Variables• Clusters colored by actual lemma:

TurkishTurkishMalteseMalteseStructVAE: Tree-structured Latent
Variable Models for Semi-supervised
Semantic Parsing
Pengcheng Yin, Chunting Zhou, Junxian He, Graham Neubig(ACL 2018)

What About More Complicated Structure?
Semantic Parsing: Transducing natural language utterances (e.g., queries) intomachine-executable formal meaning representations (e.g., logical form, source code)

Domain-Specific
Meaning Representations

General-Purpose
Programming Languages

Show me flights from Pittsburgh to
Washington

Sort my_list in descending order

lambda $0 e (and (flight $0)(from $0
san_Francisco:ci)(to $0 washington:ci))
lambda-calculus logical form

sorted(my_list,
reverse=True)
PythonResearch Issue
Neural Models are Data Hungry

Data Collection is Costly
Copy the content of file 'file.txt' to file 'file2.txt'
shutil.copy('file.txt','file2.txt')

Get a list of words `words` of a file 'myfile'
words = open('myfile').read().split()

Purely supervised neuralsemantic parsing models requirelarge amounts of training data

Check if all elements in list `mylist` are the samelen(set(mylist)) == 1

Collecting parallel training
data costs
and[Yin et al., 2018] 1700 USD for 3K Python code generation examples[Berant et al., 2013] 3000 USD for 5.7K question-to-logical form examples

Semi-supervised Semantic Parsing
Limited Amount of Labeled Data
Sort my_list in descending order

Extra Unlabeled Utterances
Get a list of words `words` of a file 'myfile'sorted(my_list, reverse=True)
Convert a list of integers into a single integer
Copy the content of file 'file.txt' tofile 'file2.txt'
shutil.copy('file.txt’,'file2.txt')
Check if all elements in list `mylist` are
the samelen(set(mylist)) == 1

Format a datetime object `when` toextract date only
Swap values in a tuple/list in list `mylist`
BeautifulSoup search string 'Elsie' inside
tag 'a'Convert string to lowercase

Meaning Representations as Tree-structured Latent Variables
Structured Latent Semantic Space

Prior

sorted(my_list, reverse=True)
Latent Meaning Representation(Abstract Syntax Trees)

Inference Model

Reconstruction Model

Posterior inference
corresponds tosemantic parsing

Sort my_list in descending orderSemi-supervised Learning with StructVAE
Labeled Data{

Unlabeled Data,{}

Structured Latent Semantic Space

}

Prior

Supervised Objective(,) Labeled Data

Unsupervised Objective

Inference Model

Reconstruction Model

Unlabeled Data

Sort my_list in descending orderp() = ∫ p(

|

) p(

)

StructVAE: VAEs with Tree-structured Latent Variables
Prior

Inference Model

Neural semantic parser

Neural sequence-to-sequence model Neural Language Model(use linearized trees as inputs)

Variational approximation of the marginal likelihood
Unsupervised Objective

Unlabeled Data

[Miao and Blunsom, 2016]

How does extra unlabeled data help learning?

Supervised Objective(,) Labeled Data

Training Examples[Miao and Blunsom, 2016]

How does extra unlabeled data help learning?
Unsupervised Objective

Unlabeled Data

The learning signal

SampledPrior

Learning signal acts as the tuning weights of gradients received by different
sampled latent meaning representations from the inference model[Miao and Blunsom, 2016]

How does extra unlabeled data help learning?
Sort my_list in descending orderPriorsorted(my_list)
1sorted(my_list,
reverse=True)
2

Prior

sorted(my_list,
descending=True)
3

Prior

Learning favors sampled latent meaning representations that both:– Faithfully encode the semantics of the utterance -> high reconstruction score– Are succinct and natural -> high prior probability

The Inference Model: a Transition-based Parser
Inference Model

A transition-based parser that transduces natural language
utterances into Abstract Syntax Trees

Gr ammar Spe c i f i c at i on
stmt

expr

FunctionDef(identifiler name,arguments args, stmt* body)| Expr(expr value)Call(expr func, expr* args,

Tr a ns i t i o n Sy s t e m

Abst r act Synt ax Tr ee

ApplyConstr(Expr)

Expr

ApplyConstr(Call)Call

keyword* keywords)| Name(identifier id)| Str(string id)

ApplyConstr(Name)

I nput Ut t e r anc e

Sort my_list in descending order

...

Name

sortedmy_listKeyword...

GenToken(sorted)Name[Yin and Neubig, 2017; Rabinovich et al. 2017]

Datasets
Django Python Code Generation
Task

ATIS Semantic Parsing Task

Call the function _generator, join
the result into a string, return the
resultShow me flights from San
Francisco to Washington

return ‘’.join(_generator())

lambda $0 e(and (flight $0)(from $0 san_Francisco:ci)(to $0 washington:ci))

Research Questions• RQ1 Does StructVAE outperforms purely supervised semantic parsers
with extra unlabeled data?
• RQ2 Can we get some empirical evidence about why StructVAE works?

