A Mathematical Framework for a Distributional
Compositional Model of Meaning
Stephen Clark
University of Cambridge Computer Laboratory
Stanford University
10 May 2013
Intro
Sentences in Google
DisCo Models of Meaning
Set-Theoretic
Lexical
Syntax
2
Intro
Set-Theoretic
Lexical
Syntax
Motivation
• Two success stories:
• distributional vector-based models of lexical meaning
• compositional logic-based models of sentence meaning
• Can we combine these approaches to give a vector-based
semantic model of phrases and sentences?
• A fundamental new problem in natural language semantics
DisCo Models of Meaning
3
Intro
Set-Theoretic
Lexical
Syntax
Interdisciplinary Endeavour
• Collaboration with the Oxford Computational Linguistics and
Quantum groups
• B. Coecke, E. Grefenstette† , S. Pulman, M. Sadrzadeh†
• Linguistics, semantics, logic, category theory, quantum logic, . . .
† thanks to Ed and Mehrnoosh for some of the slides
DisCo Models of Meaning
4
Intro
Set-Theoretic
Lexical
Today’s Talk
• Recap on set-theoretic approaches to semantics
• Distributional models of word meaning
• Categorial grammar syntax
• A compositional distributional model (in theory)
• Brief description of some empirical work
DisCo Models of Meaning
Syntax
5
Intro
Set-Theoretic
Lexical
Syntax
Formal (Montague) Semantics
• The dominant approach in linguistics and the philosophy of
language (Lewis, Montague, 1970s)
• Characterised by the use of logic as the semantic formalism
• A successful model of compositionality based on Frege’s principle
DisCo Models of Meaning
6
Intro
Set-Theoretic
Lexical
Syntax
Formal (Montague) Semantics
S → NP VP : VP 0 (NP 0 )
The dog sleeps
• dog 0 picks out an individual in some model
• sleep 0 is a relation (the set of individuals who sleep in the model)
• The dog sleeps 0 is true if dog 0 is in sleep 0 and false otherwise
DisCo Models of Meaning
7
Intro
Set-Theoretic
Lexical
Syntax
Formal (Montague) Semantics
S → NP VP : VP 0 (NP 0 )
The dog sleeps
• dog 0 picks out an individual in some model
• sleep 0 is a relation (the set of individuals who sleep in the model)
• The dog sleeps 0 is true if dog 0 is in sleep 0 and false otherwise
• Meanings of words and sentences have different semantic types
DisCo Models of Meaning
7
Intro
Set-Theoretic
Lexical
Semantics in GOFAI (and Semantic Web)
• First-order predicate calculus
• Well-defined inference procedures
• Efficient theorem provers
• Knowledge encoded as ontologies
DisCo Models of Meaning
Syntax
8
Intro
Set-Theoretic
Lexical
Syntax
Shortcomings of the Traditional Approach
Regular coffee breaks diminish the risk of getting Alzheiemers and
dementia in old age.
Three cups of coffee a day greatly reduce the chance of developing
dementia or alzheimers later in life.
DisCo Models of Meaning
9
Intro
Set-Theoretic
Lexical
Syntax
Shortcomings of the Traditional Approach
Regular coffee breaks diminish the risk of getting Alzheiemers and
dementia in old age.
Three cups of coffee a day greatly reduce the chance of developing
dementia or alzheimers later in life.
• Semantic similarity is difficult to model using traditional methods
• Similarity is at the heart of many NLP and IR problems
• Evidence from cognitive science that similarity is part of humans’
conceptual models
DisCo Models of Meaning
9
Intro
Set-Theoretic
Lexical
Syntax
Distributional and Semantic Similarity
• You shall know a word by the company that it keeps. (Firth,‘57)
• Distributional hypothesis: the meaning of a word can be
represented by the distribution of words appearing in its contexts
