Supporting Polyrepresentation in a Quantum-inspired
Geometrical Retrieval Framework
Ingo Frommholz
Department of Computing
Science
University of Glasgow
ingo@dcs.gla.ac.uk
Birger Larsen
Royal School of Library and
Information Science
Copenhagen, Denmark
blar@db.dk
Benjamin Piwowarski
Department of Computing
Science
University of Glasgow
bpiwowar@dcs.gla.ac.uk
Mounia Lalmas
Department of Computing
Science
University of Glasgow
mounia@dcs.gla.ac.uk
Peter Ingwersen
Royal School of Library and
Information Science
Copenhagen, Denmark
pi@db.dk
Keith van Rijsbergen
Department of Computing
Science
University of Glasgow
keith@dcs.gla.ac.uk
ABSTRACT
The relevance of a document has many facets, going beyond
the usual topical one, which have to be considered to satisfy
a user's information need. Multiple representations of doc-
uments, like user-given reviews or the actual document con-
tent, can give evidence towards certain facets of relevance.
In this respect polyrepresentation of documents, where such
evidence is combined, is a crucial concept to estimate the
relevance of a document. In this paper, we discuss how a ge-
ometrical retrieval framework inspired by quantum mechan-
ics can be extended to support polyrepresentation. We show
by example how dierent representations of a document can
be modelled in a Hilbert space, similar to physical systems
known from quantum mechanics. We further illustrate how
these representations are combined by means of the tensor
product to support polyrepresentation, and discuss the case
that representations of documents are not independent from
a user point of view. Besides giving a principled framework
for polyrepresentation, the potential of this approach is to
capture and formalise the complex interdependent relation-
ships that the dierent representations can have between
each other.
Categories and Subject Descriptors
H.3.3 [Information Storage and Retrieval]: Information
Search and Retrieval
General Terms
Theory
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Keywords
Quantum-inspired model, Polyrepresentation
1. INTRODUCTION
When users seek for information, their decision about
whether a document is useful usually depends on more di-
mensions than the usual IR system's estimation of topical
relevance [25]. Besides its content, this decision involves dif-
ferent contextual aspects of a document, like for instance,
writing style, understandability, authority, which are con-
cerned with the non-topical usefulness of a document [4].
We use an example of a book store scenario to illustrate
this. Consider a user whose goal is to nd \good introduc-
tions to quantum mechanics". The user might visit an online
bookstore, where it is common to nd dierent representa-
tions of documents, for example, abstracts, full texts, user-
given tags, editorial and user-given reviews, user-given rat-
ings and structured bibliographical metadata (e.g. author,
title, number of pages and publication date). These dier-
ent representations potentially answer dierent facets of the
information need. Title, abstract, tags or full text can pro-
vide evidence about the topicality of a book, but reviews
may also be used to determine if the book is about \quan-
tum mechanics". Another facet of the information need is
that the user seeks for an\introduction"to the topic; we may
get hints about this from the abstract, or the title, but to
a lesser degree from the full text (\introduction" appears in
many texts as a chapter heading and does not mean than the
document is an introduction to a given eld). Finally, the
evidence about the quality of the book (\good") may come
from ratings and reviews. These dierent representations
may interplay and have dierent importance during search.
For example, the user may judge one book relevant based
on given representations (for instance the author is known
to be an authority in the eld), and the relevance of another
book based on dierent representations (e.g., the document
has high ratings).
The above example underlines the fact that by having and
considering dierent representations of a document { from
dierent contexts { we are able to address dierent aspects
of the usefulness of a document. Naturally, we want to com-
bine the given evidence to get a more accurate estimation