StructVAE v.s. Baselines
Inference model as supervised parser
Self Training (semi-supervised baseline)
StructVAE
The gap is much more
obvious when we use
a mediocre parser ☺

all available training utterances as unlabeled dataWhy does StructVAE work?
• For each unlabeled utterance, compute the learning signal
for gold samples and other (imperfect) samples
Avg.=-5.12

Avg.=2.59

Other Samples

Gold Samples

Case Studies
Join p and cmd into a file path, substitute it for fqφ(|Learning Signal

Prior

Parser Score)f = os.path.join(p, cmd)

-1.00

-24.33

-2.00

9.14

p = path.join(p, cmd)

-8.12

-27.89-20.96-9.47

Split string pks by ‘,’ , substitute the result for primary_keys

qφ(|

Learning Signal

Prior

Parser Score)

primary_keys = pks.split(’,’)-2.38-10.24-11.392.05

primary_keys = pks.split + ’,’-1.83-20.41-14.87-2.60

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

Carnegie Mellon University

HMM for Part-of-Speech Induction
z1

z2

z3

x1

x2

x3

The

cat

sat

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Carnegie Mellon University

Gaussian HMM for POS Induction
z1

z2

z3

x1

x2

x3

The

cat

sat

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Carnegie Mellon University

Latent Embeddings w/ Neural Projectionzi ⇠ Markov Structure

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Carnegie Mellon University

Dependency Model with Valence

DT

NN

IN

NN

The cat stopped a

dog

in

Paris

DT

NN

VBD

!43

[Klein and Manning 2004]

Carnegie Mellon University

Dependency Model with Valence

DT

NN

DT

VBD

!44

NN

IN

NN

[Klein and Manning 2004]

Carnegie Mellon University

Dependency Parse Induction
from POS

?
DT

NN

DT

VBD

!45

NN

IN

NN

Carnegie Mellon University

Grammar Induction from Raw Text?
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Carnegie Mellon University

Grammar Induction from Raw Text

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The cat stopped a

dog

in

Paris

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Carnegie Mellon University

Latent Embeddings w/ Neural Projectionzi ⇠ Syntax Model

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Markov prior

Syntax
Model

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NeuralProjector

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DMV prior
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Carnegie Mellon University

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xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>!49

Carnegie Mellon University

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 ; ⌘, )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>!50

Carnegie Mellon University

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=">AAACP3icZVG7TsNAEFzzJrxpaSwQEkUUOTRAh0RDCYIAEo6i82WdnLiHdbeGBMufQAtfxQfwHVSIhnOSgsBKp52b3T3NzSaZFI6i6COYmZ2bX1hcWq6trK6tb2xurd44k1uOLW6ksXcJcyiFxhYJkniXWWQqkXibPJxV9dtHtE4YfU3DDNuK9bRIBWfkqavnjuhs7kWNaBThf9CcgD2YxEVnKziMu4bnCjVxyZy7b0YZtQtmSXCJZS3OHWaMP7Ae3nuomULXLkZay3DfM90wNdYfTeGI/T1RMOUUo77vrJKbqlUMGSNd3XdRX1WpemZ0d0OV1BNVr5qsS90fIZQetwuhs5xQ87GONJchmbDyJewKi5zk0APGrfBfCXmfWcbJuzclIlFlrRZrfOJGKaa7RTwoizhRxaAsp3kc8+h573Lzr6f/QeuwcdKILiNYgh3YhQNowhGcwjlcQAs49OAFXuEteA8+g6/xOmaCyV62YSqC7x+BKbPG</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=">AAACS3icZVBNSwMxEM1Wq7V+tXr0slgED6VsRVBvohePClYL3VKy6awN5mNJZtWy7F/wqr/Ku//Dk3gw2/Zg60CYlzdvwsuLEsEtBsGnV1paLq+sVtaq6xubW9u1+s6d1alh0GFaaNONqAXBFXSQo4BuYoDKSMB99HhZzO+fwFiu1S2OE+hL+qB4zBnFggphwAe1RtAKJuX/B+0ZaJBZXQ/q3lE41CyVoJAJam2vHSTYz6hBzgTk1TC1kFD2SB+g56CiEmw/m5jN/QPHDP1YG3cU+hP270ZGpZUUR05ZNDs3KxjUWtimU+FIFq14ZnK3Yxk1I9ksRMbGdsEIxqf9jKskRVBs6iNOhY/aL4Lxh9wAQzF2gDLD3Vd8NqKGMnTxzZmIZF6thgqemZaSqmEWvuRZGMnsJc/neZjy4HiXcnsx0/+gc9Q6awU3x43zi1ncFbJH9skhaZMTck6uyDXpEEZG5JW8kXfvw/vyvr2fqbTkzXZ2yVyVyr/6pLTN</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=">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</latexit>!51

