TESI DOCTORAL - La Salle

φ (NMI)
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0.8
0.6
0.4
0.2
WINE rbr−corr−e1
Baseline
PCA
ICA
NMF
RP
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3 4 5 6 7 8 9 10 11 12 13
dimensions
(a) Clustering results on the Wine
data set.
φ (NMI)
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0.8
0.6
0.4
0.2
Chapter 1. Framework of the thesis
miniNG rbr−corr−e1
Baseline
PCA
ICA
NMF
RP
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50 100 150 200 250 300 350 390
dimensions
(b) Clustering results on the miniNG
data set.
Figure 1.5: Illustration of the data representation indeterminacy on the clustering results
of the (a) Wine and (b) miniNG data sets clustered by the rbr-corr-e1 algorithm.
of the data 5 . In all cases, the desired number of clusters k is set to the number of classes
defined by the ground truth of each data set.
The results of these clustering processes are presented in figure 1.5, which displays
the normalized mutual information (φ (NMI) ) values between these clustering solutions and
the ground truth of each data collection. It can be observed that, in the Wine data set
(figure 1.5(a)), the clustering solution obtained operating on the original representation
is worse (i.e. it attains a lower φ (NMI) score) than all but three of the feature-extraction
based representations. In particular, the maximum value of φ (NMI) is attained using an
8-dimensional PCA transformation of the original data. However, these results cannot be
generalized. In fact, it is the baseline object representation that yields the best results
when the same clustering algorithm is applied on the miniNG data set (see figure 1.5(b)).
If we analyze the data representation that yields the best clustering results across the
12 unimodal data sets described in appendix A.2, we observe a rather even distribution:
baseline (23% of the times), LSA (31%), NMF (31%) and RP (15%), which somehow
reinforces the notion that no intrinsically superior data representation exists—see appendix
B.1 for more experimental results regarding data representation clustering indeterminacies.
Moreover, notice the remarkable influence of the data representation dimensionality on
the value of φ (NMI) , i.e. it is not only important to select the right type of representation,
but its dimensionality is also a critical factor. Although there exist several approaches for
determining the most natural dimensionality of the data (dnat) –such as spectrum of eigenvectors
in PCA (Duda, Hart, and Stork, 2001) or reconstruction error in NMF (Sevillano
et al., 2009)–, it is not trivial to ensure that any clustering algorithm will yield its best
performance when operating on the dnat-dimensional representation of the data.
To sum up, this modest example tries to demonstrate the relevance of the data representation
indeterminacy, as an incorrect choice in data representation may ruin the results
of a clustering process.
5 For a detailed description of the clustering algorithms, the data sets and the data representations employed
in the experimental sections of this thesis, see appendices A.1, A.2 and A.3, respectively.
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