=Paper=
{{Paper
|id=None
|storemode=property
|title=Using pattern structures to support information retrieval with Formal Concept Analysis
|pdfUrl=https://ceur-ws.org/Vol-1058/paper2.pdf
|volume=Vol-1058
|dblpUrl=https://dblp.org/rec/conf/ijcai/CodocedoLAN13
}}
==Using pattern structures to support information retrieval with Formal Concept Analysis==
https://ceur-ws.org/Vol-1058/paper2.pdf
Using pattern structures to support information
retrieval with Formal Concept Analysis
Vı́ctor Codocedo1? , Ioanna Lykourentzou1,2?? , Hernán Astudillo3 , and Amedeo
Napoli1
1
LORIA - CNRS - INRIA - Univesité de Lorraine, BP 239, 54506 Vandœuvre-les-Nancy.
victor.codocedo@loria.fr, amedeo.napoli@loria.fr,
2
Centre de Recherche Public Henri Tudor - 29, avenue John F. Kennedy L-1855
Luxembourg-Kirchberg, Luxembourg
ioanna.lykourentzou@tudor.lu
3
Universidad Técnica Federico Santa Marı́a - Avenida España 1680 - Valparaı́so, Chile
hernan@inf.utfsm.cl
Abstract. In this paper we introduce a novel approach to information retrieval
(IR) based on Formal Concept Analysis (FCA). The use of concept lattices to
support the task of document retrieval in IR has proven effective since they allow
querying in the space of terms modelled by concept intents and navigation in the
space of documents modelled by concept extents. However, current approaches
use binary representations to illustrate the relations between documents and terms
(“document D contains term T”) and disregard useful information present in doc-
ument corpora (“document D contains X references to term T”). We propose us-
ing pattern structures, an extension of FCA on multi-valued and numerical data,
to address the above. Given a set of weighted document-term relations, a concept
lattice based on pattern structures is built and explored to find documents satisfy-
ing a given user query. We present the meaning and capabilities of this approach,
as well as results of its application over a classic IR document corpus.
Keywords: Formal Concept Analysis, Interval Pattern Mining, Information Retrieval
1 Introduction
Information retrieval (IR), is a problem of lasting interest for the research community.
Among the tasks comprising the IR domain, document retrieval (i.e. the search and
ranking of documents that are relevant to an original user query from a given document
corpus) is one of the most popular in the field given its importance in everyday routines.
In the wide spectrum of techniques applied to support document retrieval, formal con-
cept analysis (FCA) has gained interest in the last years [3–6, 16] because of its robust
framework and the qualities of a concept lattice.
Formal concept analysis (FCA) is a mathematical formalism used for data analysis
and classification, which relies on the dualistic understanding of concepts as consisting
?
Part of the Quaero Programme, funded by OSEO, French State agency for innovation.
??
Supported by the National Research Fund, Luxembourg, and cofunded under the Marie Curie
Actions of the European Commission (FP7-COFUND).
of an extent (the objects that belong to the concept) and an intent (the attributes that
those objects share) organized in a lattice structure called a concept lattice [7]. We
refer to FCA-based information retrieval as CL4IR which stands for“concept lattices
for information retrieval”.
In a typical CL4IR approach, a binary table of documents and terms is created
and then, using FCA algorithms, the respective concept lattice is created. This lattice
contains several formal concepts, each defined by a set of documents (extent) and the
set of terms that they share (intent). Thus, the lattice provides a multiple hierarchical
classification of documents and terms which can be navigated and searched as an index,
to retrieve concepts that are “close” or “similar” to the original query concept. In this
way a CL4IR system exploits the connections of concepts within the lattice to find
relevant documents for a given query and takes advantage of the lattice structure to
enrich the answer in different ways: by navigating the lattice to look for approximate
answers through generalizations and specifications of the original query concept’s intent
[2, 14], by enriching the term vocabulary with a thesaurus [5] or by directly integrating
external knowledge sources [16]. Nevertheless, current CL4IR systems are restricted
by the binary nature of their data (a document can either contain a given term or not).
Consequently, they can work only with Boolean-like queries, which is an important
limitation w.r.t. other IR approaches such as vector-space ranking methods [13] that
allow partial-matching documents to be considered as possible answers.
