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№1 слайд![](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img0.jpg)
№2 слайд![Bra-ket notation Dirac,](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img1.jpg)
Содержание слайда: Bra-ket notation (Dirac, 1939)
– vector (Hilbert) space, - field
- pure state (vector, or operator ) – ket
- effect of state (dual vector,
dual operator Hermitian conjugate) – bra
Inner product of and is
№3 слайд![Outer product Outer product](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img2.jpg)
Содержание слайда: Outer product
Outer product for is a operator:
Arbitrary can be written
in a basis for and for :
,
where
№4 слайд![Eigenvectors and eigenvalues](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img3.jpg)
Содержание слайда: Eigenvectors and eigenvalues
, if is an orthogonal basis in which is diagonal.
- are eigen vectors
- are eigen values
Easy to check:
№5 слайд![Density operator matrix If -](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img4.jpg)
Содержание слайда: Density operator (matrix)
If - are pure states,
- are probabilities over them, then
is a dense operator
Positive operator:
for all
Theorem: is a density operator iff it’s a positive Hermitian operator with trace =1.
№6 слайд![Trace inner product and are](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img5.jpg)
Содержание слайда: Trace inner product
and are density matrices same dimension
and
№7 слайд![Distributional Semantics You](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img6.jpg)
Содержание слайда: Distributional Semantics
“You shall know the word by the company it keeps” (Firth)
Obtain meaning high dimensional vector representations from large corpora automatically
Compositionality
DS can not be applied for entire sentence (lack of frequency)
Entailment
entails if the meaning of a word is included in the meaning of a word (is-a ) - subsumption relation
non symmetric
№8 слайд![Distributional Inclusion](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img7.jpg)
Содержание слайда: Distributional Inclusion Hypothesis
If is semantically narrower than , then a significant number of salient distributional features of are also included in the feature vector of :
Hypothesis 1: If => then all the characteristic features of is expected to appear in .
Hypothesis 2: If all the characteristic features of appear in , then => .
№9 слайд![Category Theory A monoidal](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img8.jpg)
Содержание слайда: Category Theory
A monoidal category C is a category consisting of the following:
a functor called the tensor product
an object called the unit object
a natural isomorphism whose components
are called the associators
a natural isomorphism whose components are called the left unitors
a natural isomorphism whose components are called the right unitors
№10 слайд![Category Theory The objects](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img9.jpg)
Содержание слайда: Category Theory
The objects of the category are thought to be types of systems
A morphism is a process that takes a system of type to a system of type .
for and , is the composite morphism that takes a system of type into a system of type by applying the process after .
Morphisms of type are called elements of .
№11 слайд![Compact closed categories A](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img10.jpg)
Содержание слайда: Compact closed categories
A monoidal category is compact closed if for each object , there are also left and right dual objects and , and morphisms
that satisfies
The maps of compact categories are used to represent correlations, and in categorical quantum mechanics they model maximally entangled states.
№12 слайд![Graphical calculus](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img11.jpg)
Содержание слайда: Graphical calculus
№13 слайд![Graphical calculus](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img12.jpg)
Содержание слайда: Graphical calculus
№14 слайд![Compositional Distributional](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img13.jpg)
Содержание слайда: Compositional Distributional Model
Pregroup grammars (Lambek)
A partially ordered monoid consists of:
a set
a monoid multiplication operator satisfying the condition
for all
and thу monoidal unit where for all
a partial order on
№15 слайд![Pregroup Lambek, A pregroup](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img14.jpg)
Содержание слайда: Pregroup (Lambek, 2001)
A pregroup is a partially ordered monoid in which each element has both a left adjoint and a right adjoint such that
and
Adjoints have properties:
Uniqueness: Adjoints are unique
Order reversal: If then and
The unit is self adjoint:
Multiplication operation is self adjoint: and
Opposite adjoints annihilate:
Same adjoints iterate:
№16 слайд![Pregroup grammar means](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img15.jpg)
Содержание слайда: Pregroup grammar
means ( reduces to )
“John likes Mary”
“John” and “Mary” assigned to type (noun)
“likes” is assigned to compound type
“likes” takes a noun from the left and from the right, and returns a sentence
№17 слайд![Basic types noun declarative](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img16.jpg)
Содержание слайда: Basic types
: noun : declarative statement
: infinitive of the verb : glueing type
№18 слайд![Pregroups as compact closed](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img17.jpg)
Содержание слайда: Pregroups as compact closed categories
is a concrete instance of a compact closed category
Test snake identities:
…
№19 слайд![Examples John likes Mary](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img18.jpg)
Содержание слайда: Examples
“John likes Mary”
“John does not like Mary”
№20 слайд![- finite dimensional vector](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img19.jpg)
Содержание слайда: - finite dimensional vector space
- finite dimensional vector spaces over the base field R together with linear maps, form a monoidal category
FVect as a compact closed category.
№21 слайд![categorical representation of](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img20.jpg)
Содержание слайда: – categorical representation of meaning space
№22 слайд![](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img21.jpg)
Содержание слайда: “From-the-meanings-of-words-to-the-meanings-of-the-sentence” map
Let be a string of words, each with a meaning space representation . Let be a pregroup type such that . Then the meaning vector for the string is:
,
where is defined to be the application of the compact closure maps obtained from the reduction to the composite vector space .
№23 слайд![Example John likes Mary It](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img22.jpg)
Содержание слайда: Example: “John likes Mary”
It has the pregroup type
vector representations and
The morphism in corresponding to the map is of type:
From the pregroup reduction we obtain the compact closure maps . In this translates into:
№24 слайд![Example John likes Mary](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img23.jpg)
Содержание слайда: Example: “John likes Mary”
№25 слайд![Example](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img24.jpg)
Содержание слайда: Example
№26 слайд![Example](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img25.jpg)
Содержание слайда: Example
№27 слайд![Example](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img26.jpg)
Содержание слайда: Example
№28 слайд![Example](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img27.jpg)
Содержание слайда: Example
№29 слайд![Example](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img28.jpg)
Содержание слайда: Example
№30 слайд![Readings Esma Balkir. Using](/documents_6/c4ab13e7e70f2f49996f2bac8fe6f97f/img29.jpg)
Содержание слайда: Readings
Esma Balkir. Using Density Matrices in a Compositional Distributional Model of Meaning. // Master thesis. University of Oxford. 2014
Joachim Lambek. Type grammars as pregroups. Grammars, 4(1):21{39, 2001.