Carnegie Mellon University

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=">AAACSnicZVDLSgMxFM3UV63P6tLNYBFclDIVQd2JblxWtCo4pWTSO20wjyG5o63DfIJb/So/wO9wJW7MtF1YvRDuybnnhpMTJYJbDIIPrzQ3v7C4VF6urKyurW9sVrdurE4NgzbTQpu7iFoQXEEbOQq4SwxQGQm4jR7Oi/ntIxjLtbrGUQIdSfuKx5xRdNTVc5d3N2tBIxiX/x80p6BGptXqVr2DsKdZKkEhE9Ta+2aQYCejBjkTkFfC1EJC2QPtw72DikqwnWzsNff3HNPzY23cUeiP2d8bGZVWUhw4ZdHszKxgUGth606FA1m04pnx3Y5kVI9kvRAZG9s/RjA+7mRcJSmCYhMfcSp81H6Ri9/jBhiKkQOUGe6+4rMBNZShS2/GRCTzSiVU8MS0lFT1snCYZ2Eks2Gez/Iw4cHxLuXm30z/g/ZB46QRXB7WTs+mcZfJDtkl+6RJjsgpuSAt0iaM9MkLeSVv3rv36X153xNpyZvubJOZKs3/AE42tHw=</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=">AAACX3icZVDfSxtBEN5cW42xarRQKL4shoKFEC4iqPgS7EsfFUwVvHDsbebMkv1x7M5p4uX+mb7af6iv/iXuJXlodGCZb7/5Zpj5kkwKh2H4rxZ8+Phpbb2+0dj8vLW909zd++1Mbjn0uZHG3ibMgRQa+ihQwm1mgalEwk0y/lnVbx7AOmH0NU4zGCh2r0UqOENPxc2v2WE0icXsKRbnESBr0ygbiR9xsxV2wnnQ96C7BC2yjMt4t3YUDQ3PFWjkkjl31w0zHBTMouASykaUO8gYH7N7uPNQMwVuUMwPKOl3zwxpaqx/Gumc/b+jYMophiOvrJJbqVUMGiNd26twpKpUjZn/3VQl7US1K5F1qXuzCKang0LoLEfQfLFHmkuKhlZm0aGwwFFOPWDcCn8K5SNmGUdv6coSiSobjUjDIzdKMT0soklZRIkqJmW5ysOCB897l7tvPX0P+keds054ddzqXSztrpN9ckAOSZeckB75RS5Jn3AyI3/IM/lbewnqwXbQXEiD2rLnC1mJ4Nsr8rC5AQ==</latexit>⌘

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ei<latexit sha1_base64="86pgXmDcE2XVlZKOeiFEsb42X04=">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</latexit>
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xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>!52

Carnegie Mellon University

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

1

1

@f
(xi )|zi ; ⌘) det
@xi

x

f

1

(e; )

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

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

xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>!53

Carnegie Mellon University

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 ; ⌘, )= p(f<latexit sha1_base64="vUGVam4zX8tReom4E38JGVl+oAE=">AAACS3icZVDLSgMxFM3UV63P6tLNYBFclDIVQd0V3bisYFXolJJJ77SheQzJHbUM8wtu9avc+x+uxIWZtgurF8I9OffccHKiRHCLQfDhlZaWV1bXyuuVjc2t7Z3d6t6d1alh0GFaaPMQUQuCK+ggRwEPiQEqIwH30fiqmN8/grFcq1ucJNCTdKh4zBnFggoBaX+3FjSCafn/QXMOamRe7X7VOwkHmqUSFDJBre02gwR7GTXImYC8EqYWEsrGdAhdBxWVYHvZ1GzuHzlm4MfauKPQn7K/NzIqraQ4csqi2YVZwaDWwtadCkeyaMUz07udyKgeyXohMja2f4xgfN7LuEpSBMVmPuJU+Kj9Ihh/wA0wFBMHKDPcfcVnI2ooQxffgolI5pVKqOCJaSmpGmThc56Fkcye83yRhxkPjncpN/9m+h90ThoXjeDmtNa6nMddJgfkkByTJjkjLXJN2qRDGBmRF/JK3rx379P78r5n0pI339knC1Va+QETXLTa</latexit><latexit sha1_base64="hDcZgH5iScA9U7++4o6kHOLgWK0=">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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=">AAACS3icZVC7TsMwFHXKq5RXCyNLRIXEUFUpC7AhWBiLRAGJVJXj3rRW/YjsG6CK8gus8FXs/AcTYsBpM1C4knWPzz3XOj5RIrjFIPjwKkvLK6tr1fXaxubW9k69sXtrdWoY9JgW2txH1ILgCnrIUcB9YoDKSMBdNLks5nePYCzX6ganCfQlHSkec0axoEIY8EG9GbSDWfn/QacETVJWd9DwjsOhZqkEhUxQax86QYL9jBrkTEBeC1MLCWUTOoIHBxWVYPvZzGzuHzpm6MfauKPQn7G/NzIqraQ4dsqi2YVZwaDWwracCseyaMUzs7udyqgVyVYhMja2f4xgfNrPuEpSBMXmPuJU+Kj9Ihh/yA0wFFMHKDPcfcVnY2ooQxffgolI5rVaqOCJaSmpGmbhc56Fkcye83yRhzkPjncpd/5m+h/0jttn7eA6aJ5flHFXyT45IEekQ07IObkiXdIjjIzJC3klb9679+l9ed9zacUrd/bIQlVWfgD5ZLTJ</latexit>