DisCo Models of Meaning
10
Intro
Set-Theoretic
Lexical
Syntax
Distributional and Semantic Similarity
• dog and cat are related semantically:
dog and cat both co-occur with big, small, furry, eat, sleep
• ship and boat have similar meanings:
ship and boat appear as the direct object of the verbs sail, clean,
bought; as the object of the adjectives large, clean, expensive
DisCo Models of Meaning
11
Intro
Set-Theoretic
Lexical
Syntax
Window Methods
• In window methods the context is a fixed-word window either side
of the target word
• For each target word a vector is created where each basis vector
corresponds to a context word
• Coefficient for each basis is a (weighted) frequency of how often
the context word appears with the target word
• Our compositional framework is agnostic towards the word vectors
DisCo Models of Meaning
12
Intro
Set-Theoretic
Lexical
Vector Space for Window Method
furry
6
cat
dog
stroke
DisCo Models of Meaning
- pet
Syntax
13
Intro
Set-Theoretic
Lexical
Syntax
Example Output
• introduction: launch, implementation, advent, addition,
adoption, arrival, absence, inclusion, creation, departure,
availability, elimination, emergence, use, acceptance, abolition,
array, passage, completion, announcement, . . .
DisCo Models of Meaning
14
Intro
Set-Theoretic
Lexical
Syntax
Example Output
• evaluation: assessment, examination, appraisal, review, audit,
analysis, consultation, monitoring, testing, verification, inquiry,
inspection, measurement, supervision, certification, checkup, . . .
DisCo Models of Meaning
15
Intro
Set-Theoretic
Lexical
From Words to Sentences
s1
6
man killed dog
man murdered cat
?
- s3
man killed by dog
s2
DisCo Models of Meaning
Syntax
16
Intro
Set-Theoretic
Lexical
Syntax
What Semantics?!
• A semantics of similarity
• How to incorporate inference, logical operators, quantification,
etc. is an interesting question . . .
DisCo Models of Meaning
17
Intro
Set-Theoretic
Lexical
Categorial Grammar
interleukin − 10
inhibits
production
NP
(S \NP )/NP
NP
S \NP
S
DisCo Models of Meaning
Syntax
18
Intro
Set-Theoretic
Lexical
Syntax
A Simple CG Derivation
interleukin − 10
inhibits
production
NP
(S \NP )/NP
NP
S \NP
S
>
forward application
DisCo Models of Meaning
>
19
Intro
Set-Theoretic
Lexical
Syntax
A Simple CG Derivation
interleukin − 10
inhibits
production
NP
(S \NP )/NP
NP
S \NP
S
>
<
forward application
backward application
DisCo Models of Meaning
>
<
20
Intro
Set-Theoretic
Lexical
Pregroup Grammar Derivation
Google
bought
Microsoft
NP
NP r · S · NP l
NP
S \NP
S
DisCo Models of Meaning
Syntax
21
Intro
Set-Theoretic
Lexical
Pregroup Grammar Derivation
Google
bought
Microsoft
NP
NP r · S · NP l
NP
NP r · S
S
DisCo Models of Meaning
Syntax
22
Intro
Set-Theoretic
Lexical
Pregroup Grammar Derivation
Google
bought
Microsoft
NP
NP r · S · NP l
NP
NP r · S
S
DisCo Models of Meaning
Syntax
23
Intro
Set-Theoretic
Lexical
Pregroup Reduction
Google
NP
DisCo Models of Meaning
bought
NP^r S NP^l
Microsoft
NP
Syntax
24
Intro
Set-Theoretic
Lexical
Syntax
Pregroup Algebra
• A pregroup is a partially ordered monoid in which each element a
has a left adjoint al and a right adjoint ar such that
al · a → 1,
DisCo Models of Meaning
a · ar → 1
25
Intro
Set-Theoretic
Lexical
Syntax
Pregroups for Linguistics
• The monoid is the set of grammatical types: NP , NP r , NP l ,
NP rr , NP ll , S , PP , . . .
• The monoid operator (·) is just juxtaposition
• The unit of the monoid (1) is the empty string
DisCo Models of Meaning
26
Intro
Set-Theoretic
Lexical
Syntax
