General Technique for Estimating Number of Groups for Spectral Clustering

Determining the number of groups or dimension of a feature space related to an initial dataset projected to null-space of the Laplace-Beltrami-type operators is a fundamental problem of applications exploiting a spectral clustering techniques. This paper theoretically focuses on generalizing and providing minor comments to a previous work by Bruneau et al., who proposed modification of the Bartlett test that is commonly used in the principal component analysis, to estimate the number of groups related to normalized spectral clustering approaches. The generalization is based on a relation between the distributions of the spectrum associated with a covariance matrix and graph Laplacian, which allows us to use the modified Bartlett test for unnormalized spectral clustering as well. Other comments follow previous works by Lawley and James, which allow us testing subsets of eigenvalues by involving likelihood ratio statistic and linkage factors. Solving issues arising from limits of floating-point arithmetic are demonstrated on benchmarks employing spectral clustering for $2$-phase volumetric image segmentation. On a same problem, analysing spectral clustering in divide-merge settings is presented.