Figure 1: Dierent representations and cognitive overlaps.
The intersection R1, R2 and R3 denes the total cognitive
overlap; all other intersections between representations es-
tablish a partial cognitive overlap.
of the usefulness of a document. One strategy of doing so
is to apply the principle of polyrepresentation of documents,
which aims to generate and exploit the cognitive overlap be-
tween dierent representations of documents { those doc-
uments that lie in this overlap are assumed to be relevant
given a user's information need [9, 10].
We regard polyrepresentation as a key principle to satisfy
a user's information need. This cognitive viewpoint allows
all components in IR to be considered in a single coherent
and consistent theoretical framework. It is thus holistic in
its intention and has a strong focus on exploiting dierent
contexts in IR. The principle of polyrepresentation has, how-
ever, so far not been developed to a point where a strong
and complete formal mathematical IR framework encom-
passes the principle in its entirety. One reason might be its
inherent complexity with its many factors and interdepen-
dencies, which goes beyond what typical IR models cover.
The goal of the paper is to develop a mathematical formal-
ism that takes a much larger range of phenomena into con-
sideration than current IR models. We do so by extending to
polyrepresentation the geometrical IR framework presented
in [21]. This framework is based on the idea of exploiting
the quantum mechanics formalism for information retrieval
as was suggested in [28]. Besides investigating how such a
framework can be extended to support polyrepresentation,
we will also show how it is possible to model interdependen-
cies between the representations. Due to the relationship
between geometry and probability theory outlined in [28],
the proposed framework is also probabilistic by nature.
The remainder of the paper is structured as follows. In
the next section we rst give an overview of the principle
of polyrepresentation and its implied requirements. In Sec-
tion 3 we brie
y introduce the quantum-inspired IR frame-
work that we extend in the two next sections. In Section 4 we
give examples how individual representations can be mod-
eled. To support polyrepresentation, the representations
need to be combined by creating the cogntive overlap known
from the polyrepresentation principle, also considering de-
pendencies between representations. These aspects are dis-
cussed in Section 5. Subsequently, we review related works
and conclude.
To illustrate various concepts in this paper, our examples
in this paper will all be inspired by the task of searching in
a book store website.
2. POLYREPRESENTATION
In the principle of polyrepresentation all components in
Information Retrieval (IR) are regarded as being the result
of cognitive transformations of the knowledge structures of
the involved actors [9, 10]. That is, documents1 are seen
as representations of their authors' ideas, retrieval mod-
els and systems as representations of their designers' ideas,
as well as information behaviour including issued queries
and interaction with IR systems as representations of users'
needs etc. Also later interpretations of a given document by
other authors, e.g., in reviews, citations, twitter feeds etc,
are regarded as representations but with dierent cognitive
origins. Figure 1 illustrates how the intersection of dier-
ent representations maybe seen to create so-called cognitive
overlaps. The principle of polyrepresentation hypothesises
that documents retrieved by representations that are more
dierent from each other in cognitive origin and time have
higher probability of being relevant. Thus as an example,
if Figure 1 is taken to illustrate sets of documents retrieved
using three cognitively dierent representations, we would
expect the documents in the total cognitive overlap between
R1, R2 and R3 to have a higher probability of being rel-
evant than those in partial overlaps, e.g., between R2 and
R3, or those outside the overlaps.
As mentioned above, so far there is no formal model which
covers the principle of polyrepresentation in its entirety. In
the following we analyse some of the main features a formal
model of polyrepresentation should encompass based on the
extensive account in [9]. Basically, we deal with the polyrep-
resentation of documents in this work; some of the addressed
aspects are:
 Flexible combination of representations, e.g. fusion of
representations (Boolean and weighted cognitive over-
laps); diusion of representations and selection of rep-
resentations given a certain context;
 Temporal aspects and dynamic changes over time in
relation to representations, for instance changes/drifts
in the user's information need and new interpreta-
tions of documents arising subsequently from other
agents (e.g., categorisation/indexing/tagging, in-links
and anchor text, social bookmarking, reviews, ratings,
annotations, etc.);
 Dierent semantic document levels, e.g., sub-
document level (akin to element/XML retrieval, log-
ical document structure) and groups/clusters of docu-
ments;
 Although the representations may be kept separate
and processed independently, it is a fact that they are
interdependent and contextual to each other, which is
a further aspect to consider.
The
exible combination of representations, also taking into
account possible interdependencies, and temporal aspects
regarding new interpretations are described in Section 5.
The discussion of our basic framework and the examples
of it, which can be found in sections 3 and 4, respectively,
covers dynamic information need changes as well as how
dierent semantic document levels can be addressed. As a
more technical issue, a polyrepresentation framework also
needs to handle several heterogeneous textual (e.g., doc-
ument content, reviews and annotations) and non-textual
1By documents we mean \physical (digital) entities in a va-
riety of media", that is information objects including text
documents as dened in [10]