Pregroups for Linguistics
• Partial order encodes the derivation relation;
for the earlier derivation/reduction:
NP · (NP r · S · NP l ) · NP → 1 · (S · NP l ) · NP → 1 · S · 1 = S
DisCo Models of Meaning
27
Intro
Set-Theoretic
Lexical
Syntax
Pregroups for Linguistics
• Partial order encodes the derivation relation;
for the earlier derivation/reduction:
NP · (NP r · S · NP l ) · NP → 1 · (S · NP l ) · NP → 1 · S · 1 = S
• At an abstract mathematical level (Category Theory), the algebra
of pregroups and vector spaces can be seen as equivalent
DisCo Models of Meaning
27
Intro
Set-Theoretic
Lexical
Syntax
Category Theory
• Pregroups form a compact closed category, with the types as
objects, derivation arrows as morphisms, juxtaposition as tensor,
and the under-links as the ‘cups’ of composition
DisCo Models of Meaning
28
Intro
Set-Theoretic
Lexical
Syntax
Category Theory
• Pregroups form a compact closed category, with the types as
objects, derivation arrows as morphisms, juxtaposition as tensor,
and the under-links as the ‘cups’ of composition
• Vector spaces form a compact closed category, with vector spaces
as objects, linear maps as morphisms, tensor product as tensor,
and tensor contraction as the ‘cups’ of composition
DisCo Models of Meaning
28
Intro
Set-Theoretic
Lexical
Syntax
Category Theory
• Pregroups form a compact closed category, with the types as
objects, derivation arrows as morphisms, juxtaposition as tensor,
and the under-links as the ‘cups’ of composition
• Vector spaces form a compact closed category, with vector spaces
as objects, linear maps as morphisms, tensor product as tensor,
and tensor contraction as the ‘cups’ of composition
v
Ψ
• Similar pictures can be drawn for quantum protocols
DisCo Models of Meaning
w
29
Tensor Semantics
Empirical
Predicate-Argument Semantics
man
bites
dog
NP
NP r · S · NP l
NP
man0 λx.λy bites0 (x, y) dog 0
NP r · S
S
DisCo Models of Meaning
Conclusion
30
Tensor Semantics
Empirical
Predicate-Argument Semantics
man
bites
dog
NP
NP r · S · NP l
NP
man0 λx.λy bites0 (x, y) dog 0
NP r · S
λy bites0 (dog 0 , y)
S
Function application
DisCo Models of Meaning
Conclusion
31
Tensor Semantics
Empirical
Predicate-Argument Semantics
man
bites
dog
NP
NP r · S · NP l
NP
man0 λx.λy bites0 (x, y) dog 0
NP r · S
λy bites0 (dog 0 , y)
S
bites0 (dog 0 , man0 )
Function application
DisCo Models of Meaning
Conclusion
32
Tensor Semantics
Empirical
Vector-Space Semantics?
man
bites
dog
NP
NP r · S · NP l
NP
man0 λx.λy bites0 (x, y) dog 0
NP r · S
λy bites0 (dog 0 , y)
S
bites0 (dog 0 , man0 )
• What are the semantic types of the vectors?
• What is the equivalent of function application?
DisCo Models of Meaning
Conclusion
33
Tensor Semantics
Empirical
Adjective Noun Combinations
red
car
N · Nl N
N
DisCo Models of Meaning
Conclusion
34
Tensor Semantics
Empirical
Adjective Noun Combinations
red
car
N · Nl N
N
DisCo Models of Meaning
Conclusion
35
Tensor Semantics
Empirical
Conclusion
Adjective Noun Combinations
red
car
N · Nl N
N
• Adjective is a function
• How are functions represented in linear algebra? (B&Z, 2010)
• Functions are matrices (Linear Maps)
DisCo Models of Meaning
35
Tensor Semantics
Empirical
Conclusion
Adjective Noun Combinations
red
car
N · Nl N
N
• Adjective is a function
• How are functions represented in linear algebra? (B&Z, 2010)
• Functions are matrices (Linear Maps)
• How do functions combine with arguments in linear algebra?
• Matrix multiplication
DisCo Models of Meaning
35
Tensor Semantics
Empirical
Conclusion
Matrix Multiplication






