Spectral Clustering Algorithm for the Allometric Extension Model

The spectral clustering algorithm is often used as a binary clustering method for unclassified data by applying the principal component analysis. When investigating the theoretical properties of the spectral clustering algorithm, existing studies have tended to invoke the assumption of conditional homoscedasticity. However, this assumption is restrictive and, in practice, often unrealistic. Therefore, in this paper, we consider the allometric extension model in which the directions of the first eigenvectors of two covariance matrices and the direction of the difference of two mean vectors coincide. We derive a non-asymptotic bound for the error probability of the spectral clustering algorithm under this allometric extension model. As a byproduct of this result, we demonstrate that the clustering method is consistent in high-dimensional settings.