Towards a Geometrical Model for Polyrepresentation of Information Objects
Ingo Frommholz and C. J. van Rijsbergen
Department of Computing Science
University of Glasgow
fingojkeithg@dcs.gla.ac.uk
Abstract
The principle of polyrepresentation is one of the
fundamental recent developments in the field of
interactive retrieval. An open problem is how to
define a framework which unifies different as-
pects of polyrepresentation and allows for their
application in several ways. Such a framework
can be of geometrical nature and it may embrace
concepts known from quantum theory. In this
short paper, we discuss by giving examples how
this framework can look like, with a focus on in-
formation objects. We further show how it can be
exploited to find a cognitive overlap of different
representations on the one hand, and to combine
different representations by means of knowledge
augmentation on the other hand. We discuss the
potential that lies within a geometrical frame-
work and motivate its further development.
1 Introduction
One of the promising recent developments in information
retrieval (IR) is the idea of polyrepresentation, which came
up as a consequence of cognitive theory for interactive IR
[Ingwersen and Ja¨rvelin, 2005]. The basic idea is that en-
tities may be interpreted or represented in different func-
tional and cognitive ways. Finding relevant documents
goes along with finding the cognitive overlap of function-
ally or cognitively different information structures.
We can regard polyrepresentation w.r.t. information ob-
jects [Skov et al., 2006]. For instance, a Web document
can be represented by its content (which reflects the au-
thors view on the document). Nowadays, it is common
that users annotate a document in several ways. Annota-
tions may be, for instance, comments, opinions, tags or rat-
ings. Such annotations provide a cognitively different rep-
resentation of a document, in this case reflecting the users’
view on it. Another form of polyrepresentation considers
the user’s cognitive state [Kelly et al., 2005] and different
search engines [Larsen et al., 2009]. The former one in-
cludes the work task, the perceived information need, the
experience and the domain knowledge, and others. The lat-
ter one sees different search engines as different reflections
of the cognitive view of its designers on the retrieval prob-
lem. One of the conclusions from evaluating all these facets
of polyrepresentation is that the more positive evidence is
coming from different representations, the more likely is
the object in the cognitive overlap relevant to a given infor-
mation need.
The experiments on polyrepresentation suggest that
search effectiveness can benefit from a retrieval model
which explicitly supports the principle of polyrepresenta-
tion. What is missing so far is a unified view which in-
corporates the different facets of polyrepresentation which
allows for determining cognitive overlaps, but can go even
beyond. For instance, different representations may be
combined, as it is possible with knowledge augmentation
(see below), to create a new representation. A unified view
for polyrepresentation should also consider the combina-
tion of the concept of polyrepresentation with the dynam-
ics arising from interactive retrieval. Such a view can be
based on a geometrical model, as it was discussed in [van
Rijsbergen, 2004]. A growing number of geometrical mod-
els, inspired by quantum theory, were introduced recently.
For example, Piwowarski and Lalmas propose a geometri-
cal framework which takes into account the evolution of the
user’s information need (represented as a vector in a Hilbert
space) [Piwowarski and Lalmas, 2009]. So far, the concept
of polyrepresentation has not been discussed in this model.
The considerations presented here are a first attempt
to describe the notion of polyrepresentation (in particular
w.r.t. information objects) in a geometrical way. They are a
starting point for further discussion into how we can com-
bine the idea of polyrepresentation and geometrical mod-
els, possibly inspired by quantum theory.
2 Towards a Geometrical Model
Our discussion starts with an example of how document
features in different representations can be expressed geo-
metrically. A representation of a document can be based on
a set of distinct features. Such features can be topical, like
the appearance of a term in a document, or non-topical, for
example the document genre or the page rank. Documents
may have static features (like terms and their weights), but
they can also be dynamic (e.g., a property which shows
whether a document was presented to the user or not). In
general, we assume that for a feature f we can estimate the
probability Pr(f jd) that we observe the feature given that
we observed d. Similarly, Pr(f jd) = 1 Pr(f jd) denotes
the probability that we do not observe the feature.
Before we continue our considerations by giving an ex-
ample, we introduce the notation, which is used in quantum
mechanics as well.
