Developing Additive Spectral Approach to Fuzzy Clustering

An additive spectral method for fuzzy clustering is presented. The method operates on a clustering model which is an extension of the spectral decomposition of a square matrix. The computation proceeds by extracting clusters one by one, which allows us to draw several stopping rules to the procedure. We experimentally test the performance of our method and show its competitiveness. In spite of the fact that many relational fuzzy clustering algorithms have been developed already [1,2,3,4,12], most of them are ad hoc and, moreover, they all involve manually specified parameters such as the number of clusters or threshold of similarity without providing any guidance for choosing them. We apply a model-based approach of additive clustering, combined with the spectral clustering approach, to develop a novel relational fuzzy clustering method that is both adequate and supplied with model-based parameters helping to choose the right number of clusters. We assume the data in the format of what is called similarity or relational data, that is a matrix W = (wtt′), t, t′ ∈ T , of similarity indexes wtt′ , between objects t, t′ from a set of objects T . We further assume that this similarity values are but manifested expressions of some hidden relational patterns which can be represented by fuzzy clusters. We propose to formalize a relational fuzzy cluster as represented by two items: (i) a membership vector u = (ut), t ∈ T , such that 0 ≤ ut ≤ 1 for all t ∈ T , and (ii) an intensity μ > 0 that expresses the extent of significance of the pattern corresponding to the cluster. With the introduction of the intensity, applied as a scaling factor to u, it is the product μu that is a solution rather than its individual co-factors. Given a value of the product μut, it is impossible to tell which part of it is μ and which ut. To resolve this, we follow a conventional scheme: let us constrain the scale of the membership vector u on a constant level, for example, by a condition such as ∑ t ut = 1 or ∑ t u 2 t = 1, then the remaining factor will define the value of μ. The latter normalization better suits the criterion implied by our fuzzy clustering method and, thus, is accepted further on. S.O. Kuznetsov et al. (Eds.): RSFDGrC 2011, LNAI 6743, pp. 273–277, 2011. c © Springer-Verlag Berlin Heidelberg 2011 274 B. Mirkin and S. Nascimento To make the cluster structure in the similarity matrix sharper, we apply the spectral clustering approach to pre-process a raw similarity matrix W into A by using the so-called normalized Laplacian transformation as related to the popular clustering criterion of normalized cut [6]. This criterion relates to the minimum non-zero eigenvalue of the Laplacian matrix. To change this to the maximum eigenvalue, we further transform this to its pseudo-inverse matrix, which also increases the gaps between eigenvalues. Our additive fuzzy clustering model follows that of [11,7,10] and involves K fuzzy clusters that reproduce the pseudo-inverted Laplacian similarities att′ up to additive errors according to the following equations: