Clustering algorithms are a useful tool to explore data structures and have been employed in many disciplines. The focus of this paper is the partitioning clustering problem with a special interest in two recent approaches: kernel and spectral methods. The aim of this paper is to present a survey of kernel and spectral clustering methods, two approaches able to produce nonlinear separating hypersurfaces between clusters. The presented kernel clustering methods are the kernel version of many classical clustering algorithms, e.g., K-means, SOM and neural gas. Spectral clustering arise from concepts in spectral graph theory and the clustering problem is configured as a graph cut problem where an appropriate objective function has to be optimized. An explicit proof of the fact that these two paradigms have the same objective is reported since it has been proven that these two seemingly different approaches have the same mathematical foundation. Besides, fuzzy kernel clustering methods are presented as extensions of kernel K-means clustering algorithm.

In this paper, data dimensionality estimation methods are reviewed. The estimation of the dimensionality of a data set is a classical problem of pattern recognition. There are some good reviews [1] in literature but they do not include more recent developments based on fractal techniques and neural autoassociators. The aim of this paper is to provide an up-to-date survey of the dimensionality estimation methods of a data set, paying special attention to the fractal-based methods.

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Kernel Methods are algorithms that, by replacing the inner product with an appropriate positive definite function, implicitly perform a nonlinear mapping of the input data into a high-dimensional feature space. In this paper, we present a kernel method for clustering inspired by the classical K-Means algorithm in which each cluster is iteratively refined using a one-class Support Vector Machine. Our method, which can be easily implemented, compares favorably with respect to popular clustering algorithms, like K-Means, Neural Gas, and Self-Organizing Maps, on a synthetic data set and three UCI real data benchmarks (IRIS data, Wisconsin breast cancer database, Spam database).

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