Transcript Document

The Geometry of IR
Keith van Rijsbergen
(lost in Hilbert space!)
Tampere 15th August, 2002
© CvR
1
“Unscripted” comments I
States
Observables
Measurement
=> Reality?
Projection Postulates
Cognitive State Changes
© CvR
2
“Unscripted” comments II
(quoting John von Neumann)
However, all quantum mechanical probabilities are defined by
inner products of vectors. Essentially if a state of a system is given
by one vector, the transition probability in another state is the
inner product of the two which is the square of the angle between
them. In other words, probability correspond precisely to introducing the angles geometrically. Furthermore, there is only one
way to introduce it. The more so because in the quantum mechanical
machinery the negation of a statement, so the negation of a statement
which represented by a linear set of vectors, correponds to the
orthogonal complement of this linear space.
Unsolved problems in mathematics, typescript, September, 1954
© CvR
3
What is this talk about?
Not about quantum computation.
see Nielsen and Chuang, CUP, 2000
Not about Logic
see Engesser and Gabbay, AI, 2002
• History (von Neumann, Dirac, Schroedinger)
• Motivation (complementarity)
• Duality (Syntax/Semantics)
• Measurement (Incompatibility)
• Projections (subspaces)
• Probability (inner products)
• IR application (feedback, clusters, ostension)
© CvR
4
© CvR
5
© CvR
6
Images not Text: how might that
make a difference?
no visual keywords (yet)
- tf/idf issue
aboutness revisable (eg Maron)
relevance revisable (eg Goffman)
feedback requires salience
aboutness -> relevance -> aboutness
© CvR
7
This is not new!
Goffman, 1969: ‘..that the relevance of the information
from one document depends upon what is already known
about the subject, and in turn affects the relevance of other
documents subsequently examined.’
Maron, : ‘Just because a document is about the subject
sought by a patron, that fact does not imply that he would
judge it relevant.’
© CvR
8
Maron’s theory of indexing
…..in the case where the query consists of single
term, call it B, the probability that a given document
will be judged relevant by a patron submitting B
is simply the ratio of the number of patrons who submit
B as their query and judge that document as relevant,
to the number of patrons, who submit B as their search
query
© CvR
9
In 1949 D.M Mackay wrote a paper ‘Quantal
aspects of scientific information’, SER, vol 41, no.314,
in which he alluded to using the quantum mechanics
paradigm to IR
© CvR
10
Expectation Catalogue
It (y-function) is now the means for predicting probability of
measurement results. In it is embodied the
momentarily-attained sum of theoretically based
future expectation, somewhat as laid down in a
catalogue. It is the relation-and-determinacy-bridge
between measurements and measurements......
It is, in principle, determined by a finite number of suitably
chosen measurement on the object.....Thus the catalogue
of expectations is initially compiled.
Schrödinger, 1935 &1980
© CvR
11
Hypotheses
Cluster Hypothesis: closely associated documents tend to be
relevant to the same requests. (1971)
[co-ordination is positively correlated with external relevance,
Jackson, 1969]
Association Hypothesis: If an index term is good at discriminating
relevant from non-relevant documents then any closely associated
index term is also likely to be good at this. (1979)
[co-occurrence of terms within documents is a suitable measure
of similarity between terms, Jackson,1971]
© CvR
12
Navigation - Browsing
T-space
D-space
© CvR
13
DUALITY
Direct file/Inverted file Statespace/Space of Projections
d’ = (x,y,z,u,v,w)d” =(u,v,w,k,l,m)
[[u]] = {d’,d”}; [[x]] = {d’}; [[m]] = {d”}
Boolean Logic: [[ux]] = {d”}; [[xm]] ={d’,d’’}
Quantum Logic: [[ux]] = same; [[xm]] = different
© CvR
14
The mathematics you need
Hilbert space (complex!!!)
•inner product
<x|y>
•norms
||x||2 = <x|x>
•operator (linear)
<x|A|y>
•Hermitian
A*=A
•trace
tr(A) =Saii
•eigenvalues
Ax = lx
© CvR
15
Crash course on Dirac notation
|x> : vector (called ket)
<x| = |x>*: functional (bra)
<x|y> = (row vector)(column vector)= S xi*yi
|x><y| : linear operator
|x><x| : a projector onto ray x
tr(|x><y|) = <x|y>
I = S |i><i| : universal projector
© CvR
16
Hierarchy of Projectors
P0 = f; Pn = I
P1 = |1><1|
P2 = |1><1| +|2><2|
.
.
.
Pn = |1><1| +…+|n><n|
© CvR
17
Summary
Relevance/Aboutness
Observables
Queries
Operators
State function
Documents
Operators can be applied to state function; and
operators can be decomposed into projectors.
A=
© CvR
S aP
i
i
18
‘That is the relevance or irrelevance of
a given retrieved document may affect
the user’s current state of knowledge
resulting in a change of the user’s
information need, which may lead to
a change of the user’s perception/
interpretation of the subsequent
retrieved documents….’ Borlund, 2000
© CvR
19
Relevance/Aboutnes
is
Interaction/User dependent
N
Y
N
T
© CvR
Y
R
T
Y
N
T
20
probability as inner product
|t><t||r><r||t><t| =|t> <t|r><r|t> <t|
= <t|r><r|t> |t><t|
= |<t|r>|2 |t><t|
= cos2f |t><t| (in real Hilbert space)
© CvR
21
|t=0>
|r=0>
x
|r=1>
|t=1>
© CvR
22
An operator T is of trace-class provided that T is positive
(<x|T|x>  0, x) and trace of T is finite (S <ei|T|ei>)
T is a density operator if T is trace-class and tr(T) = 1
T = S aiPi is a density operator if 0  ai and S ai = 1
© CvR
23
Theorem
Let m be any measure on the closed subspaces of a
separable (real or complex) Hilbert space H of dimension
at least 3. There exists a positive self-adjoint operator T
of trace class such that, for all closed subspace L of H,
m(L) = Tr(TPL)
If m is to be a probability measure, thus requiring that
m(H) = 1, then Tr(T) =1, that is, T is a density operator.
© CvR
24
Conditional Probability
P(LA|LB) = tr(PBDPBPA) / tr(DPB)
Note that PA could be E -> F
© CvR
25
What is T? – without blinding you with science
-Relevance Feedback ( a mixture with log weights)
-Pseudo relevance feedback (a mixture with similarity weights)
-Clustering (superposition of members?)
-Ostension (a history)
© CvR
26
Conclusions?
Is it worth it? Does it matter?
- images
- logic/probability/information/vectors
- language
© CvR
27
Useful References
Readings in Information Retrieval,Morgan
Kaufman, Edited by Sparck Jones and Willett
Advances in Information Retrieval: Recent
Research from CIIR, Edited by Bruce Croft.
Information Retrieval: Uncertainty and
Logics,Advanced Models for the Representation
and Retrieval of Information, Edited by Crestani,
Lalmas, Van Rijsbergen.
Finding out about, Richard Belew.
© CvR
28