2.1 Notation
We give a short introduction to the Dirac notation, which
we are going to use in the following. A vector x in a real1 n-
dimensional Hilbert space H can be written as a so-called
1Our considerations can be expanded to complex Hilbert
spaces, but for the time being it is sufficient to assume that H
is spanned over R

ket in Dirac notation:
jx i =
0
B
@
x1
...
xn
1
C
A
with x1; : : : xn 2 R. Transposed vectors are represented
as a bra, that is hx j = (x1; : : : ; xn). Based on this we
can define an inner product between two vectors x and y as
hx j y i =
Pn
i=1 yixi if we assume a canonical basis.
Besides inner products, we can also define an outer prod-
uct as jx i hy j = xyT , which yields a square n  n ma-
trix in our case. Each such matrix can be regarded as a
linear transformation or operator. Projectors are idem-
potent, self-adjoint linear operators; they can be used to
project vectors onto subspaces. For example, let je0 i =
(1; 0)T and je1 i = (0; 1)
T be the base vectors of a two-
dimensional vector spaceH, and jx i = (x1; x2)
T a vector
in H. Then P = je0 i he0 j is a projector onto the one-
dimensional subspace spanned by je0 i; P jx i = (x1; 0)
T
is the projection of jx i onto that subspace. jjxjj =pPn
i x
2
i denotes the norm of a vector, and jjxjj = 1
means the vector is a unit vector. If je0 i ; : : : ; jen i form
an orthonormal basis of a Hilbert space, then tr(T) =Pn
i=1 hei jT jei i is called the trace of the matrix T. It
is the sum of the diagonal elements of T.
2.2 Polyrepresentation of Information Objects
Representing Document Features
Our basic idea is to encode every feature in a qubit (quan-
tum bit). A qubit is a two-dimensional subspace whose
base represents two possible disjoint states j0 i = (1; 0)T
and j1 i = (0; 1)T . We give some examples of how a doc-
ument feature, in particular a term, can be expressed as a
qubit.
Let us assume we have a probabilistic indexer which as-
signs two probabilities to each term w.r.t. its correspond-
ing document: Pr(tjd) is the probability that document d
could be indexed with t, and Pr(tjd) = 1 Pr(tjd) is the
probability that it could not. Let j0t i and j1t i be the base
Figure 1: A term feature of d in a qubit
vectors of the qubit for term t. If we set  =
p
Pr(tjd)
and  =
p
Pr(tjd), then jd i = 
j1t i + 
j0t i is
a unit vector (length 1). The situation is depicted in Fig-
ure 1 with Pr(tjd) = 0:8 and Pr(tjd) = 0:2 resulting in
jd i = (d1; d2)
T = (
p
0:8;
p
0:2)
T
in this qubit.
Retrieval with Polyrepresentation Example
Let us assume that we have a collection consisting of two
terms, t1 and t2, and a document d with a user comment
(annotation) a attached to it, so we have two cognitively
different representations of the same document. We denote
these two representations by two vectors, jdc i for the con-
tent view and jda i for the annotation view on d. We give
an example of how we can derive a simple well-known re-
trieval function from our representation, namely the tradi-
tional vector space model (VSM) which measures the simi-
larity between a document and query vector in a term space.
In order to support this, we need to transform our represen-
tation based on qubits into the classical vector space repre-
sentation where the terms are the base vectors. One way to
achieve this is to create a new vector jd0c i = (d
0
1; d
0
2)
T with
d01 = j1t1 i h1t1 j jdc i (the projection of jdc i onto j1t1 i)
and d02 = j1t2 i h1t2 j jdc i. jd
0
c i is then a vector in the clas-
sical term space known from the VSM. We can create jd0a i
out of jda i analogously. The new situation is depicted in
Fig. 2. We can see that in contrast to the classical VSM,
where the document is represented by only one vector, we
now have two vectors for d, namely jd0c i and jd
0
a i. We
further assume that
P2
i=1 Pr(tijd) = 1 =
P2
i=1 Pr(tija),
which means that jd0c i and jd
0
a i are unit vectors. We can
Figure 2: Document representation and query
represent a content query as a normalised vector jq i in the
term space. We measure the similarity between the query
and the two document representations by applying the trace
function: tr(jq i hq j jdc i hdc j) = j hq j dc i j2. This equals
cos2  (where is the angle between jdc i and jq i) because
we assume all vectors to be normalised. The beauty of this
is that the resulting similarity value can be interpreted as
a probability; see [van Rijsbergen, 2004, p. 83] for further
details. By calculating the similarity between jq i and jda i
analogously, we get two probabilities, one for the content
representation and one for the representation of the docu-
ment by its comment. These probabilities can now be used
to determine the cognitive overlap of these representations
and they can be combined to calculate the final score of d
w.r.t. q.
