Random walk distances in data clustering and applications

Abstract

In this paper, we develop a family of data clustering algorithms that combine the strengths of existing spectral approaches to clustering with various desirable properties of fuzzy methods. In particular, we show that the developed method “Fuzzy-RW,” outperforms other frequently used algorithms in data sets with different geometries. As applications, we discuss data clustering of biological and face recognition benchmarks such as the IRIS and YALE face data sets.

Appendix

The procedure commonly employed in the literature for minimizing the FCM functional (6) is an alternating directions scheme, originally proposed by Bezdek et al. (1984). For completeness, we provide a listing of the algorithm below. More details can be found in Gan et al. (2007) and Bezdek et al. (1984), among others.

The convergence criterion in line $6$ is usually chosen to be of the form $||U^{(r)}-U^{(r-1)}||<\varepsilon$ for some pre-specified $\varepsilon >0$ (Gan et al. 2007). Here, $U^{(r)}$ and $U^{(r-1)}$ denote the values of the fuzzy membership matrix $U=(u_{ij})_{i\le k, j\le n}$ in the $r$ and $r-1$ iteration of the loop, respectively.

Now, the $k$ clusters of data points may be decided in terms of thresholding with respect to the membership matrix, or sometimes data points can be assigned to clusters based on their maximal membership values.