1

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

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

xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>!54

Carnegie Mellon University

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>

log p(x) = log pGHMM (f

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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="whzOwvpgac/k7KeUY3/DRCrw/7o=">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</latexit>1@f1
p(f (xi )|zi ; ⌘) det
@xi
!55

X

log det

@f

1

@xi

1 when f is not invertible<latexit sha1_base64="mEEGp2ktLOyduAzcmi2vdTnECNI=">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</latexit><latexit sha1_base64="NOdjYPNrY5zneVpTueePUz9eQrc=">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</latexit>

1f<latexit sha1_base64="0TvUX8MzRC5izm1cmvW9PX4J8iQ=">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</latexit>

1

(e; )

Carnegie Mellon University

Why Invertible
Example of Markov priormax log pGHMM (f (x))<latexit sha1_base64="QrUGXd8IoThzpkp7A2ek+zngVOU=">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</latexit>xf (e; )
<latexit sha1_base64="FsElmiZYiYnGj5o7UHjArdk6Bk8=">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</latexit>?

Information Loss

!56

Carnegie Mellon University

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>

ei<latexit sha1_base64="REkAECskl6vTPQb6KrE5yd3HUvc=">AAACDXicZVDLSgMxFM3UV62vqks3wSK4KGUqgrorunFZwbGFdiiZNNOGJpMhuSOUof8gbvU7XIlbv8HP8A/MTGdh2wvhnpx7bjg5QSy4Adf9cUpr6xubW+Xtys7u3v5B9fDoyahEU+ZRJZTuBsQwwSPmAQfBurFmRAaCdYLJXTbvPDNtuIoeYRozX5JRxENOCViq2w9kymYDPqjW3IabF14FzQLUUFHtQfW3P1Q0kSwCKogxvaYbg58SDZwKNqv0E8NiQidkxHoWRkQy46e53xk+s8wQh0rbEwHO2f8bKZFGEhhbZdbMwixjQClh6lYFY5m17Jn8bqYyqAeynom0Cc2SEQiv/ZRHcQIsonMfYSIwKJxlg4dcMwpiagGhmtuvYDommlCwCVZsRs3lRFaBd9G4abgPl7XWbRFWGZ2gU3SOmugKtdA9aiMPUSTQK3pD786L8+F8Ol9zackpdo7RQjnff2gonSg=</latexit>f

Inverse
Projection<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

(xi )

<latexit sha1_base64="fjhIxwws58GsuvPWxj5BWS9qhG8=">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</latexit>
sha1_base64="T5O2W5fdBe9YVbRkg7tBseU/cmg=">AAACI3icZVBNLwRBEK3xbX0tjiQ6REKy1owLbsLFkcQisWvS09tjO7qnJ901YjOZix8jrvwOJ3FxcPAT/AM9uw4+KunUq1evOlUvSqWw6Ptv3sDg0PDI6Nh4ZWJyanqmOjt3anVmGG8wLbU5j6jlUiS8gQIlP08NpyqS/Cy6Pij7ZzfcWKGTE+ymvKXoVSJiwSg6KqwuxmHejFTeTDuiKC7zjaBYK+vbIhTrYXXFr/u9IP9B8A1W9jY/lggAHIXVz2Zbs0zxBJmk1l4EfoqtnBoUTPKi0swsTym7plf8wsGEKm5bee+Kgqw6pk1ibdxLkPTYnxM5VVZR7DhlmeyvXsmg1tLWnAo7qkzlN73adlVUi1StFBkb2z+LYLzTykWSZsgT1t8jziRBTUrHSFsYzlB2HaDMCHcKYR1qKEPna8V5FPx15D9obNV36/6x82of+jEGC7AMaxDANuzBIRxBAxjcwQM8wpN37z17L95rXzrgfc/Mw6/w3r8AAwenQQ==</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>!57