In this article we present a novel CL4IR approach, which deals with numerical
datasets, i.e. document-term relations, where a document is annotated by a term with a
certain weight. This approach provides CL4IR systems with an extended query space,
on which vector-space ranking methods can be adapted and applied. In parallel, this
approach retains the main advantage of using lattices as the document search index,
which is the provision and exploration potential of the complete query space. Our ap-
proach is based on the pattern structures framework, an extension of FCA to deal with
complex data [7]. Given a numerical table representing weighted associations between
documents and terms, we apply pattern structures to build the extended query space,
while we also introduce steps for reducing and simplifying the document search within
the constructed query space. We illustrate our approach through running example on
a classical IR dataset and by comparing our results, in terms of precision and recall,
to those reported in the literature. Furthermore we provide a discussion on the mean-
ing and capabilities of the proposed approach. The remainder of the paper is organized
as follows. Section 2 provides an introduction to the use of formal concept analysis
for document retrieval. Section 3 describes the pattern structure framework and details
the proposed CL4IR approach which can be applied on numerical datasets. Section 4
presents the experiments. Finally, Section 5 concludes the paper.
2 Concept Lattices for Information Retrieval
The setting of a typical concept lattice for information retrieval (CL4IR) application
is given by a formal context K = (D, T, I) made of a set of documents D, set of
terms T and an incidence relation I = {(di , tj )} indicating that document di contains
term tj . Table 1 illustrates a document-term formal context created from a corpus of 9
documents and 12 terms.
computer
response
interface
system
human
survey
minor
graph
time
EPS
user
tree
d1 x x x
d2 x x x x x x
d3 x x x x
d4 x x x
d5 x x x
d6 x
d7 x x
d8 x x x
d9 x x x
q* x x
*
Grey row represents the query.
Table 1: A term-document formal context including the query q.
Given a user query q = {t1 , t2 ...t|q| } , the document retrieval task consists in return-
ing a set of documents ordered by “relevance” w.r.t. the query q. In CL4IR systems a
query can be represented as a virtual document containing the set of terms {t1 , t2 ...t|q| }.
Then, the query is inserted in the formal context as another object and the incidence re-
lation set I is updated to include the relations of the virtual query-document and its
terms. The formal context becomes Kq = (D + {q}, T, I + {(q, ti )i..|q| }).
The standard procedure to find “relevant” documents within the concept lattice con-
sists in identifying the query concept (which is defined as the object concept of the
virtual object q and denoted by γ(q) = ((q 0 )0 , q 0 )) and concepts related to the query
concept (for example, its superconcepts) which can provide further results. We refer to
the later concepts as “answer concepts”. For the formal context in Table 1, consider the
query q with terms “graph” and “tree” (grey row). Figure 1 shows the concept lattice
derived from this formal context (including the query). The query concept corresponds
to concept 17 and contains in its extent documents d7 and d8 which satisfy the query
and can be retrieved to the user. The superconcepts of the query concept (concepts 7
and 8) contain documents d6 and d9 which can also be retrieved. Different relevance
measures can be used to rank the retrieved documents. For example, the topological
distance within the lattice between the query concept and the “answer concepts” (i.e.
concepts partially satisfying the query) can be calculated and in this case documents d7
and d8 are at distance 0 (more relevant), while d6 and d9 are at distance 1 (less rele-
vant). Other such measures include semantic distance, extent intersection, and Jaccard
similarity [2, 15, 5].
More generally, the concept lattice defines a query space where each formal concept
C can be considered as a conjunctive Boolean query (i.e. a query where the constraint
Fig. 1: Concept lattice in reduced notation derived from a document-term formal
context including the query.
is given by the conjunction of the attributes in the intent of C) and a combination of
formal concepts provides disjunction and negation (e.g. The union of concepts 7 and 8
in Figure 1 satisfies the disjunctive query “graph” or “tree”). Unfortunately, the binary
case is the “ideal world”. In most real-world datasets the relation between a document
and a term is built w.r.t. a measure such as frequency, distance or weight involving a
range of numerical values [13].