We have seen that we can derive a well-known retrieval
function from our representation. But what is the benefit
from expressing features as qubits, when at least for term
features we could have created a term space without in-
troducing them? To answer this, we will now give an ex-
ample of the possible combination of different representa-
tions, which relies on the proposed description of features
as qubits.
Combination of Representations and Knowledge
Augmentation
One may wonder whether the representation of features as
a qubit is too redundant, since at least for the term fea-
tures we also store Pr(tjd), the probability that a document
cannot be indexed with a certain term. While in general
for other features it might be useful to store this probabil-
ity as well, it can be incorporated in a senseful way when
we want to combine different representations to create a

new one. This happens when for example we apply the
concept of knowledge augmentation. Here, we augment
our knowledge about an object with other objects which
are connected to it, according to the probability that we
actually consider them (see, e.g., [Frommholz and Fuhr,
2006] for a discussion of knowledge augmentation with an-
notations). Knowledge augmentation basically means to
propagate features and their weights from connected ob-
jects to the one under consideration. A simple example
shall illustrate knowledge augmentation in a geometrical
framework. Again, we have a document d and an associ-
ated annotation a. We want to augment d with a, which
means to propagate all term probabilities in a and also d
to an augmented representation of d, denoted d. In d,
the terms (features) and probabilities of d and a are ag-
gregated. Along with a goes Pr(cad), the probability that
we consider a when processing d. One can think of this
as a propagation factor2. We can store this probability in
a qubit as discussed above; the corresponding vector is
jc i = (c1; c2)
T =
p
Pr(cad)
j1 i +
p
1 Pr(cad)
j0 i.
Based on this, we can now propagate a term t from d and
a to d as follows. Qubits can be combined by means of
tensor products, and we perform knowledge augmentation
by calculating the tensor product of jd i, jc i and ja i:
jd i = jd i
jc i
ja i =
0
B
B
B
B
B
B
B
B
@
d1
c1
a1
d1
c1
a2
d1
c2
a1
d1
c2
a2
d2
c1
a1
d2
c1
a2
d2
c2
a1
d2
c2
a2
1
C
C
C
C
C
C
C
C
A
jd i, which represents d, is a vector in an 8-dimensional
space. The first element of jd i expresses the event that
we index d with t (d1) and consider a (c1) and a is indexed
with t (a1). The fifth element denotes the case that we do
not index d with t (d2) and consider a (c1) and a is in-
dexed with t (a1). Similarly for the other 6 elements. Each
base vector thus represents a possible event, and all these
events are disjoint. In fact, the resulting vector represents a
probability distribution over these events and is thus a unit
vector.
How can we now calculate the probability Pr(tjd) that
we observe t in the augmented representation d? We ob-
serve t in the augmented representation in the following
five cases: when we observe it in d, and when we do not ob-
serve it in d, but consider a and observe t there. These are
exactly the events described by the first 5 elements of jd i.
These elements contribute to Pr(tjd), whereas the last 3
elements of jd i determine Pr(tjd). To get Pr(tjd), we
project jd i to the subspace spanned by the first 5 base vec-
tors, and calculate the trace the projection. If Pt is such a
projector, then Pr(tjd) = tr(jd i hd jPt). Similarly for
Pr(tjd). Having achieved both probabilities, we can store
them in a qubit as discussed above, and repeat the proce-
dure for the other terms. Note that in this example, we
combined a term-based representation with another term-
based representation, but we are not bound to this. We can
also combine topical and non-topical representations of a
document in a similar way.
2A discussion of this probability is beyond the focus of this
paper. It might be system-oriented, e.g. determined by the number
of comments, or user-oriented, for instance by rating comments as
important or less important.