document representations (like ratings). Examples of how
such representations can be modelled are given in Section 4.
The selection of features represents the ones we regard
as the most important in working with polyrepresentation
on the document side. Apart from these features, we can
also identify others, which we do not address directly in
this work, including dierent levels of representation of the
users' information need and exploitation of the user context,
and keeping track of the representation origin (origin/actor,
time). A deeper discussion of these features and their in-
tegration into the framework presented here are subject to
future work. Such an integration is in our view possible.
In this paper, we do not deal with the issue to relate a
facet of an information need to one or more dierent repre-
sentations. For example, for a user typing \good introduc-
tions to quantum mechanics", we would need to distinguish
three dierent facets (\good", \introductions" and \quantum
mechanics") and map them appropriately to the rating and
topical spaces (e.g., title/content and comments). We there-
fore assume that, through either an interface or a sophisti-
cated algorithm, we are able to analyse and assign the user's
request appropriately.
3. QUANTUM-INSPIREDGEOMETRICAL
IR FRAMEWORK
We describe the geometrical framework proposed in [21],
upon which our work is based. In this framework, the IR
system is initially uncertain about the user's information
need (IN), and two dynamics modify the system view of
the user's IN. Firstly, when the user interacts with the sys-
tem, for instance by typing, rening a query or browsing,
the system view of the user's IN becomes more and more
specic, i.e., the uncertainty of the system about the IN is
reduced. We refer to this dynamic process as (D1). Sec-
ondly, the IN may change from a user point of view ; if a user
gathers more knowledge in the information seeking process,
the IN may become more specic or may drift, for instance
when the user's perceived information need changes. We
refer to this process as (D2). Supporting this process sat-
ises the requirement on a formal polyrepresentation model
for addressing changing information needs, as stated in the
previous section.
The postulate made in [21] is that this interaction can
be captured using the connection between probabilities and
geometry present in the quantum physics formalism, whose
connection with IR has been discussed in [28, ch. 6]. This
work was applied to the topical relevance facet of the infor-
mation need [19]. From the next section onwards, we show
how we extend it to cover dierent facets of relevance.
3.1 Information Need Space
The assumption underlying the framework of [21] is that
there exists an Information Need space where any user's
\pure" IN can be represented from an IR system point of
view. \Pure" in this sense means that the user's IN is com-
pletely dened, i.e., if the IR system knew the user's pure
state, then it would exactly know what the user is look-
ing for, and return the documents that are relevant to that
user's IN.
This view is motivated from quantum mechanics that pos-
tulates that associated with each physical system is a space,
the state space. Formally, this state space is a Hilbert space
|ϕ〉
O
d
3
O
d
2
O
d
1
Figure 2: Projections onto subspaces in a 3-dimensional vec-
tor space with a state vector j'i
H (a vector space with an inner product). Following the
Dirac notation used in quantum mechanics, we will denote
a vector as j'i. A physical system (e.g., a photon) whose
state is known is completely described by a state vector j'i,
which is a unit vector (i.e. its norm k'k is 1) in the space
H.
By analogy, in the IR framework proposed by [21], the
state corresponds to the IN of a user, expressed in a state
space which we call the IN space. In the following, we rst
describe how to compute quantities of interest for an IR sys-
tem (e.g. the probability of relevance), and then show how
to update the state given some observation (i.e., interaction
with the user).
In the quantum formalism, any event is dened by a sub-
space. Hence, in the IN space, for each document d, we can
dene a subspace Od corresponding to the event \the docu-
ment d is relevant". If we let j'i be the user's IN state, the
probability Pr(Rjd; ') of the document d being relevant to
the user's IN j'i is dened as the square of the length of
the projection of the vector j'i onto the subspace Od [28],
which adheres to the denition of probabilities in quantum
mechanics.
For example, let us assume we have three documents d1,
d2 and d3. In Figure 2 we can see the corresponding docu-
ment subspaces, Od
1
, Od
2
and Od
3
, and the state vector j'i
representing the user's pure information need. In this state,
the resulting ranking would be d3, d1, d2 as the length of the
projection of j'i onto Od
3
is greater than onto Od
1
, which
is greater than the length of the projection onto Od
2
. The
actual probability of relevance for, say, d3 is the square of
the length of the projection of j'i onto Od
3
.
Returning to Section 2, we discussed the need to support
dierent semantic document levels for polyrepresentation.
This can be achieved by utilising the above description of
documents (respectively their relevance) as subspaces. For
instance, rening a document into further subspaces is a
means to re
ect the logical document structure [22]. A
paragraph could be a low dimensional subspace of the IN
space; the subspace associated with a section the paragraph
is found in would then contain the paragraph subspace. The
union of document subspaces can be used to represent doc-
ument groups or clusters.
3.2 Uncertain States and User Interaction
At the beginning of the search, we cannot assume that the
IR system knows exactly about the user's IN. To capture this

