A survey of kernel and spectral methods for clustering

Abstract

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.

DifFUZZY: a fuzzy clustering algorithm for complex datasets

“…As briefly addressed in Section 2.4, it involves an understanding of the mixing time (see, e.g., [17]) of the random walk defined in (8) for specific types of graphs. In particular, for a given data set, the performance of the developed method relies on the parameter α determining the diffusion distance in (9). Computational experimentation with test data sets reveals that the optimal choice of α tends to be robust for a broad variety of data set geometries.…”

“…The matrix P can be thought of as a transition matrix whose rows all sum to 1, and whose entry P i,j corresponds to the probability of jumping from the node (data point) i to the node j in one time step. The j-th component of the vector P α e, which is used in (9), is the probability of a random walk ending up in the j-th node, j = 1, 2, . .…”

“…Further examples are shown in the Supplementary Material. A classical example where the K-means algorithm fails (Filippone et al [9]) is shown in Fig. 2(a).…”

Modeling Tumor Growth for Kidney Cancer Based on Nuclei Clusters of Pathology Slides

“…For example, two main methods are defined as partitioning and hierarchical clustering, where an optimization rule applied to define clusters for the former type and a recursive approach which results in dendograms is introduced for the latter [7]. K-means clustering defines closeness as the metric for similarity to group data sets into clusters.…”

“…Structure of clusters is quantified by a variety of methods reported in literature [7], [8], [9]. Typically, the validity of clusters is evaluated by either the dispersion of data each cluster contains, or the data separation between clusters, or both [8].…”