Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure

In this paper we study the properties of a kurtosis matrix and propose its eigenvectors as interesting directions to reveal the possible cluster structure of a data set. Under a mixture of elliptical distributions with proportional scatter matrix, it is shown that a subset of the eigenvectors of the fourth-order moment matrix corresponds to Fisher's linear discriminant subspace. The eigenvectors of the estimated kurtosis matrix are consistent estimators of this subspace and its calculation is easy to implement and computationally efficient, which is particularly favourable when the ratio n/p is large.