uncertainty, we introduce a further probability distribution,
the probability pi that the system is in a state j'ii. By doing
so we allow for the IN to be in one of a set of dierent states
with a given certain probability. In this case, we say that the
user's IN is in a mixed state. This can be formally expressed
by dening an ensemble S = f(pi; j'ii)g of states j'ii (where
each of them represents a pure IN) and their corresponding
probability pi, with
P
i pi = 1. The IR system assumes the
user's IN is j'ii with probability pi . When the IR system
knows the user's information need with certainty, then the
ensemble is reduced to only one state j'i, which in this case
is called a pure state.
Given an ensemble S, we can compute the probability
of any event, like the relevance of a document, by apply-
ing the law of total probability. For example, say that the
system assumes that it is in state j'1i with probability p1
or in state j'2i with probability p2, so the mixed state is
described by S1 = f(p1; j'1i) ; (p2; j'2i)g. Then the proba-
bility that a document d is relevant given the current state
is Pr(Rjd; S1) = p1
Pr(Rjd; '1) + p2
Pr(Rjd; '2), where
Pr(Rjd; 'i) is again the square of the length of the projec-
tion of j'ii onto the subspace Od. In general, given a mixed
state described by the ensemble S = f(pi; j'ii)g, we have
Pr(Rjd; S) =
X
i
pi
Pr(Rjd; 'i): (1)
As said, a mixed state re
ects the system's uncertainty
about the user's IN if this is underspecied, which is often
the case in an information seeking scenario [17]. Initially, be-
fore any user interaction has taken place, the system state
is a mixture of all possible INs with a probability that de-
pends for instance on the popularity of an IN. Upon user
interaction, the system state may become more specic or
react to a drift in the information need.
User interaction can be used to reduce this uncertainty,
using another well-known concept from quantum mechan-
ics, measurement, which is comparable to probabilistic con-
ditionalisation. Measurement uses again the subspace that
describes an observed event (e.g., the user has judged this
document as relevant) and acts upon an ensemble S in a
geometric way. Without entering into technical details, it
involves projecting and renormalising each vector j'i into
the subspace dening the event, and updating the dierent
probabilities pi. In practice, it means that after measure-
ment, all the vectors in the ensemble belong to the subspace
that denes the observed event.
3.3 Example
Let us illustrate how measurement supports (D1) and
(D2) by considering an ensemble of ve possible states and
an event O1 as depicted in Figure 3a.
The state vectors of the ensemble are freely distributed in
the 3-dimensional space. After measurement, the ensemble
is projected onto the 2-dimensional subspace O1, which is
shown in Figure 3b. In this measurement, we can illustrate
the two dierent dynamics. (D1) is supported because the
IR system is now \less uncertain" about the user's IN due to
the fact that the ensemble is now bound to the 2-dimensional
plane. For example, the vector j'4i has been removed from
the ensemble, while j'2i that was belonging to the subspace
O1 has been kept. (D2) is supported since j'1i, j'3i and
j'5i have been changed through projection.
The framework dened here does not make any assump-
|ϕ
3
〉
|ϕ
5
〉
|ϕ
4
〉
|ϕ
2
〉
|ϕ
1
〉
O
1
(a) Before measurement
|ϕ
3
〉
|ϕ
5
〉
|ϕ
2
〉
|ϕ
1
〉
O
1
(b) After measurement
Figure 3: Eects of measurement. Vectors orthogonal to
the subspace are eliminated, vectors already in the subspace
remain unchanged, the rest is projected onto the subspace
and renormalised.
tion on the geometry of the underlying vector space, nor
on how an \information need" is actually dened. Provided
that the hypothesis of the framework are correct, that is
that there exists a IN space where all the possible INs can
be dened and that it is possible to dene subspaces for
any event of interest, this framework oers the possibility to
describe geometrically the whole IR search process. In the
following, we show that within this framework it is possi-
ble support a polyrepresentation of documents by modelling
dierent aspects of relevance. We show how the quantum
framework can relate in a complex manner these dierent
aspects through a so-called tensor product of Hilbert spaces.
4. SINGLE REPRESENTATIONS
We discuss some examples of spaces that re
ect dierent
aspects of relevance, or said otherwise, that deal with dif-
ferent representations of a document. As our goal is to in-
tegrate these dierent representations, each of them should
satisfy our hypotheses about the IN space dened in Sec-
tion 3. To this end, we associate with each distinct repre-
sentation a representation space, which is a Hilbert space
where a state re
ects the component of the IN related to
the representation.
In order to be compatible with the basic interactive frame-
work introduced in the previous section, each space should
be able to support the dynamic processes (D1) and (D2) by
means of measurement. It should also allow for the calcula-
tion of the probability of relevance regarding the correspond-
ing representation, applying Equation 1 and the projection
of state vectors onto document subspaces. The geometrical
description of representations can make use of the
exibil-
ity that comes with the quantum formalism (for instance by
using non-orthogonality) and its relationship to probability
theory.
Dening a distinct space for each representation (instead
of one that covers all representations at once) is not only
simpler, since we can focus on the peculiarities of each rep-
resentation, but it is also necessary, since with polyrepresen-
tation we want to be able to consider
exible combinations
of representations, as stated in Section 2. Another advan-
tage is that as we can compute the probability of relevance
given any representation separately with Equation 1, we can

use any strategy based on polyrepresentation to combine the
dierent probabilities.
In this section, we present examples of spaces that are
useful for an online book store and fulll the above require-
ments. As each potential application comes with dierent
representations of various kind, we restrict ourselves to ex-
amples. Therefore, it is the goal of this section to give the
reader an impression of the potential of the quantum for-
malism, not to dene an exhaustive set of representation
spaces.
In a book store scenario we deal with dierent types of rep-
resentations, which basically can be textual (e.g., abstract,
title or comments) or non-textual (e.g., bibliographic meta-
data or ratings). In this section, we discuss representation
spaces for both types of representations. In the case of non-
textual information, we chose two important representations
of dierent types, namely the author and the rating.
4.1 Textual Representations
The most common representations of a document in IR are
textual, and given for instance by the full text of a document,
the document title, annotations or comments attached to
the document, but also by user-given tags. In IR, a generic
way of representing textual content is in form of vectors
which lie in a space where each dimension corresponds to
a term; the term vector may then contain for instance the
associated term weights. However, recent work indicates
that it is benecial to represent textual content by more
than just one vector [3]. Furthermore, a document may
be relevant to more than one information need [23], which
also suggests a more ne-grained representation than with
just one vector, potentially re
ecting the logical document
structure. The idea therefore is to represent textual content
by more than one vector, i.e., a subspace.
A Hilbert space representation for documents and queries
based on the term space was proposed and evaluated in [19,
20], where the term space roughly corresponds to the topical
representation space of the user's IN. We brie
y outline this
representation here.
The (simplifying) assumption is that each document is
composed of a set of excerpts, each one answering a specic
(topical) IN. This is depicted in Figure 4 where a document
is covered up by the excerpts. Each excerpt corresponds to
a \pure" IN, it is thus possible to associate each of them
with a unit vector juii in the space. The next hypothesis
made is that the relevance of a document can be represented
as a subspace that spans the set of vectors fjuiig, or said
Figure 4: Extraction of IN vectors from a text
|car crash〉 (IN)
|jupiter crash〉 (IN)
|jupiter〉 (Term)
|car〉 (Term)
|crash〉 (Term)
Figure 5: Textual representation { IN space superimposed
on a term space
otherwise by the minimal subspace that contains all of these
vectors. By doing so, it ensures that if a user is in one of
the states juii, then the probability that the document is
relevant is 1 (since the projection of juii onto the subspace
is juii itself).
This type of construction, where we dene the subspace
associated with the relevance of a document as the minimal
subspace containing all the INs for which this document is
relevant, is a general one and we will apply it to the dierent
representation spaces we describe in the next sections.
In the case of topical relevance, the topical IN space can be
approximated by a term space where each term corresponds
to one dimension [19, 20], as in the vector space model (see,
e.g., [24]) widely used in IR. An IN is thus described as a set
of weights, one for each term. More precisely, each vector
juii is built from the terms of the corresponding excerpt, i.e.
has non null components for the terms within the excerpt.
Figure 5 illustrates the set up of this topical/term space.
Here, two information needs are shown, \Jupiter crash" (de-
scribing for instance a recent comet crash on Jupiter) and
\car crash". The terms \jupiter" and \crash" make up the
former IN, while the latter is composed of the terms \car"
and \crash". In this example, the IN vectors are not nec-
essarily orthogonal, which motivates the use of a quantum
probability framework.
A more elaborate discussion of the text/topical represen-
tation can be found in [19, 20].
4.2 Non-Textual Representations
We give examples for two important non-textual represen-
tations in a book store scenario, namely authors and ratings.
4.2.1 Author Space
Our rst example deals with users searching for a book
from a specic author. In this section, we propose several
possibilities of increasing complexities. To simplify our dis-
cussion, we assume the user is looking for one specic au-
thor, not several.
A rst possibility is to associate with each author a dis-
tinct dimension of the author space, where the relevance of
a document is the subspace spanned by the dierent vectors
of the book authors. In that case, a pure IN corresponds to
one of the author vectors, and only documents authored by
this specic person are relevant.
With this representation, we are using a standard prob-

