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2.2. Related work on consensus functions
crisp partitions with different number of clusters, although the desired number of clusters
k in the consensus clustering solution is a necessary parameter for the execution of their
consensus functions. The proposals presented in this work are based on the computation of
a probabilistic mapping for solving the cluster correspondence problem–instead of the oneto-one
classic cluster mapping– which allows combining partitions with different number of
clusters avoiding the addition of dummy clusters as in (Dimitriadou, Weingessel, and Hornik,
2002). In particular, three ways for deriving such probabilistic mapping based on the idea
of cumulative voting are presented. The construction of the consensus clustering solution is
a two-stage procedure: firstly, based on the cumulative vote mapping, a tentative consensus
is derived as a summary of the cluster ensemble maximizing the information content in
terms of entropy. And secondly, the extraction of the final consensus clustering solution
is obtained by applying an agglomerative clustering algorithm that minimizes the average
generalized Jensen-Shannon divergence within each cluster.
As already mentioned, solving the cluster correspondence problem paves the way for the
application of voting strategies for combining the outcomes of multiple clustering processes.
This issue is the central focus of (Boulis and Ostendorf, 2004), which presented several
methods for finding the correspondence between the clusters of the individual partitions in
the cluster ensemble. Two of these proposals are based on linear optimization techniques,
which are applied on an objective function that measures the degree of agreement among
clusters. In contrast, the third cluster correspondence method is based on Singular Value
Decomposition, and it sets cluster correspondences based on cluster correlation. All these
methods operate on a common space where the clusters of the distinct partitions (which
can be either crisp or fuzzy) are represented by means of cluster co-association matrices.
2.2.2 Consensus functions based on graph partitioning
The work by Strehl and Ghosh on consensus clustering based on graph partitioning is
probably one of the most classic references in the field of cluster ensembles (Strehl and
Ghosh, 2002). To our knowledge, they were the first to formulate the consensus clustering
problem in an information theoretic framework –i.e. the consensus clustering solution
should be the one maximizing the mutual information with respect to all the individual
partitions in the cluster ensemble–, a path followed by other authors in subsequent works
(Fred and Jain, 2003). In view of its prohibitive cost when formulated as a combinatorial
optimization problem in terms of shared mutual information, the authors propose
three clustering combination heuristics based on deriving a hypergraph representation of
the cluster ensemble —all of which require the desired number of clusters k as one of their
parameters. The first consensus function (called Cluster-based Similarity Partitioning Algorithm
or CSPA) induces a pairwise object similarity measure from the cluster ensemble
(as in (Fred and Jain, 2002a)), obtaining the consensus partition by reclustering the objects
with the METIS graph partitioning algorithm (Karypis and Kumar, 1998). For this
reason, we have enclosed the CSPA consensus function in both the graph partitioning and
object co-association categories in table 2.1. In the second clustering combiner proposed
in (Strehl and Ghosh, 2002) –named HGPA for HyperGraph Partitioning Algorithm–, the
cluster ensemble problem is posed as the partitioning of a hypergraph where hyperedges
represent clusters into k unconnected components of approximately the same size, cutting
a minimum number of hyperedges. And the third consensus function (Meta-CLustering
Algorithm or MCLA) views the clustering integration process as a cluster correspondence
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