3 Discussion
We have seen examples for polyrepresentation of informa-
tion objects in a unified geometrical framework. Document
features, be it content features expressed as terms, or non-
topical ones, can be represented with the help of qubits
which encode the probabilities that a certain feature can
be observed or not. In this way, we can integrate different
representations of documents in one model, calculate their
relevance and use this information to compute the cogni-
tive overlap. Different representations of documents may
also be combined, as we have seen for knowledge augmen-
tation. This way, we can exploit the polyrepresentation of
information objects to obtain a higher-level representation.
This simple example can of course not properly define a
whole geometrical framework. This paper is not meant to
deliver such a definition, but to undertake a first step to-
wards it and to further motivate it. The following discus-
sion shall reveal what we potentially gain when we further
specify a geometrical framework which also includes inspi-
rations coming from quantum mechanics.
We showed an example with different representations
of information objects. In fact, also a polyrepresentation
of search engines is potentially possible within our frame-
work. How different retrieval models (like the generalised
vector space model, the binary independent retrieval model
or a language modelling approach) can be described geo-
metrically is reported in [Ro¨lleke et al., 2006]. It is in prin-
cipal possible to transfer these ideas into our framework,
although it has yet to be specified which further knowledge
(like relevance judgements) needs to be incorporated. An-
other extension of the model might also introduce polyrep-
resentation w.r.t the user’s cognitive state, which may be
represented as a vector similar to information objects.
The framework discussed in this paper may be used to
support other models which indirectly apply polyrepresen-
tation. An example is the Lacostir model introduced in
[Fuhr et al., 2008]. This model aims at the integration
and utilisation of layout, content and structure (and thus
polyrepresentation) of documents for interactive retrieval.
The core part of the model consists of certain operations
and their resulting system states. For instance, a selection
operator (basically a query) lets the user choose relevant
documents. Once relevant documents are determined, the
user can select suitable representations thereof with a pro-
jection operator. An organisation operator can be applied
by the user to organise the projected representations, for
instance in a linear list or graph. With the visualisation op-
erator, the user can choose between possible visualisations
of the organised results. During a session, the user can at
any time modify these operators. To support this model,
an underlying framework must be capable of handling the
different states the user and the system can be in as well as
the transitions between them. It also needs to deal with the
polyrepresentation of information objects. A geometrical
framework can potentially handle the different representa-
tions and the dynamics in such a system. At least the selec-
tion and projection operators might be mapped to geometri-
cal counterparts, whereas the organisation and visualisation
operators may benefit from a geometrical representation of
system states as vectors.
While we used some notations borrowed from quantum
mechanics, the examples so far are purely classical, but
with a geometrical interpretation. They give us a clue of the
tight relation between geometry and probability theory and
show the potential to embrace existing models in one uni-

fied framework. However, we did not touch any concepts
used in quantum mechanics yet, like entanglement or com-
plex numbers. For instance, different representations of an
information object can be related, a property which we ap-
ply with knowledge augmentation. This relationship may
also be expressed by using one state vector per feature and
document, but with a different basis for each representa-
tion. Different representations may be entangled, and such
property could easily be included in our model. An open
question therefore is how the relationship between different
representations should be modelled.
4 Related Work
The idea of using geometry for information retrieval, go-
ing far beyond the VSM, was formulated in [van Rijsber-
gen, 2004]. In this book, the strong connection between
geometry, probability theory and logics is expressed. The
examples in this paper are further inspired by Melucci’s
work reported in [Melucci, 2008]. Here, contextual fac-
tors (which may be different representations of information
objects, but also reflect the user’s point of view) are ex-
pressed as subspaces. Information objects and also queries
are represented as vectors within these subspaces. Given
this representation, a probability of context can be com-
puted. This resembles the idea sketched in this paper, but
the approach is not focused on polyrepresentation of ob-
jects. In the model presented by Piwowarski and Lalmas,
a user’s information need is represented as a state vector
in a vector space which may for instance be set up by
(possibly structured) documents [Piwowarski and Lalmas,
2009]. Initially, less is known about the actual informa-
tion needs. Each user interaction gains more knowledge
about her information need, which lets the state vector col-
lapse until the information need is expressed unambigu-
ously. Schmitt proposes QQL, a query language which in-
tegrates databases and IR [Schmitt, 2008]. In his work,
he makes use of qubits as the atomic unit of retrieval val-
ues and interrelates quantum logic and quantum mechanics
with database query processing. Further approaches about
the relation of quantum theory and IR are reported in the
proceedings of the Quantum Interaction symposium (see,
e.g., [Bruza et al., 2009]).