|miller〉
|jones〉
|smith〉
Figure 6: Author IN vectors
ability framework. However, it can be argued that a user
interested in a document by a specic author may also be
interested in documents by another, related author. This
may for instance be the case when two authors have a co-
author relationship or work in the same eld. This could
also be based on the style of the author (in the case of nov-
els). The information about how close two authors should be
could be constructed by using the content of authors' books,
but this could be also extracted from the purchase behaviour
of a book store's clients { people who bought books from an
author A might often also buy books from an author B.
To allow such dependencies between authors, a second
possibility is to allow vectors to be non orthogonal. This
is depicted in Figure 6, where three dierent author vec-
tors are shown; we assume that \Jones" and \Smith" share
many characteristics and should hence be represented as
non-orthogonal vectors. Let us imagine that a book has
the authors Smith and Miller. This book would be (author)
relevant for an IN jsmithi (probability of 1), but dierently
to the rst possibility, it would be also relevant to an author-
IN jjonesi, although with a lesser probability. This prob-
ability can be tuned since it depends on the angle between
the jjonesi and jsmithi vectors: The smaller the angle, the
higher the probability.
The relationship between Jones and Smith might be more
complex than the one described above. Say, for example,
Jones is interested in probabilistic logics and also in inter-
active retrieval, whereas Smith' interests are probabilistic
logics on the one hand and theoretic models in information
retrieval on the other hand. Smith may be a former PhD
student of Jones with probabilistic logics as PhD topic, so
that both share many publications on this topic, and both
may have published further articles on probabilistic logics,
although not as co-authors. So Smith and Jones are very re-
lated when it comes to probabilistic logics, and users looking
for documents about probabilistic logics by Jones may likely
be interested in the further work performed by Smith on that
topic. On the other hand, when a user seeks for documents
about interactive retrieval by Jones, Smith' publications are
likely to be not relevant.
This motivates our third and last possibility of represen-
tation, where an author is represented as a subspace. More
precisely, an author can be associated with a set of author
INs. With our Jones/Smith example, a possible representa-
tion is depicted in Figure 7. In this gure, a book authored
by Jones can be either an answer to a user looking for the
logic writings of Jones or his interactive IR writings. In the
latter case (interactive IR), a book written by Smith would
have a zero probability of being relevant, whereas in the
former one (logics) it would have a non zero probability.
4.2.2 Rating Space
In e-stores, ratings re
ect the user's opinion about the
quality of an object. They are often given on a rating scale,
ranging for instance from zero star (\very bad") to 5 stars
(\very good"). It is common to compute an average value of
the ratings and categorise the average onto the given rating
scale or a more ner grained discrete one (for example, the
average could be \3 1/2 stars").
We assume that users generally prefer higher-rated doc-
uments, and that the ratings can be mapped on a set of
ordinal scaled labels flig. For example, in an online store
it may be possible to give, say, 0 to 2 stars, where 0 stars
are mapped to the label bad, 1 star is mapped to the label
average and 2 stars means good. We can then establish an
order on these labels: good > average > bad.
As in the case of independent authors, we can set up a
Hilbert space for such a representation. In the above exam-
ple, this would result in a 3-dimensional vector space with
the orthogonal vectors jgoodi, javeragei and jbadi, where
each vector corresponds to the minimum rating the user
wants for a book.
To re
ect the order of labels, a \good" book would thus
be represented as the whole 3-dimensional subspace (as it is
relevant to any rating-IN), an\average"book would be the 2-
dimensional subspace spanned by javeragei and jbadi, and
a \bad" book would be the 1-dimensional subspace corre-
sponding to jbadi. A pure rating-IN can then be interpreted
as a threshold. For example, if the rating-IN is javeragei,
this means that for the user average and good books are
interesting; the javeragei vector is contained in both the
subspaces for \good" and \average" books.
As for authors, it might be interesting to make the dif-
ferent vectors non-orthogonal, since if the user rating-IN is
jgoodi, an average book is better than a bad book. In prac-
tice (not described here), it is possible to set a probabil-
ity that a user is satised with an average/bad book given
its rating-IN, and to compute the non-orthogonal jgoodi,
javeragei and jbadi vectors in a three dimensional space
that satisfy those probabilities.
5. COMBINING THE EVIDENCE
In the last section we gave examples of single represen-
tation spaces. In each of these spaces, a state can change
according to the dynamics dened in Section 3, and it is
|JonesIIR〉
|JonesLogics〉
|SmithLogics〉
Figure 7: Author/Topic Space