Carnegie Mellon University

Learning with Inverse Projectionei(1)hi,l = xi,l<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><latexit sha1_base64="ecOX4VbbnENfsay0Sy5ZWNwjfdg=">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</latexit>(1)
hi,r = xi,r + g(xi,l )ei<latexit sha1_base64="REkAECskl6vTPQb6KrE5yd3HUvc=">AAACDXicZVDLSgMxFM3UV62vqks3wSK4KGUqgrorunFZwbGFdiiZNNOGJpMhuSOUof8gbvU7XIlbv8HP8A/MTGdh2wvhnpx7bjg5QSy4Adf9cUpr6xubW+Xtys7u3v5B9fDoyahEU+ZRJZTuBsQwwSPmAQfBurFmRAaCdYLJXTbvPDNtuIoeYRozX5JRxENOCViq2w9kymYDPqjW3IabF14FzQLUUFHtQfW3P1Q0kSwCKogxvaYbg58SDZwKNqv0E8NiQidkxHoWRkQy46e53xk+s8wQh0rbEwHO2f8bKZFGEhhbZdbMwixjQClh6lYFY5m17Jn8bqYyqAeynom0Cc2SEQiv/ZRHcQIsonMfYSIwKJxlg4dcMwpiagGhmtuvYDommlCwCVZsRs3lRFaBd9G4abgPl7XWbRFWGZ2gU3SOmugKtdA9aiMPUSTQK3pD786L8+F8Ol9zackpdo7RQjnff2gonSg=</latexit><latexit sha1_base64="GqPvuAJfnZZO3NgGi6xcpTtqYcE=">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</latexit>f

Inverse
Projection<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=">AAACI3icZVA9T8MwEHX4LOUrwAgSEQgJpFISFmCrYGEsEqWV2hI5rkOt2nFkXxBVlIUfgxiBn8DMhFgYGFjZ+Ac4LQMfJ1n37t076+4FMWcaXPfVGhkdG5+YLEwVp2dm5+bthcUzLRNFaI1ILlUjwJpyFtEaMOC0ESuKRcBpPegd5f36JVWayegU+jFtC3wRsZARDIby7ZXQT1uBSFtxl2XZebrtZZt5fZX5bMu3192yOwjnP/C+wXpl5311+/Huo+rbn62OJImgERCOtW56bgztFCtghNOs2Eo0jTHp4QvaNDDCgup2OrgiczYM03FCqcyLwBmwPydSLLTA0DXKPOlfvZwBKbkuGRV0RZ7ybwa17ougFIhSLlI61H8WgXC/nbIoToBGZLhHmHAHpJM75nSYogR43wBMFDOnOKSLFSZgfC0aj7y/jvwHtd3yQdk9MV4domEU0DJaQ5vIQ3uogo5RFdUQQdfoFt2jB+vGerKerZehdMT6nllCv8J6+wKDuqmq</latexit>
sha1_base64="T5O2W5fdBe9YVbRkg7tBseU/cmg=">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</latexit>

1

(xi )

