Non-unique cluster numbers determination methods based on stability in spectral clustering

Recently, a large amount of work has been devoted to the study of spectral clustering—a simple yet powerful method for finding structure in a data set using spectral properties of an associated pairwise similarity matrix. Most of the existing spectral clustering algorithms estimate only one cluster number or estimate non-unique cluster numbers based on eigengap criterion. However, the number of clusters not always exists one, and eigengap criterion lacks theoretical justification. In this paper, we propose non-unique cluster numbers determination methods based on stability in spectral clustering (NCNDBS). We first utilize the multiway normalized cut spectral clustering algorithm to cluster data set for a candidate cluster number $k$ . Then the ratio value of the multiway normalized cut criterion of the obtained clusters and the sum of the leading eigenvalues (descending sort) of the stochastic transition matrix is chosen as a standard to decide whether the $k$ is a reasonable cluster number. At last, by varying the scaling parameter in the Gaussian function, we judge whether the reasonable cluster number $k$ is also a stability one. By three stages, we can determine non-unique cluster numbers of a data set. The Lumpability theorem concluded by Meil $breve{a}$ and Xu provides a theoretical base for our methods. NCNDBS can estimate non-unique cluster numbers of the data set successfully by illustrative experiments.