5 Conclusion and Future Work
In this short paper, we showed by an example how polyrep-
resentation of information objects can be realised geometri-
cally. The goal is to undertake a first step towards a unified
framework for polyrepresentation, which is missing so far.
The example also shows how we can geometrically com-
bine different representations to a new one. A subsequent
discussion reveals some of the further possibilities coming
from a geometrical approach.
We will also investigate the integration of existing
quantum-inspired models into the framework, like the ones
reported in [Piwowarski and Lalmas, 2009] or [Melucci,
2008], which do not deal with polyrepresentation yet.
These models may thus be extended with the ideas that
came up in the discussion so far, like knowledge augmen-
tation and the possible entanglement of representations.
6 Acknowledgements
Our work was supported by the EPSRC project Renais-
sance3 (grant number EP/F014384/1).
3http://renaissance.dcs.gla.ac.uk/
References
[Bruza et al., 2009] Peter Bruza, Donald Sofge, William
Lawless, Keith van Rijsbergen, and Matthias Klusch,
editors. Proceedings of the Third International Sympo-
sium on Quantum Interaction (QI 2009), LNCS, Heidel-
berg et al., March 2009. Springer.
[Frommholz and Fuhr, 2006] Ingo Frommholz and Nor-
bert Fuhr. Probabilistic, object-oriented logics for
annotation-based retrieval in digital libraries. In M. Nel-
son, C. Marshall, and G. Marchionini, editors, Proc. of
the JCDL 2006, pages 55–64, New York, 2006. ACM.
[Fuhr et al., 2008] Norbert Fuhr, Matthias Jordan, and
Ingo Frommholz. Combining cognitive and system-
oriented approaches for designing IR user interfaces.
In Proceedings of the 2nd International Workshop on
Adaptive Information Retrieval (AIR 2008), October
2008.
[Ingwersen and Ja¨rvelin, 2005] P. Ingwersen and
K. Ja¨rvelin. The turn: integration of information
seeking and retrieval in context. Springer-Verlag New
York, Inc., Secaucus, NJ, USA, 2005.
[Kelly et al., 2005] Diane Kelly, Vijay Deepak Dollu, and
Xin Fu. The loquacious user: a document-independent
source of terms for query expansion. In Proceedings of
the SIGIR 2005, pages 457–464, New York, NY, USA,
2005. ACM.
[Larsen et al., 2009] Birger Larsen, Peter Ingwersen, and
Berit Lund. Data fusion according to the principle of
polyrepresentation. Journal of the American Society for
Information Science and Technology, 60(4):646–654,
2009.
[Melucci, 2008] Massimo Melucci. A basis for informa-
tion retrieval in context. Information Processing &Man-
agement, 26(3), June 2008.
[Piwowarski and Lalmas, 2009] Benjamin Piwowarski
and Mounia Lalmas. Structured information retrieval
and quantum theory. In Bruza et al. [2009], pages
289–298.
[Ro¨lleke et al., 2006] Thomas Ro¨lleke, Theodora
Tsikrika, and Gabriella Kazai. A general matrix
framework for modelling information retrieval. In-
formation Processing and Management, 42(1):4–30,
2006.
[Schmitt, 2008] Ingo Schmitt. QQL: A DB&IR query lan-
guage. The International Journal on Very Large Data
Bases, 17(1):39–56, 2008.
[Skov et al., 2006] Mette Skov, Birger Larsen, and Peter
Ingwersen. Inter and intra-document contexts applied in
polyrepresentation. In Proceedings of IIiX 2006, pages
97–101, New York, NY, USA, 2006. ACM.
[van Rijsbergen, 2004] C. J. van Rijsbergen. The Geom-
etry of Information Retrieval. Cambridge University
Press, New York, NY, USA, 2004.