(1)
hi,l

1

(1)
hi,r

<latexit sha1_base64="BSeNPzIye1eC45VcyN+Ot1HADN0=">AAACGnicZVDLSgMxFM3UV62vapdugkWoUMqMCOqu6MZlBWsLbS2ZNNOGJpMhuSOUYT5F3Op3uBK3bvwM/8BM24VtL4R7cu654eT4keAGXPfHya2tb2xu5bcLO7t7+wfFw6NHo2JNWZMqoXTbJ4YJHrImcBCsHWlGpC9Yyx/fZvPWM9OGq/ABJhHrSTIMecApAUv1i6WuL5NR2k94FYv0Kal4Z2m/WHZr7rTwKvDmoIzm1egXf7sDRWPJQqCCGNPx3Ah6CdHAqWBpoRsbFhE6JkPWsTAkkpleMjWf4lPLDHCgtD0h4Cn7fyMh0kgCI6vMmlmYZQwoJUzVqmAks5Y9M72bifSrvqxmIm0Cs2QEgqtewsMoBhbSmY8gFhgUzoLCA64ZBTGxgFDN7VcwHRFNKNg4CzYjbzmRVdA8r13X3PuLcv1mHlYeHaMTVEEeukR1dIcaqIkomqBX9IbenRfnw/l0vmbSnDPfKaGFcr7/AHw0oVI=</latexit>
sha1_base64="d20bZPfnQOhStvpbd1JYhfyuJWM=">AAAB93icZVDLSgMxFM3UVx2r1rWbYBFclDJ1o+4ENy4rOLbQDiWTybSheQzJHaEMXQtu/Q5X4sf4Gf6BmWkXtr0Q7sm554aTE2eCWwiCH6+2s7u3f1A/9I8a/vHJabPxYnVuKAupFtoMYmKZ4IqFwEGwQWYYkbFg/Xj2UM77r8xYrtUzzDMWSTJRPOWUgKN642Yr6ARV4W3QXYEWWtW4+TtKNM0lU0AFsXbYDTKICmKAU8EW/ii3LCN0RiZs6KAiktmoqGwu8KVjEpxq444CXLH/NwoirSQwdcqy2bVZyYDWwradCqaybOUz1d3OZdyOZbsUGZvaDSOQ3kYFV1kOTNGljzQXGDQuI8EJN4yCmDtAqOHuK5hOiSEUXHC+i6i7Gcg2CK87d53gKUB1dI4u0BXqoht0jx5RD4WIogS9ow/vzfv0vpZJ1rxVpGdorbzvP0dVlyg=</latexit>
sha1_base64="UqJBtKil1Ol91WcJviTDNRmxj54=">AAACD3icZVDLSgMxFL3js9aqVZdugkVQKGXGjboT3LisYG2hrSWTZtpgMhmSO0IZ5lPErX6HK/EL/Az/wEzbhbYXwj0599xwcsJECou+/+2trK6tb2yWtsrblZ3dvep+5cHq1DDeYlpq0wmp5VLEvIUCJe8khlMVSt4On26KefuZGyt0fI+ThPcVHcUiEoyiowbVw16osnE+yESdyPwxOw3O8kG15jf8aZFlEMxBDebVHFR/ekPNUsVjZJJa2w38BPsZNSiY5Hm5l1qeUPZER7zrYEwVt/1saj4nJ44Zkkgbd2IkU/bvRkaVVRTHTlk0+29WMKi1tHWnwrEqWvHM9G4nKqyHql6IjI3sghGMLvuZiJMUecxmPqJUEtSkCIoMheEM5cQByoxwXyFsTA1l6OIsu4yCxUSWQeu8cdXw73wowREcwykEcAHXcAtNaAGDCbzCG7x7L96H9zkLc8Wbp3oA/8r7+gWMo5/N</latexit>
sha1_base64="3qhYlX9qLp+05qt4Pk5toEMfBus=">AAACGnicZVDLTgIxFO34RHyhLN00EhNMCJlxo+6Iblxi4ggJIOmUDjS000l7x4RM5lOMW/0OV8atGz/DP7ADLARu0tzTc89tTk8QC27AdX+ctfWNza3twk5xd2//4LB0dPxoVKIp86kSSrcDYpjgEfOBg2DtWDMiA8Fawfg2n7eemTZcRQ8wiVlPkmHEQ04JWKpfKncDmY6yfsprWGRPadU7z/qlilt3p4VXgTcHFTSvZr/02x0omkgWARXEmI7nxtBLiQZOBcuK3cSwmNAxGbKOhRGRzPTSqfkMn1lmgEOl7YkAT9n/GymRRhIYWWXezMIsZ0ApYWpWBSOZt/yZ6d1MZFALZC0XaROaJSMQXvVSHsUJsIjOfISJwKBwHhQecM0oiIkFhGpuv4LpiGhCwcZZtBl5y4msAv+ifl13791K42YeVgGdoFNURR66RA10h5rIRxRN0Ct6Q+/Oi/PhfDpfM+maM98po4Vyvv8AevShTg==</latexit>

1<latexit sha1_base64="M6AXFmCURF5oagaaj1n7FJbNekI=">AAACGnicZVDLSgMxFM3UV62vapdugkWoUMqMCOqu6MZlBWsLbS2ZNNOGJpMhuSOUYT5F3Op3uBK3bvwM/8BM24VtL4R7cu654eT4keAGXPfHya2tb2xu5bcLO7t7+wfFw6NHo2JNWZMqoXTbJ4YJHrImcBCsHWlGpC9Yyx/fZvPWM9OGq/ABJhHrSTIMecApAUv1i6WuL5NR2k94Fev0Kal4Z2m/WHZr7rTwKvDmoIzm1egXf7sDRWPJQqCCGNPx3Ah6CdHAqWBpoRsbFhE6JkPWsTAkkpleMjWf4lPLDHCgtD0h4Cn7fyMh0kgCI6vMmlmYZQwoJUzVqmAks5Y9M72bifSrvqxmIm0Cs2QEgqtewsMoBhbSmY8gFhgUzoLCA64ZBTGxgFDN7VcwHRFNKNg4CzYjbzmRVdA8r13X3PuLcv1mHlYeHaMTVEEeukR1dIcaqIkomqBX9IbenRfnw/l0vmbSnDPfKaGFcr7/AIYqoVg=</latexit>=+<latexit sha1_base64="J710iawjlDt3Kcch2yC+lHTgmcU=">AAACE3icZVBNSwMxEM3Wr7p+VT16WSyCh1K2XtRDsejFYwVrC+1astlsG5pslmRWKEv/hnhV/4YnETx5Fn+G/8Bs24NtB8K8vHkzzDw/5kyD6/5YuaXlldW1/Lq9sbm1vVPY3bvTMlGENojkUrV8rClnEW0AA05bsaJY+Jw2/cFVVm8+UKWZjG5hGFNP4F7EQkYwGOq+40se6KEwKa2OuoWiW3bH4SyCyhQULz7tavz6bde7hd9OIEkiaASEY63bFTcGL8UKGOF0ZHcSTWNMBrhH2wZGWFDtpeOtR86RYQInlMq8CJwx+78jxUILDH2jzJKeqWUMSMl1yaigL7KUjRn/s4NKvihlIqVDPbcIhGdeyqI4ARqRyR5hwh2QTuaQEzBFCfChAZgoZk5xSB8rTMD4aBuPKvOOLILGSfm87N64xdolmkQeHaBDdIwq6BTV0DWqowYiSKEn9IxerEfrzXq3PibSnDXt2UczYX39ASoHoxk=</latexit><latexit
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<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