possible to compute a probability of relevance. However, we
have not yet described how to combine the evidence of the
dierent representations into a single framework, which is
central to every polyrepresentation approach.
At rst, the underlying assumption is that the represen-
tations are independent. As the geometrical framework in
Section 3 has a probabilistic interpretation given by Equa-
tion 1, we can describe the creation of the (total) cognitive
overlap from these representations purely probabilistic. In
order to support partial overlaps and weighted representa-
tions, a requirement stated in Section 2, we need to intro-
duce an extra dimension to our representation spaces. All
this is discussed in Section 5.1. Then, in Section 5.2, we
show how we can go beyond this simple probabilistic model
by exploiting further the quantum formalism, allowing us to
drop the assumption that representations are independent.
5.1 Total and Partial Cognitive Overlaps
The total cognitive overlap introduced in Section 2 re-
quires that the dierent representations of a relevant docu-
ment should all be relevant to the user's IN.
If we suppose that the representations are independently
in
uencing relevance, then we can apply the probabilistic
interpretation of our framework; following Griths [8], we
dene the probability of a document to be relevant as the
product of the probabilities of the document to be relevant
in each representation. Formally, we write
Pr(Rjd) =
Y
i
Pr(Rjd; Si) (2)
where Pr(Rjd; Si) is the probability of relevance in the ith
representation space and is computed with Equation 1.
The creation of the total cognitive overlap has some short-
comings, as a document can easily have a zero probability
of relevance. More precisely, it is sucient that the doc-
ument it not relevant in one representation to get a value
of zero. This problem is stressed when we dynamically add
new representation spaces. Furthermore, some kinds of rep-
resentations seem to be more promising than others, and the
representations should be carefully mixed and weighted [26].
Another important argument comes from the user side { a
user may be interested in some representations more than in
others, and may change her mind at another point in time.
This suggests that the relative importance of the represen-
tation should be
exible enough so that it can evolve with
time.
It is therefore desirable to loosen the total cognitive over-
lap requirement, and to ask for a level of coordination be-
tween representations: documents should also be retrieved
even if they are not in the total cognitive overlap, but failing
to be relevant w.r.t. certain representations.
We discuss a possible solution to introduce weights for
representations into our framework. Our basic idea is that
if the user does not care about a certain representation, all
documents should be fully relevant in this representation.
This can be achieved by adding an extra dimension, that we
call the\don't care" dimension, in each representation space.
The associated vector is denoted ji , and it is orthogonal
to all other IN vectors and subspaces.
In order to introduce a weight, we can simply state that
the user's IN is in a mixed state, and that with a probabil-
ity  the user's IN is \don't care". We can then distribute
the remaining 1  probability among the possible states
(a) Separable state
(b) Non-separable state
Figure 8: Separable and non-separable state example
of the user's IN. If we take our author space for example,
the system initially does not know what author the user is
looking at and assumes a priori that the user does not con-
sider authors at all with probability . Having Jones, Smith
and Miller as possible authors, the initial ensemble would be
1
3 ; jjonesi


;

1
3 ; jsmithi


;

1
3 ; jmilleri


; (; ji)

.
Regarding document subspaces, we require that each such
subspace includes the \don't care" dimension, which makes
each document \relevant" to the don't care \need". This
has the following eect. If the \don't care" dimension has
the weight 1 and the representation is therefore in the pure
state ji, we would have Pr(Rjd; S = ji) = 1 for every
document d, because ji is always contained in the document
subspace. According to Equation 2, this would mean that
the representation is completely ignored.
Note that we can go back to a total cognitive overlap by
giving a zero probability, i.e.  = 0, to the \don't care"
IN in all the representations. All other cases mean that
we gradually relax the obligation of the document to be
in overlaps containing the particular representation by as-
signing increasingly higher probabilities to the \don't care"
dimension.
Since ji is a valid state vector, through interaction it is
possible (e.g., using projections as described in Section 3) to
adapt automatically the weight  associated to this state.
It is beyond the scope of this paper to explain how.
5.2 Interdependent Representations
So far we assumed that the dierent representations are
independent, but from a user point of view, this assump-
tion does not necessarily hold. For example, a user might
think that documents written by Jones are always of high