A document corpus can be defined as a term-document matrix A = [aij ], where
terms ti ∈ T are in rows, documents dj ∈ D in columns and each cell aij of the matrix
represents the “value” of the term ti in the document dj , given by a function val(dj , ti )
(weight, frequency, etc.). In order to work with this kind of datasets, a CL4IR system
can resort to interordinal scaling [8] by simply assigning an incidence relation when a
term in a document has a value within a given range, i.e. I = {(d, t)|val(d, t) > 0}).
However, interordinal scaling could greatly increase the complexity for IR tasks [12] as
it induces redundancy as shown in [11]. To the best of the authors’ knowledge, a CL4IR
system directly dealing with a weighted term-document dataset is not yet reported in the
FCA nor in IR literature. In the following, we present a method and an implementation
of a CL4IR approach dealing with numerical datasets.
3 CL4IR with many-valued datasets
3.1 Pattern structure framework
Here, we introduce the pattern structure framework firstly described in [7].
A pattern structure K = (G, (P, u), δ) is a generalization of a formal context. In K,
G is a set of objects, (P, u) is a semi-lattice of object descriptions and δ : G → P is a
mapping associating a description to an object. The description of an object g ∈ G is a
vector of intervals v = h[li , ri ]ii∈{1..|M |} , where v ∈ P , li , ri ∈ R and li ≤ ri .
In (P, u) the similarity operator u applied to va = h[li1 , ri1 ]i and vb = h[li2 , ri2 ]i
yields the convex hull va u vb = h[min(li1 , li2 ), max(ri1 , ri2 )]i where i ∈ {1..|M |}. The
associated subsumption relation is defined as va u vb = va ⇐⇒ va v vb .
A Galois connection between ℘(G) (powerset of G) and (P, u) is defined as fol-
lows:
d
X = g∈X δ(g) ; v = {g ∈ G|v v δ(g)}
where X represents the common description to all objects in X while v represents
the set of objects respecting the description v. A pair (X, v) such as X = v and
v = X is called a interval pattern concept (ip-concept) with extent X and pattern
intent v. Ip-concepts can be ordered in an interval pattern concept lattice (ip-concept
lattice). Algorithms for computing ip-concepts from an interval pattern structure are
proposed in [11, 7].
3.2 CL4IR based on pattern structures
A document corpus or a term-document matrix, as described at the end of Section 2, can
naturally be represented as a many-valued context [8] K = (D, T, W, I), where W =
{val(dj , ti )}∀dj ∈D,ti ∈T and I = (dj , ti , wk ); f (dj , ti ) = wk , wk ∈ W . Table 2 shows
an example containing 9 documents (white rows) and 12 terms. The value in a cell
represents the “relative frequency” of a term in a document, i.e. the ratio between the
amount of times a term appears in a document and the total amount of terms occurrences
in the document. Like in the binary case, a query q = {t1 , t2 , ..t|q| } is considered as
a virtual document and included in the many-valued context which becomes Kq =
(D + {q}, T, W, I + {val(q, ti )}∀ti ∈q ). The cells of the query contain also a “relative
frequency” value. The query q = {“graph”, “tree”} is illustrated in the grey row in
Table 2 (e.g. val(q, graph) = 1/2 = 0.5).
To deal with Kq , we define the pattern structure as (D + {q}, (P, u), δ) where inter-
val patterns in P contain the interval-vector representation of documents in |T | dimen-
sions (one for each term). The mapping δ(d) = h[val(d, ti ), val(d, ti )]i∈[1..|T |] i assigns
an interval pattern representation to a document (or the virtual query-document) con-
sisting of a zero-length interval for each term existing in T at the value of the term in the
document (e.g. in Table 2, δ(d1 ) = h[0.33, 0.33][0.33, 0.33][0.33, 0.33][0, 0] . . . [0, 0]i,
where [0, 0] is represented by [−]). The similarity operator u applied to two interval
patterns returns the convex hull between their document representations. From the pat-
tern structure we construct the ip-concept lattice representing the query space which
will be used to retrieve documents in a similar way as binary approaches.
The query concept is still considered as the object concept of q. However the se-
mantic of the query space changes. While in the binary case the query space represents
a pool of Boolean query possibilities, here the query space can be considered as a vec-
tor space where the query is grouped with documents having similar representations.