0
1

.
.

1
1
1

<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>!58[Dinh et al. 2014]

Carnegie Mellon University

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>

ei,r

ei,l
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sha1_base64="ZndHDGIBuRmtjyQp3+WY268Rqpk=">AAACGnicZVBNTwIxEJ31E/AL5ehlIzHxQMjiRb0RvXjExBUSINgtXWhot5t21oRs+CEejFf9HZ6MVy/+DP+BXeAgMEkzr2/etDMviAU36Hk/ztr6xubWdi5f2Nnd2z8oHh49GJVoynyqhNKtgBgmeMR85ChYK9aMyECwZjC6yerNJ6YNV9E9jmPWlWQQ8ZBTgpbqFUudQIm+GUub0g5yycykVyx7VW8a7iqozUG5no+fHwGg0Sv+dvqKJpJFSAUxpl3zYuymRCOngk0KncSwmNARGbC2hRGxv3TT6fAT99QyfTdU2p4I3Sn7vyMl0kiCQ6vMklmoZQwqJUzFqnAos5Q9M71nW1UCWclE2oRmaRAML7spj+IEWURnc4SJcFG5mVFun2tGUYwtIFRzu4pLh0QTitbOgvWotuzIKvDPq1dV7856dQ2zyMExnMAZ1OAC6nALDfCBwhhe4Q3enRfnw/l0vmbSNWfeU4KFcL7/AOnOpC4=</latexit><latexit sha1_base64="ILR7g1A/jBsoHtYqP1JEuivgHQ4=">AAACEnicZVDLSgMxFM34rPVVdekmWAQXpUxFUHdFNy4rOLbQDiWTZtrQZDImd4QyzGeIW/0OV+LWH/Az/AMz0y5seyHck3PPDScniAU34Lo/zsrq2vrGZmmrvL2zu7dfOTh8NCrRlHlUCaU7ATFM8Ih5wEGwTqwZkYFg7WB8m8/bz0wbrqIHmMTMl2QY8ZBTApbye4FMWdZPeQ3rrF+punW3KLwMGjNQRbNq9Su/vYGiiWQRUEGM6TbcGPyUaOBUsKzcSwyLCR2TIetaGBHJjJ8WpjN8apkBDpW2JwJcsP83UiKNJDCyyryZuVnOgFLC1KwKRjJv+TPF3UxkUAtkLRdpE5oFIxBe+SmP4gRYRKc+wkRgUDgPCA+4ZhTExAJCNbdfwXRENKFgYyzbjBqLiSwD77x+XXfvL6rNm1lYJXSMTtAZaqBL1ER3qIU8RNETekVv6N15cT6cT+drKl1xZjtHaK6c7z8AuJ8Q</latexit>(1)hi,l = xi,l