R11
R21
R31
R41
R51
DisCo Models of Meaning
R12
R22
R32
R42
R52
R13
R23
R33
R43
R53
−−−−→
red car
−
→
car
RED
R14
R24
R34
R44
R54
R15
R25
R35
R45
R55






c1
c2
c3
c4
c5




 = 






rc1
rc2
rc3
rc4
rc5






36
Tensor Semantics
Empirical
Conclusion
Matrix and Vector Types






R11
R21
R31
R41
R51
DisCo Models of Meaning
R12
R22
R32
R42
R52
RED
−
→
car
−−−−→
red car
N⊗N
N
N
R13
R23
R33
R43
R53
R14
R24
R34
R44
R54
R15
R25
R35
R45
R55






c1
c2
c3
c4
c5




 = 






rc1
rc2
rc3
rc4
rc5






37
Tensor Semantics
Empirical
Conclusion
Matrix and Vector Types
−
→
car
−−−−→
red car
N⊗N
N
N
l
N
N
RED
N ·N






R11
R21
R31
R41
R51
DisCo Models of Meaning
R12
R22
R32
R42
R52
R13
R23
R33
R43
R53
R14
R24
R34
R44
R54
R15
R25
R35
R45
R55






c1
c2
c3
c4
c5




 = 






rc1
rc2
rc3
rc4
rc5






38
Tensor Semantics
Empirical
Syntactic Types to Tensor Spaces
man
bites
dog
NP NP r · S · NP l NP
N N⊗S ⊗N
DisCo Models of Meaning
N
Conclusion
39
Tensor Semantics
Empirical
Syntactic Types to Tensor Spaces
man
bites
dog
NP NP r · S · NP l NP
N N⊗S ⊗N
DisCo Models of Meaning
N
Conclusion
40
Tensor Semantics
Empirical
Syntactic Types to Tensor Spaces
man
bites
dog
NP NP r · S · NP l NP
N
DisCo Models of Meaning
N ⊗S ⊗N
N
Conclusion
41
Tensor Semantics
Empirical
Syntactic Types to Tensor Spaces
man
bites
dog
NP NP r · S · NP l NP
N
DisCo Models of Meaning
N⊗S⊗N
N
Conclusion
42
Tensor Semantics
Empirical
Conclusion
Syntactic Types to Tensor Spaces
man
bites
dog
NP NP r · S · NP l NP
N
N⊗S⊗N
N
• What is the sentence space (different to the noun space)?
DisCo Models of Meaning
42
Tensor Semantics
Empirical
Conclusion
Meaning Vectors as Tensors
S
N
N
DisCo Models of Meaning
N
N
43
Tensor Semantics
Empirical
Conclusion
Tensors
• Rank 1 – vector:
→
−
v ∈A=
X
−
Civ →
ai
i
• Rank 2 – matrix:
M ∈A⊗B =
X
→
−
−
M →
Cij
ai ⊗ bj
ij
• Rank 3 – cuboid:
R∈A⊗B⊗C =
X
→
− −
−
R →
Cijk
ai ⊗ bj ⊗ →
ck
ijk
• Rank n:
T ∈ V1 ⊗ . . . ⊗ Vn =
X
α1 ...αn
DisCo Models of Meaning
−→
−−→
CαT1 ...αn βα1 1 ⊗ . . . ⊗ βαnn
44
Tensor Semantics
Empirical
Conclusion
Tensor contraction
• Rank 0 × rank 0: field multiplication
• Rank 0 × rank n: scalar multiplication
• Rank 1 × rank 1: inner product (dot product)
• Rank 2 × rank 1: matrix-vector multiplication
• Rank 2 × rank 2: matrix multiplication
• ...
The general vector reduction mechanism is just a generalisation of
these familiar tensor contractions
DisCo Models of Meaning
45
Tensor Semantics
Empirical
Conclusion
Multi-Linear Algebra
S
N
N
N
S
N
N
S
DisCo Models of Meaning
N
46
Tensor Semantics
Empirical
Type Reductions
man
bites
dog
NP NP r · S · NP l NP
N
N⊗S⊗N
N
NP r · S
S
DisCo Models of Meaning
Conclusion
47
Tensor Semantics
Empirical
Type Reductions
man
bites
dog
NP NP r · S · NP l NP
N
N⊗S⊗N
N
NP r · S
N⊗S
S
Tensor contraction via inner products
Objects ‘get smaller’ (as they do in formal semantics)
DisCo Models of Meaning
Conclusion
48
Tensor Semantics
Empirical
Type Reductions
man
bites
dog
NP NP r · S · NP l NP
N
N⊗S⊗N
N
NP r · S
N⊗S
S
S
Tensor contraction
DisCo Models of Meaning
Conclusion
49
Tensor Semantics
Empirical
Type Reductions
man
bites
dog
NP NP r · S · NP l NP
N
N⊗S⊗N
N
NP r · S
N⊗S
S
S
Verbs only have operator semantics
Nouns only have contextual semantics
DisCo Models of Meaning
Conclusion
50
Tensor Semantics
Empirical
Conclusion
Summary of Tensor Semantics
Meaning of a sentence
w1 · · · wn
with the grammatical structure
p1 · · · pn →α s
is:
−
w−1−·−·−
·−
w→
n
:=
→ ⊗ ··· ⊗ −
F (α)(−
w
w→
1
n)
• F (α) is Montague’s homomorphic passage (Frege’s principle) in
the form of a linear map
DisCo Models of Meaning
51
Tensor Semantics
Empirical
Conclusion
A Real Example
In
l
S · S · NP
an
l
Oct.
NP[nb] · N
M isanthrope
l
at
r
Revitalized Classics
N · Nl
,
N
N ·N
review
l
l
0
Celimene ,
N
s
NP · NP[nb] · N l
N
T ake
&
the
Arts
played
attributed
N
−RRB− ,
RRB
by
N · Nl
to
−LRB−
in
the
, NP[nb] · N l
W indy
role
of
N
NP r · NP · NP l
was
, NP r · S [dcl] · S [pss]l · NP
NP
Christina Haag
N · Nl
NP r · NP · S [dcl]l
N
S r · NP rr · NP r · S · NP l N · N l
Cattrall ,
NP r · S · S l · NP r NP r · S [pss] · PP l PP · NP l
DisCo Models of Meaning
NP[nb] · N l
Stage
, NP r · S [pss] S r · NP rr · NP r · S · NP l
mistakenly
l
Goodman T heatre
r
NP r · S [dcl] · NP l N P [nb] · N l
Leisure
T he
NP · NP · NP
N
, S r · S · S l · S S r · S · S l · S Sr · S
N
of
r
Chicago
NP · NP · NP
N
City
N ·N
19
l
N
.
.
52
Tensor Semantics
Empirical
Conclusion
Learning Matrices
red car
red balloon
red chair
red shoe
...
DisCo Models of Meaning