For example, consider the first three columns in Table 3 where each row represents an
ip-concept. Concept 1 is the query concept which in its extent includes documents d7
and its interval pattern (intent) only includes zero-length intervals in all 12 dimensions,
making the description of the query identical to the description of d7 . Concept 2 is a
superconcept of 1, whose extent contains d7 and d8 . This time, there are only 9 zero-
length intervals in all 12 dimensions. Concept 2 is less similar to the query than concept
1 w.r.t 3 dimensions. Following with concept 3, we can see that the later is less similar
to the query than concept 1 w.r.t. 4 dimensions. We get in this way a “natural” ranking
of the concepts.
In order to rank ip-concepts we rely on the notion of maximal distance within an
interval pattern. For illustrating this notion, we will use the geometrical interpretation
of patterns already introduced in [11]. Let us consider the 2-dimensional case with two
ip-concepts in Figure 2, namely Z1 = ({q, A, B, C}, h[2, 7][2, 7]i) (clear rectangle)
and Z2 = ({q, A, B, C, D}, h[1, 7][1, 7]i) (dark rectangle). For ranking an ip-concept
Zi w.r.t. the query, we will consider the “maximal distance” possible between any two
objects in the extent of Zi , which in the case of Z1 is between objects q and C and for
Z2 is between q and D. Thus, this distance is actually the Euclidean distance between
the edges of the interval vector. Table 3 presents the retrieved ip-concepts for the query
in Table 2 ranked by maximal distance in column 4.
computer
response
interface
system
human
survey
minor
graph
time
EPS
user
tree
d1 0.33 0.33 0.33 0 0 0 0 0 0 0 0 0
d2 0 0 0.16 0.16 0.16 0.16 0.16 0 0.16 0 0 0
d3 0 0.25 0 0.25 0.25 0 0 0.25 0 0 0 0
d4 0.25 0 0 0 0.5 0 0 0.25 0 0 0 0
d5 0 0 0 0.33 0 0.33 0.33 0 0 0 0 0
d6 0 0 0 0 0 0 0 0 0 1 0 0
d7 0 0 0 0 0 0 0 0 0 0.5 0.5 0
d8 0 0 0 0 0 0 0 0 0 0.33 0.33 0.33
d9 0 0 0 0 0 0 0 0 0.33 0 0.33 0.33
a
q 0 0 0 0 0 0 0 0 0 0.5 0.5 0
Fig. 2: Two interval patterns in
Table 2: Many-valued document term the
context (including query). query space.
a
Grey row represents the query concept.
Id Extent Pattern intent max dist
1 q, d7 * h[−][−][−][−][−][−][−][−][−][0.5, 0.5][0.5, 0.5][−]i 0
2 q, d7 , d8 h[−][−][−][−][−][−][−][−][−][0.33, 0.5][0.33, 0.5][0, 0.33]i 0.408
3 q, d7 , d8 , d9 h[−][−][−][−][−][−][−][−][0, 0.33][0, 0.5][0.33, 0.5][0, 0.33]i 0.704
4 q, d6 , d7 h[−][−][−][−][−][−][−][−][−][0.5, 1][0, 0.5][−]i 0.707
5 q, d2 , d7 h[−][−][0, 0.16][0, 0.16][0, 0.16][0, 0.16][0, 0.16][−][0, 0.16][0, 0.5][0, 0.5][−]i 0.808
6 q, d3 , d7 h[−][0, 0.25][−][0, 0.25][0, 0.25][−][−][0, 0.25][−][0, 0.5][0, 0.5][−]i 0.866
7 q, d1 , d7 h[0, 0.33][0, 0.33][0, 0.33][−][−][−][−][−][−][0, 0.5][0, 0.5][−]i 0.909
8 q, d5 , d7 h[−][−][−][0, 0.33][−][0, 0.33][0, 0.33][−][−][0, 0.5][0, 0.5][−]i 0.909
9 q, d4 , d7 h[0, 0.25][−][−][−][0, 0.5][−][−][0, 0.25][−][0, 0.5][0, 0.5][−]i 0.935
*
Grey row represents the query concept.
Table 3: Extents and Intents of concepts in Figure 2 presenting the cosine similarity
between its edges ([−] represents the zero-length interval [0, 0]).