=

+

···

···

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sha1_base64="tt7sHfQoCO2k4YfwNUEEi7Nc9MQ=">AAACE3icZVDNSgMxEJ6tf7X+VT16WSyCh1K2XtRDsejFYwVrC20t2TTbhiabJZkVylLwKcSrPocnETz5AD6Gb2C29WDbgTBfvvlmmPn8SHCDnvftZJaWV1bXsuu5jc2t7Z387t6dUbGmrE6VULrpE8MED1kdOQrWjDQj0hes4Q+v0nrjgWnDVXiLo4h1JOmHPOCUoKXu274SPTOSNiWVcTdf8EreJNxFUP4DhYvPXOURAGrd/E+7p2gsWYhUEGNaZS/CTkI0cirYONeODYsIHZI+a1kYEslMJ5lsPXaPLNNzA6XtC9GdsP87EiKNJDiwyjSZmVrKoFLCFK0KBzJN6ZjJPz2o6MtiKtImMHOLYHDWSXgYxchCOt0jiIWLyk0dcntcM4piZAGhmttTXDogmlC0PuasR+V5RxZB/aR0XvJuvEL1EqaRhQM4hGMowylU4RpqUAcKGp7hBV6dJ+fNeXc+ptKM89ezDzPhfP0CDOChpQ==</latexit><latexit sha1_base64="REkAECskl6vTPQb6KrE5yd3HUvc=">AAACDXicZVDLSgMxFM3UV62vqks3wSK4KGUqgrorunFZwbGFdiiZNNOGJpMhuSOUof8gbvU7XIlbv8HP8A/MTGdh2wvhnpx7bjg5QSy4Adf9cUpr6xubW+Xtys7u3v5B9fDoyahEU+ZRJZTuBsQwwSPmAQfBurFmRAaCdYLJXTbvPDNtuIoeYRozX5JRxENOCViq2w9kymYDPqjW3IabF14FzQLUUFHtQfW3P1Q0kSwCKogxvaYbg58SDZwKNqv0E8NiQidkxHoWRkQy46e53xk+s8wQh0rbEwHO2f8bKZFGEhhbZdbMwixjQClh6lYFY5m17Jn8bqYyqAeynom0Cc2SEQiv/ZRHcQIsonMfYSIwKJxlg4dcMwpiagGhmtuvYDommlCwCVZsRs3lRFaBd9G4abgPl7XWbRFWGZ2gU3SOmugKtdA9aiMPUSTQK3pD786L8+F8Ol9zackpdo7RQjnff2gonSg=</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>ei(1)hi,r = xi,r + g(xi,l )<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><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>f

Inverse
Projection<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 )

+

=
<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="9nq30MFPFBgzWoiP8ja7QQU/IdQ=">AAACE3icZVDNSsNAEN7Uv1r/qh69BIsgWEoqgnorevFYwdhCG8tms2mX7mbD7kQooa8hXvU5PIlXH8DH8A3cpDnYdmCZb7/5Zpj5/JgzDY7zY5VWVtfWN8qbla3tnd296v7Bo5aJItQlkkvV9bGmnEXUBQacdmNFsfA57fjj26zeeaZKMxk9wCSmnsDDiIWMYDDUU9+XPNATYVJ6Nh1Ua07DycNeBs0C1FAR7UH1tx9IkggaAeFY617TicFLsQJGOJ1W+ommMSZjPKQ9AyMsqPbSfOupfWKYwA6lMi8CO2f/d6RYaIFhZJRZ0nO1jAEpua4bFYxElrIx+T87qO6LeiZSOtQLi0B45aUsihOgEZntESbcBmlnDtkBU5QAnxiAiWLmFJuMsMIEjI8V41Fz0ZFl4J43rhvO/UWtdVOYVUZH6Bidoia6RC10h9rIRQQp9Ire0Lv1Yn1Yn9bXTFqyip5DNBfW9x9LXZ/K</latexit><latexit(1)hi,l<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

1

(1)
hi,r

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xi

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1
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Carnegie Mellon University

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!60

Carnegie Mellon University

Part-of-speech Induction
100

Feature-based 80.2
75

VM

Many-to-1 %

100

80.8

75.4

75

Feature-based 72.1

74.1

68.5
62.7
50

HMM

Gaussian
HMM

50

NeuralProjection

53.8

HMM

Gaussian
HMM

Outperform feature-based SOTA
!61

NeuralProjection

Carnegie Mellon University

75

75

Oracle POS 55.1
50

60.2

55.4
49.6

25

DMV

Gaussian
DMV

NeuralProjection!62≤

Directed % on on all≤

Directed % on len <= 10

Dependency Parse Induction

50

Oracle POS 39.7

47.9

43.1

35.8

25

DMV

Gaussian
DMV

NeuralProjection

Carnegie Mellon University

Original Embedding Space
adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number!63

Carnegie Mellon University

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x

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1

(e; )

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xi<latexit sha1_base64="Fx1hTPOIk2ME1RopXl5ys2C3zr4=">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</latexit>!64Carnegie Mellon University

Projected Embedding Space w/ DMV Prior
adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number!65

Carnegie Mellon University

Projected Embedding Space w/ DMV Prior
adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number!66

Carnegie Mellon University

Projected Embedding Space w/ DMV Prior
adjective
adverbnoun singularnoun proper
noun pluralverb baseverb gerundverb past tenseverb past participle
verb 3rd singularcardinal number!67

Carnegie Mellon University

Conclusion!68

Carnegie Mellon University

Learning with Latent Linguistic
Structure• How can we harness the power of neural networks?
• NN-based learning on top of latent structured representations• How can we learn on unlabeled data?
• Structured variational auto-encoders for semi-supervised
learning• Structured priors and invertible transformations for
unsupervised learning!69