quality and therefore does not care about how they were
rated. At the same time, the user does not know the author
Smith, and thus tends to rely more on the ratings. So in one
case the ratings are considered, while in the other they are
not, showing that the author and rating representations are
not independent of each other from the user's perspective.
This is of course just one example of possible relationships
between representations as they can occur in a book store
scenario.
Before we continue the discussion, we need to brie
y intro-
duce another important geometrical concept, namely tensor
products and tensor spaces. We refer the reader to [18, ch.
2] for a more in-depth discussion. The tensor product (de-
noted
) is a way to combine dierent Hilbert spaces into a
(larger) Hilbert space. If H1 and H2 are two Hilbert spaces
of respective dimensions n and m, the tensor space H1
H2
is an n
m-dimensional Hilbert space. If j1i is a vector in
H1 and j2i is a vector in H2, then j1i
j2i is a vector
in H1
H2. Furthermore, if A and B are subspaces in H1
and H2, respectively, then A
B is a subspace in H1
H2
. Finally, the projection of j1i
j2i onto A
B is simply
the tensor product of the two projected vectors (if one of
the projections is null, then the result is the null vector in
H1
H2). Eventually, the norm jj:jj of a vector j1i
j2i is
the product of the single norms. The denition of the projec-
tion and the norm in the tensor space allows us to compute
the probability of any event as described in Section 3. Note
that these operations can be extended to tensor products of
more than two spaces.
In quantum mechanics, tensor spaces and the tensor prod-
uct are used to create composite spaces out of single com-
ponent spaces. In a tensor product of spaces, we can dis-
tinguish separable and non-separable (or entangled) states.
Separable states describe a state where the component states
are independent of each other (that is, knowing something
on a component state does not give any information about
the other component states), whereas non-separable states
are states where the composite state cannot be decomposed
anymore into independent component states.
Figure 8 shows an example of separable and non-separable
states which we will describe below. In Figure 8a, the sys-
tem assumes that the user wants a document to be either
authored by Smith or by Jones, and the rating to be good
or that the user doesn't care about the rating. The pos-
sible states in the author-rating tensor product, which we
call the polyrepresentation space, are given in the right hand
side of the gure. As illustrated, we can isolate the IN of
both representations, by operating in each representation
separately and obtaining the corresponding probability of
relevance. This is equivalent to computing the probability
of relevance as in Equation 2 { the representation states are
independent.
To cope with above situation where ratings and authors
are interdependent, we cannot regard our single represen-
tations in isolation anymore. We must consider the com-
posite polyrepresentation space. Figure 8b depicts a state
which exactly re
ects the situation given in the example.
The combined representation ensemble consists of two states
this time: the state s1 dened as jsmithi
jgoodi (good
documents by Smith), and the state s2 dened as jjonesi

ji (the user does not care about the ratings when the doc-
ument is written by Jones). The peculiarity of this state is
that we cannot isolate a state for each representation any-
State Projection Probability Non-sep.
jsmithi
jgoodi null 0 
jjonesi
jgoodi null 0
jsmithi
ji jsmithi
ji 1
jjonesi
ji jjones0i
ji kjjones0ik2 
Table 1: Probabilities of relevance for Od
1
regarding the
single state vectors in the author-rating polyrepresentation
space. jx0iis the projection of jxi onto the author subspace
O(a)d
1
. The column \Non-sep." shows the states that are
part of the non-separable mixed state (denoted by a black
square).
more. Both states shown in Figure 8 involve the same au-
thor and rating vectors, but only the state in Fig. 8a can
be broken down to separate states in the author and rating
space.
Let us see how this translates to the calculation of the
probability of relevance in the polyrepresentation space. In
Table 1, we see in the left column the states from Figure 8.
Two of them are part of the non-separable mixed state,
as shown in the right column. The relevance of a docu-
ment w.r.t. the author and ratings is dened by a subspace
Od = O
(a)
d
O
(r)
d , where O
(a)
d is the subspace in the au-
thor space and O(r)d the one in the rating space. Consider
a document d1 authored by Smith and rated \bad". O
(a)
d
1
is
then a 2-dimensional subspace in the author space spanned
by jsmithi and ji, and from the way we model relevance
in the rating space discussed in Section 4.2.2, O(r)d
1
would be
the 2-dimensional subspace spanned by jbadi and ji. We
can see that in the case of a separable state (Fig. 8a), doc-
ument d1 has a probability of 1 to be relevant to the state
jsmithi
ji (3rd row). The probability that d1 is relevant
w.r.t. jjonesi
ji (4th row) is determined by the projec-
tion of jjonesi onto Od
1
(remember that in this example
we assumed that jjonesi and jsmithi are non-orthogonal).
The nal probability of relevance for a bad book by Smith is
thus determined by the states in row 3 and 4 in the separable
case, while in the non-separable case, only the probability in
row 4 determines the relevance { due to the interdependence
between authors and ratings, the state in row 3 was ruled
out. The bad book by Smith would get a higher probability
of relevance in the separate case than in the non-separate
one. In the latter case, the probability of relevance just de-
pends on the relationship between the authors Smith and
Jones and the fact that the user does not care about the
ratings for books by Jones.
With regards to the cognitive overlap, non-separable
states and the interdependencies going along with them
give us ner control what conditions between representa-
tions must be satised for a document to be in the cognitive
overlap.
The remaining question is how a state in the polyrepre-
sentation space can become non-separable. We can assume
that we initially have a separable state, for instance a tensor
product of the initial states of the single representations. A
user might then state relationships like the ones above di-
rectly, or we may extract such interdependence from other
sources, for instance by automatically creating association
rules (like \author=Smith ) rating=good"), known from
data mining, from documents the user judged relevant dur-