3.3 Dealing with real-world datasets
Calculating a concept lattice is an expensive task which can yield a large amount of
concepts making it prohibitive for large document corpora. The scenario is worst for
pattern structures since for every concept the size of the intent is set to the whole set of
attributes adding even more complexity. Calculating the whole query space of a term-
document matrix is not advisable, since for a given query only a small part of the whole
space is required. In order to avoid a sizeable query space in each step of the retrieval
process, progressive actions to filter data are performed. In the following, we describe
the retrieval process and each action.
1. Constructing the pattern structure: The process starts with the input of a query
q = {t1 , t2 , ..., t|q| } and ends after the pattern structure containing the virtual query-
document is created. We include in the set of documents only those which contain at
least a given number of the terms provided in the query, which can be performed at a
negligible cost by firstly storing documents and terms in a relational database. The set
of terms only include those provided in the query. The minimum number of terms for a
document is left as a parameter of the process.
2. Constructing the ip-concept lattice: This step receives the pattern structure in
order to create an ip-concept lattice. A standard FCA algorithm, namely Ganter’s al-
gorithm [8] is used for this purpose. However the algorithm has been adapted for the
present task.
Many ip-concepts found in the interval pattern lattice are not useful for document
retrieval purposes. For example, the framework creates ip-concepts with documents
which do not share terms (e.g. consider the interval h[0, 1][0, 1][0, 1]i created from the
documents sharing no terms with orthogonal representations v1 = h[0, 0][1, 1][1, 1]i
and v2 = h[1, 1][0, 0][0, 0]i). We denominate these concepts non-informational.
In order to reduce the amount of non-informational concepts, we modified the u
operator in the set of ordered patterns (P, u) such as [l, r] u [0, 0] = [∗] and [l, r] u [∗] =
[∗]; ∀l, r ∈ R. The interval [∗] has been used before to indicate absence of similarity
[10]. Let Zi = (Xi , vi ) be an ip-concept, then ρ(Zi ) represents the number of intervals
different from [∗] in vi . We call ρ(Zi ) the dimensionality of Zi . For a second ip-concept
Zj = (Xj , vj ) is easy to show that (Zi ≤ Zj ⇐⇒ vj v vi ) =⇒ ρ(Zi ) ≤ ρ(Zj ).
We use a threshold of minimal dimensionality (min dim) to reduce the amount of ip-
concepts calculated. Consider this analogous to the use of a minimal support in the
construction of an iceberg lattice [18].
4 Experiments and Discussion
To test the validity of the proposed approach, we applied it on a popular IR dataset
which is openly available. We refer to this implementation as “ip-CL4IR”. The CISI
dataset4 consists of 1460 documents and 35 queries, each one containing a set of valid
answers. Documents contain text in natural language and queries are given as a set of
terms connected by Boolean operators. In our experiments, we converted documents to
collections of weighted terms and stored them in a relational database. The weighting
4
http://ftp.cs.cornell.edu/pub/smart/cisi/
measure used was term frequency-inverse document frequency (tf.idf ) [13]. Boolean
operators in the query were ignored since they do not provide meaning in the vector-
space model (except in the extended Boolean model case [17] not considered in this
work). The virtual query-document was constructed using the inverse document fre-
quencies calculated from the dataset for each of its terms.
After receiving a query, ip-CL4IR consults the database and extracts all documents
that contain at least 2 terms of the query (as described in Section 3.3, this value is a pa-
rameter of ip-CL4IR). The ip-concept lattice is computed using a minimal dimension-
ality of min dim = 2, to keep consistency w.r.t. the restriction given for the creation of
the pattern structure. The query concept is searched in the lattice and its superconcepts
are retrieved and ranked using the Euclidean distance between the boundaries of their
interval patterns. Cosine distance (instead of Euclidean distance) was also calculated
showing better results. Table 4 shows the results for 11-point precision of fixed recall
and 6 measures of precision for the top 5,10 and 20 ranked documents retrieved. Results
on an implementation based on concept lattice-based ranking (CLR) [2] using the same
dataset and a simple binnarization of the relation document-term is reported along with
our results for comparison purposes.
