TESI DOCTORAL - La Salle

2.2. Related work on consensus functions
to a hypothetical consensus clustering solution membership matrix. The goal is to find
a correspondence matrix that yields the best projection of each individual clustering on
the space defined by the consensus clustering solution. From a practical viewpoint, both
the correspondence and consensus clustering matrices are derived simultaneously, using an
EM-like approach.
The beautiful proposal of (Lange and Buhmann, 2005) introduced a consensus function
named Probabilistic Label Aggregation (PLA), which operates on soft cluster ensembles
(although it also works on crisp ones). Its rationale is as follows: given a single fuzzy
partition, a pairwise object co-association matrix is created by simply multiplicating the
membership probabilities matrix by its own transpose. Repeating this process on all the
partitions in the soft cluster ensemble and aggregating (and subsequently normalizing) the
resulting matrices gives rise to a joint probability matrix of finding two objects in the
same cluster. Neatly, the authors propose subjecting this joint probability matrix to a
non-negative matrix factorization process that yields estimates for class-likelihoods and
class-posteriors, upon which the consensus clustering solution is based. This factorization
process is posed as an optimization problem which is solved by applying the EM algorithm.
Besides the elegance of the proposed solution, this work also stands out by the fact that
it supports an out-of-sample extension that makes it possible to assign previously unseen
objects to classes of the consensus clustering solution. Moreover, the proposed method also
allows combining weighted partitions, i.e. it gives the user the chance to assign different
degrees of relevance to the cluster ensemble partitions.
A closely related proposal is the application of Non-Negative Matrix Factorization
(NMF) for solving the consensus clustering problem presented (Li, Ding, and Jordan, 2007).
In contrast to (Lange and Buhmann, 2005), the aim is to combine crisp partitions, which
imposes a series of constraints on the optimization problem that is solved via symmetric
NMF—which, from an algorithmic viewpoint, is implemented by means of multiplicative
rules. Moreover, the same approach is employed for conducting semi-supervised clustering,
a problem that lies beyond the scope of this work.
2.2.6 Consensus functions based on reinforcement learning
Reinforcement learning has also been applied for the construction of consensus clustering
solutions (Agogino and Tumer, 2006). In that work, the average φ (NMI) of the consensus
clustering solution with respect to the cluster ensemble is regarded as the reward that must
be maximized by the actions of the agents. In this case, each agent casts a vote indicating
which cluster each object should be assigned to (i.e. it operates on hard cluster ensembles).
The application of a majority voting scheme on these votes yields the consensus clustering
solution, which is iteratively refined as the agents learn how to vote so as to maximize the
average φ (NMI) . The authors highlight the ease of their approach for combining clusterings
in distributed scenarios, which makes it specially suitable in failure-prone domains.
2.2.7 Consensus functions based on interpeting object similarity as data
The work by Kuncheva, Hadjitodorov, and Todorova (2006) introduced three consensus
functions based on interpreting object similarity as data. That is, each object is represented
by n features, where the jth feature of the ith object corresponds to the co-association
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