R11
h. . .i
 R21
h. . .i

h. . .i =⇒ 
 R31
 R41
h. . .i
R51
R12
R22
R32
R42
R52
R13
R23
R33
R43
R53
R14
R24
R34
R44
R54
R15
R25
R35
R45
R55






53
Tensor Semantics
Empirical
Conclusion
Learning Matrices
red car
red balloon
red chair
red shoe
...

R11
h. . .i
 R21
h. . .i

h. . .i =⇒ 
 R31
 R41
h. . .i
R51
R12
R22
R32
R42
R52
R13
R23
R33
R43
R53
R14
R24
R34
R44
R54
• Use linear regression to learn the RED matrix (B&Z)
• RED can now be applied to unseen pairs:
−−−−−−→
−−−−−−−−−→
RED × pantaloon ⇒ red pantaloon
DisCo Models of Meaning
R15
R25
R35
R45
R55






53
Tensor Semantics
Empirical
Conclusion
Learning Matrices
tiger sleeps
cat sleeps
man sleeps
pig sleeps
...
DisCo Models of Meaning

S11
h. . .i
 S21
h. . .i

h. . .i =⇒ 
 S31
 S41
h. . .i
S51
S12
S22
S32
S42
S52
S13
S23
S33
S43
S53
S14
S24
S34
S44
S54
S15
S25
S35
S45
S55






54
Tensor Semantics
Empirical
Conclusion
Learning Matrices
tiger sleeps
cat sleeps
man sleeps
pig sleeps
...

S11
h. . .i
 S21
h. . .i

h. . .i =⇒ 
 S31
 S41
h. . .i
S51
S12
S22
S32
S42
S52
S13
S23
S33
S43
S53
S14
S24
S34
S44
S54
S15
S25
S35
S45
S55






• Use linear regression to learn the SLEEPS matrix (Gref et. al)
• SLEEPS can now be applied to unseen pairs:
−−−−−→
−−−−−−−−−−→
SLEEPS × chiwawa ⇒ chiwawa sleeps
DisCo Models of Meaning
54
Tensor Semantics
Empirical
Conclusion
Context and Compositionality
• Should the meanings of all units be contextual? (B&Z, Clarke)
• If so, what role does compositionality play?
• is it just to combat sparse data?
DisCo Models of Meaning
55
Tensor Semantics
Empirical
Conclusion
Learning Tensors
STEP 1: ESTIMATE VP MATRICES
dogs.eat.meat
dogs
cats
EAT
MEAT
cats.eat.meat
boys.eat.pie
boys
girls
EAT
PIE
girls.eat.pie
STEP 2: ESTIMATE V TENSOR
EAT
MEAT
meat
EAT
pie
EAT
PIE
training example (input)
training example (output)
function to estimate
Thanks to Marco Baroni and Ed Grefenstette for the picture
DisCo Models of Meaning
56
Tensor Semantics
Empirical
Conclusion
Disambiguation Evaluation
Nouns and intransitive verbs (Mitchell and Lapata, 2008)
Subject
Landmark
High
Low
face
fire
horse
glow
glow
draw
beam
burn
pull
burn
beam
sketch
Example judgements (score between 1 and 7):
• “the face glowed” vs. “the face beamed”
• “the face glowed” vs. “the face burned”
• “the fire glowed” vs. “the fire burned”
DisCo Models of Meaning
57
Tensor Semantics
Empirical
Conclusion
Disambiguation Evaluation
Nouns and transitive verbs (Grefenstette and Sadrzadeh, 2011)
Subject
Object
Landmark
High
Low
people
tribunal
poll
door
crime
support
try
try
show
test
judge
express
judge
test
picture
Example judgements (score between 1 and 7):
• “the people tried the door” vs. “the people tested the door”
• “the people tried the door” vs. “the people judged the door”
• “the tribunal tried the crime” vs. “the tribunal judged the crime”
DisCo Models of Meaning
58
Tensor Semantics
Empirical
Conclusion