Interpolated precision @ 11 points
0,6
ip-CL4IR CLR EM EM
11-point IAPa 0.232 0.191 0.174 CLR
0,5
MAPb 0.202 0.163 0.145 ip-CL4IR
Precision@5 0.257 0.206 0.285
Precision@10 0.251 0.174 0.257 0,4
Precision@20 0.245 0.174 0.207
Precision
Recall@5 0.032 0.049 0.057 0,3
Recall@10 0.060 0.073 0.079
Recall@20 0.146 0.112 0.123
0,2
Table 4: CISI dataset. Results 0,1
for 35 queries.
0
a 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Interpolated average precision Recall
b
Mean average precision
Fig. 3: Interpolated precision in 11 points of recall
Table 4 reports the results in 8 measures for ip-CL4IR, a reported CL4IR system
called concept lattice-based ranking (CLR) [2] and a naive approach called exact match-
ing (EM) where documents are ranked according to how many terms have in common
w.r.t. the query. The second row contains the values of the interpolated average precision
(IAP) over 11-points of recall illustrated on Figure 3. Interpolated precision in a given
recall point ri in Figure 3 indicates the best precision value in the interval [ri , ri+1 [.
From Figure 3, the interpolated precision in the recall point 0 for ip-CL4IR is the best
precision obtained in the recall interval [0, 1[ equal to 0.51. The third row contains the
values of the mean average precision (MAP) calculated over the precision values for
each valid document found in the ranked documents retrieved by a system for each
query. For example, given query if the first valid document is found in the third position
of the ranking it has a precision value of 0.3. If the second is found in the fifth position
its precision is 0.2 and the MAP is 0.25. IAP and MAP are standard information re-
trieval measures [13] to evaluate ranked results from a retrieval system. The remaining
rows present values of precision and recall in the first 5 (@5), 10 (@10) and 20 (@20)
ranked documents from each system. Boldface entries indicate the best values for the
three systems.
Values in Table 4 show a better performance of ip-CL4IR on 4 of the 8 measures
while EM is better in the remaining 4, namely precision and recall in the first 5 and
10 ranked documents. This indicates that EM is actually better to recognize documents
very close to the query, but for documents with less elements in common with the query,
EM is not very precise. This can be better appreciated in Figure 3 where the interpolated
precision values of ip-CL4IR quickly overcome those of EM which is only better in 1
of the 11 recall points. This fact is also supported by the significant difference in the
values of IAP and MAP between ip-CL4IR and EM. For the 35 queries in the dataset,
our approach took 42.23 seconds (1.2 seconds per query) to execute while for CLR
took 1550.333 (44.29 seconds per query) showing an impressive enhancement in the
computational time required to retrieve documents, a key issue in document retrieval.
Both these times include lattice construction. Using better measures which consider the
correlation among terms, or including external knowledge sources like term taxonomies
may improve greatly the quality of the answers provided by our approach. These issues
are currently planned as future work. These experiments were performed in an Intel
Xeon machine running at 2.27 GHz with 62 GB of RAM memory.
There are many perspectives for our approach, however the most important is the
full exploitation of the ip-lattice structure to improve the quality in the answers. While
our principal goal in this article is to describe a general process to directly support
numeric term-document datasets in a concept lattice-based information retrieval system,
we argue that different IR tasks (some already supported on CL4IR systems) can be also
supported on ip-CL4IR for example, document clustering [1], user feedback inclusion
[6] and recommendation [9].
5 Conclusions
In this article we introduce a CL4IR approach which is able to deal directly with numer-
ical datasets through the use of the pattern structure framework (ip-CL4IR). We provide
a method and a process to construct an interval pattern concept lattice (ip-concept lat-
tice) which can be used as a document index. We present the idea of an ip-concept
lattice as a query space which can be navigated in order to find relevant documents. We
also provide means to rank these documents using vector-based distances. The feasi-
bility of our approach is validated through its application on a popular IR dataset for
which we present precision and recall values contrasted to those reported in the lit-
erature showing a better performance in the overall list of ranked documents and an
impressive enhancement in the time needed to answer a single query.
The perspectives for our approach are numerous, ranging from the improvement of
its answers, its application on different real-world datasets, but most importantly, the
full exploitation of the lattice structure to support different IR tasks.
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