TESI DOCTORAL - La Salle

3.1. Motivation
– the computational complexity of combining a large number of individual partitions, by
means of hierarchical consensus architectures, which consist in the layered construction
of the consensus clustering solution through a hierarchical structure of low complexity
intermediate consensus processes.
– the negative bias induced by poor quality clusterings in the consensus clustering solution,
by means of a self-refining post-processing that, using the obtained consensus
clustering solution as a reference, builds a select and reduced cluster ensemble (i.e. a
subset of the original cluster ensemble), deriving a new and refined consensus clustering
upon it in a fully unsupervised manner.
Although both strategies are complementary (not in vain they can be naturally combined
giving rise to SHCA), their description and study is decoupled in the present and the
next chapters, respectively. Thus, in our description and analysis of hierarchical consensus
architectures (chapter 3), we ultimately aim to design computationally optimal consensus
architectures and, consequently, we will solely focus on aspects regarding their time complexity.
Meanwhile, the study of consensus self-refining procedures, which are presented in
chapter 4, is centered on improving the quality of the consensus solutions yielded by the
most computationally efficient consensus architectures devised in the present chapter.
The introduction, the discussion of their rationale and the theoretical description of
hierarchical consensus architectures are complemented by the presentation of multiple experiments
analyzing multiple aspects of their performance on several real data collections.
Last but not least, it is to note that although all the proposals put forward in this chapter
are focused on a hard cluster ensemble scenario, they are also applicable for fuzzy clusterings
combination.
3.1 Motivation
The construction of consensus clustering solutions is usually tackled as a one-step process,
in the sense that the whole cluster ensemble E is input to the consensus function F at once
—see figure 3.1(a). This is what we call flat consensus clustering. However, as outlined in
chapter 2, the time and space complexities of consensus functions typically scale linearly or
quadratically with the size of the cluster ensemble l –i.e. O (l w ), where w ∈{1, 2}–, which
may lead to a highly costly or even impossible execution of the consensus clustering task if
it is to be conducted on a cluster ensemble containing a large number of partitions 1 .
For this reason, a natural way for avoiding this limitation besides reducing the computational
complexity of the consensus solution creation process consists in applying the
classic divide-and-conquer strategy (Dasgupta, Papadimitriou, and Vazirani, 2006) which
basically:
– breaks the original problem into subproblems which are nothing but smaller instances
of the same type of problem
1 Moreover, the time complexity of consensus functions also depends –linearly or quadratically, see appendix
A.5– on the number of objects in the data set n and the number of clusters k of the clusterings
in the ensemble. However, as we assume that these two factors are constant for a given cluster ensemble
corresponding to a specific data set, the only dependence of concern is that referring to the cluster ensemble
size l.
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