Experimental Details (Gref et. al)
• Build context vectors using large corpus and window method
(with dimensionality reduction)
• Calculate similarity of each pair using cosine:
−−−−−−−−−−→ −−−−−−−−−−−→
• Cosine(people try door, people test door)
−−−−−−−−−−→ −−−−−−−−−−−−→
• Cosine(people try door, people judge door)
• Calculate correlation coefficient between human and cosine scores
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Tensor Semantics
Results (Gref et. al)
First Evaluation
Model
ρ
UpperBound
Multiply.nmf
Regression.nmf
Add.nmf
Verb.nmf
Regression.svd
Add.svd
Verb.svd
0.40
0.19
0.18
0.13
0.08
0.23
0.11
0.06
DisCo Models of Meaning
Empirical
Conclusion
60
Tensor Semantics
Empirical
Conclusion
Results (Gref et. al)
First Evaluation
Second Evaluation
Model
ρ
Model
ρ
UpperBound
Multiply.nmf
Regression.nmf
Add.nmf
Verb.nmf
Regression.svd
Add.svd
Verb.svd
0.40
0.19
0.18
0.13
0.08
0.23
0.11
0.06
UpperBound
Regression.nmf
Multiply.nmf
Add.nmf
Verb.nmf
Regression.svd
Add.svd
Verb.svd
0.62
0.29
0.23
0.07
0.04
0.32
0.12
0.08
DisCo Models of Meaning
60
Tensor Semantics
Empirical
Conclusion
Current Thoughts I
• What should the sentence space be?
• Should the sentence space be contextual?
• What should the learning mechanism be?
• Can current learning methods be generalised to typed tensors and
naturally occurring text?
DisCo Models of Meaning
61
Tensor Semantics
Empirical
Conclusion
Current Thoughts II
• How to deal with closed-class words (eg relative pronouns)
• How to combine distributional and symbolic methods
• find me all wild animals which might make good pets
DisCo Models of Meaning
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Tensor Semantics
Empirical
Conclusion
References
• Type-Driven Syntax and Semantics for Composing Meaning Vectors, Stephen
Clark, in OUP Quantum Physics and Linguistics: A Compositional, Diagrammatic
Discourse, Heunen, Sadrzadeh and Grefenstette (eds), 2013
• Mathematical Foundations for a Compositional Distributional Model of Meaning,
Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark, Linguistic Analysis: A
Festschrift for Joachim Lambek, van Bentham and Moortgat (eds), 2011
• Experimental Support for a Categorical Compositional Distributional Model of
Meaning, Edward Grefenstette and Mehrnoosh Sadrzadeh, In Proceedings of
EMNLP, Edinburgh, 2011
• Nouns are Vectors, Adjectives are Matrices: Representing Adjective-Noun
Constructions in Semantic Space, M. Baroni and R. Zamparelli, Proceedings of
EMNLP, Cambridge MA, 2010
• A Context-Theoretic Framework for Compositionality in Distributional Semantics,
Daoud Clarke, Computational Linguistics, 38(1):41-71, 2012.
• MultiStep Regression Learning for Compositional Distributional Semantics,
Edward Grefenstette, Georgiana Dinu, Yao-Zhong Zhang, Mehrnoosh Sadrzadeh and
Marco Baroni, Proceedings of IWCS, Potsdam, 2013
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