© CvR1 The Geometry of IR Keith van Rijsbergen Tampere 15 th August, 2002 (lost in Hilbert space!)

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The Geometry of IR

Keith van Rijsbergen

Tampere 15th August, 2002(lost in Hilbert space!)

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“Unscripted” comments I

StatesObservablesMeasurement

=> Reality?Projection PostulatesCognitive State Changes

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“Unscripted” comments II (quoting John von Neumann)

However, all quantum mechanical probabilities are defined byinner products of vectors. Essentially if a state of a system is givenby one vector, the transition probability in another state is theinner product of the two which is the square of the angle betweenthem. In other words, probability correspond precisely to intro-ducing the angles geometrically. Furthermore, there is only oneway to introduce it. The more so because in the quantum mechanicalmachinery the negation of a statement, so the negation of a statementwhich represented by a linear set of vectors, correponds to theorthogonal complement of this linear space.

Unsolved problems in mathematics, typescript, September, 1954

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What is this talk about?

Not about quantum computation.see Nielsen and Chuang, CUP, 2000

Not about Logicsee 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)

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Images not Text: how might thatmake a difference?

no visual keywords (yet)- tf/idf issue

aboutness revisable (eg Maron)relevance revisable (eg Goffman)feedback requires salienceaboutness -> relevance -> aboutness

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This is not new!

Goffman, 1969: ‘..that the relevance of the informationfrom one document depends upon what is already knownabout the subject, and in turn affects the relevance of otherdocuments subsequently examined.’

Maron, : ‘Just because a document is about the subject sought by a patron, that fact does not imply that he wouldjudge it relevant.’

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Maron’s theory of indexing

…..in the case where the query consists of singleterm, call it B, the probability that a given documentwill be judged relevant by a patron submitting Bis simply the ratio of the number of patrons who submitB as their query and judge that document as relevant,to the number of patrons, who submit B as their search query

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JPEG Image

In 1949 D.M Mackay wrote a paper ‘Quantalaspects of scientific information’, SER, vol 41, no.314,in which he alluded to using the quantum mechanicsparadigm to IR

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Expectation Catalogue

It (-function) is now the means for predicting probability ofmeasurement results. In it is embodied themomentarily-attained sum of theoretically basedfuture expectation, somewhat as laid down in acatalogue. It is the relation-and-determinacy-bridgebetween measurements and measurements......It is, in principle, determined by a finite number of suitablychosen measurement on the object.....Thus the catalogueof expectations is initially compiled.

Schrödinger, 1935 &1980

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Hypotheses

Cluster Hypothesis: closely associated documents tend to berelevant to the same requests. (1971)

[co-ordination is positively correlated with external relevance,Jackson, 1969]

Association Hypothesis: If an index term is good at discriminatingrelevant from non-relevant documents then any closely associatedindex term is also likely to be good at this. (1979)

[co-occurrence of terms within documents is a suitable measureof similarity between terms, Jackson,1971]

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Navigation - Browsing

T-space

D-space

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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: [[ux]] = {d”}; [[xm]] ={d’,d’’}Quantum Logic: [[ux]] = same; [[xm]] = different

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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) =aii

•eigenvalues Ax = x

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Crash course on Dirac notation

|x> : vector (called ket)<x| = |x>*: functional (bra)

<x|y> = (row vector)(column vector)= xi*yi

|x><y| : linear operator|x><x| : a projector onto ray xtr(|x><y|) = <x|y>

I = |i><i| : universal projector

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Hierarchy of ProjectorsP0 = Pn = I

P1 = |1><1|P2 = |1><1| +|2><2|

.

.

.Pn = |1><1| +…+|n><n|

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Summary

Relevance/Aboutness

Documents

Queries

Observables

Operators

State function

Operators can be applied to state function; andoperators can be decomposed into projectors.

A = aiPi

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‘That is the relevance or irrelevance ofa given retrieved document may affectthe user’s current state of knowledgeresulting in a change of the user’sinformation need, which may lead toa change of the user’s perception/interpretation of the subsequentretrieved documents….’ Borlund, 2000

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T T

T

RR

Y

N

Y

N

N

Y

Relevance/Aboutnes is

Interaction/User dependent

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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| = cos2 |t><t| (in real Hilbert space)

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|r=1>

|t=1>

|t=0>

|r=0> x

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An operator T is of trace-class provided that T is positive(<x|T|x> 0, x) and trace of T is finite (<ei|T|ei>)

T is a density operator if T is trace-class and tr(T) = 1

T = aiPi is a density operator if 0 ai and ai = 1

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Theorem

Letbe any measure on the closed subspaces of aseparable (real or complex) Hilbert space H of dimensionat least 3. There exists a positive self-adjoint operator Tof trace class such that, for all closed subspace L of H,

(L) = Tr(TPL)

If is to be a probability measure, thus requiring that(H) = 1, then Tr(T) =1, that is, T is a density operator.

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Conditional Probability

P(LA|LB) = tr(PBDPBPA) / tr(DPB)

Note that PA could be E -> F

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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)

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Conclusions?

Is it worth it? Does it matter?

- images- logic/probability/information/vectors- language

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Useful ReferencesReadings 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.