ing the session [29]. Such rules may directly be translated
into subspaces inducing non-separable states like those in
Fig. 8b by means of measurement.
6. RELATED WORK
There are basically three classes of related work: models,
applications and the evaluation of the polyrepresentation
principle.
Regarding (quantum) models supporting polyrepresenta-
tion, Melucci [16] proposes a dual approach to our frame-
work where a subspace is used to describe a user's informa-
tion need and a vector to represent documents. The prob-
ability that a document is relevant to a user's information
need is determined by the projection of the document vector
representation onto the corresponding IN subspace. Similar
to ours, this approach also utilises the relationship between
geometry and probability theory. In contrast to Melucci's
idea and following the notion of state vectors and dynamics
as applied in quantum mechanics, we interchanged the role
of document and user's information need in our framework.
This is motivated by the fact that the user's information
need should be represented as a dynamic component, as ad-
vocated in e.g. [10]. The approach in [16] does not consider
polyrepresentation per se, but in [2] an approach is proposed
to use Melucci's framework for combining multiple sources
of evidence. A document is still modelled as one vector in a
vector space, but the same document can be described us-
ing dierent representations, where a document vector can
be generated by a dierent vector space basis. In this work,
though dealing with multiple evidence coming from dierent
sources, no explicit relationship to the polyrepresentation
principle is discussed.
Beckers [1] discusses the possible application of polyrep-
resentation of documents to support information seeking
strategies, motivated with a book store example as in this
paper. Other works considering dierent document con-
texts, like annotation-based retrieval [7], can be interpreted
as an application of polyrepresented documents.
Since the introduction of polyrepresentation, for instance
in [9], several experiments have been performed to validate
the eectiveness of this principle regarding its various as-
pects. One aspect, which we focus on in our work, is the
polyrepresentation of documents. The results reported in
[26, 27] support the principle of polyrepresentation of doc-
uments and also show that assigning weights to dierent
representations (higher weights for those with higher preci-
sion) can be crucial to gain better eectiveness, motivating
the introduction of dierent representation weights into our
framework through the\don't care"dimension. A second as-
pect of polyrepresentation considers the dierent represen-
tations of a user's information need, which includes among
others the work task, the perceived information need, the ex-
perience, the domain knowledge and dierent query facets
[11, 12, 5, 6, 14]. A third form of polyrepresentation sees dif-
ferent search engines as dierent re
ections of the cognitive
view of their designers on the retrieval problem [13, 15]. The
main conclusion from evaluating all these facets of polyrep-
resentation is that the more positive evidence is coming from
dierent representations, the more likely is the document in
the cognitive overlap relevant to a given information need.
This strongly supports the principle of polyrepresentation
in general and also the idea to create a retrieval framework
that explicitly considers polyrepresentation.
7. CONCLUSION AND OUTLOOK
In this paper, we discussed how to introduce the prin-
ciple of polyrepresentation to a geometrical IR framework
inspired by the quantum mechanics formalism, addressing
some of the main requirements on a model for polyrepre-
sentation. First, we recap our motivation for amalgamating
a geometrical quantum formalism and polyrepresentation.
After presenting the basic framework, we showed by exam-
ple how textual and non-textual representations could be
expressed within the framework, by dening vector spaces
for the representation and subspaces of it for document rep-
resentations. We have described how these representations
beneted from the connection between geometry and prob-
ability present in the quantum formalism. To calculate the
cognitive overlap and rank documents based on the prob-
ability that they lie in this overlap, we can combine the
probabilities of relevance coming from the single represen-
tations to compute the probability that a document is in
the total cognitive overlap. A \don't care" dimension, ex-
pressing the fact that the user does not use a representation
to determine relevance, is introduced to each representation
space to relax the strict obligation that an information ob-
ject lies in the total cognitive overlap and assign weights
to each representation. Finally, we discussed how we could
handle situations in which the single representations are not
independent any more from a user's point of view. To do so,
the notion of non-separability and quantum entanglement
is introduced in our framework, giving us the possibility to
re
ect representation interdependencies when creating the
cognitive overlap.
We have succeeded in building a model that encompasses
quantum mechanics and polyrepresentation and which is a
strong platform for future work and further developments.
As a next step, we plan to experiment with the framework.
In particular, we would use the data collected in 2009-10 by
the interactive track of the INitative for the Evaluation of
XML Retrieval (INEX), where participants ran user experi-
ments using a collection consisting of a crawl of over 2 million
records (bibliographic metadata, reviews and ratings) from
the online bookseller Amazon, enriched with reviews, ratings
and tags coming from the cooperative book cataloguing tool
LibraryThing2. The continuing user experiments yield log
les capturing several kinds of user interaction (queries and
query reformulations, relevance judgements) as well as in-
sights which representations were actually considered by the
users. These representations from dierent contexts give us
the possibility to evaluate our framework and also polyrepre-
sentation w.r.t. retrieval eectiveness, using simulated user
interaction coming from these log les. The test data can
also potentially be used to validate some of the assumptions
underlying our framework, especially the non-independence
of representations from a user point of view.
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