TESI DOCTORAL - La Salle

6.5. Discussion
solutions on hard cluster ensembles (Dudoit and Fridlyand, 2003; Fischer and Buhmann,
2003; Greene and Cunningham, 2006). To our knowledge, the only voting-based consensus
function for soft cluster ensembles is the Voting-Merging Algorithm (VMA) of (Dimitriadou,
Weingessel, and Hornik, 2002), which employs a weighted version of the sum rule for
confidence voting. Moreover, all these works share a common point in that they use the
Hungarian algorithm for solving the cluster correspondence problem.
Additional techniques for cluster disambiguation developed in the consensus clustering
literature include the correspondence estimation based on common space cluster representation
by clusters clustering or Singular Value Decomposition (Boulis and Ostendorf, 2004),
the Soft Correspondence Ensemble Clustering algorithm, which is based on establishing a
soft correspondence between clusters (in the sense that a cluster of a given clustering corresponds
to every cluster in another clustering with different weights) (Long, Zhang, and
Yu, 2005), the cumulative voting approach, that, unlike common one-to-one voting schemes
(e.g. Hungarian), computes a probabilistic mapping between clusters (Ayad and Kamel,
2008), or the FullSearch, Greedy and LargeKGreedy cluster alignment algorithms (Jakobsson
and Rosenberg, 2007). The first two approaches coincide in that they can be indistinctly
applied for aligning the clusters of crisp and fuzzy partitions. Given the key importance of
the cluster disambiguation process as a prior step to voting, we plan to evaluate these alternatives
to the Hungarian method, so as to investigate their impact on both the quality of
the consensus clusterings obtained and the time complexity of the whole consensus process.
The comparative performance analysis of the four proposed consensus functions has
revealed that they constitute a feasible alternative for conducting consensus clustering processes
on soft cluster ensembles, as they are capable of yielding consensus clustering solutions
of comparable or superior quality to those obtained by state-of-the-art clustering combiners
at a reasonable computational cost. An additional appealing feature of our proposals is
that they naturally deliver fuzzy consensus clustering solutions, which makes all sense in a
soft clustering scenario —a fact other recent consensus functions for soft cluster ensembles
–as the one presented in (Punera and Ghosh, 2007)– do not consider. However, the lack
of a fuzzy ground truth has not allowed evaluating the soft consensus clusterings obtained,
which constitutes one of the future directions of research of the work conducted in this
chapter. As mentioned earlier, this would probably make the differences between the proposed
consensus functions more evident, as it would highlight the differences between the
distinct voting methods employed.
As reported earlier, the sequential application of the cluster disambiguation and the
voting processes penalizes the time complexity of our proposals, specially when they are
compared to VMA. Thus, in the future, we plan to adopt the iterative cluster alignment
plus voting strategy employed by this consensus function, which, in our opinion, will surely
reduce the execution time of the proposed voting-based consensus functions without significantly
reducing the quality of the consensus functions obtained.
Another significant conclusion is that the EAC and HGPA consensus functions yield
the lowest quality consensus clusterings, as already noticed in the vast majority of the
experiments conducted in the hard clustering scenario.
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