Classification on Stiefel and Grassmann manifolds via maximum Likelihood estimation of matrix distributions

Abstract

Matrix manifolds such as Stiefel and Grassmann manifolds have been widely used in modern computer vision. This paper is concerned with the problem of classifying such manifold-valued data, based on the maximum likelihood estimation for the parametric probability density functions defined on the manifolds. By using a new way of computing normalisation constants for the matrix Langevin distribution function, defined for the data on Stiefel manifold, and the Fisher-Bingham density function on Grassmann manifold, we proposed a simple way to estimate the parameters in the distribution and demonstrated on real world datasets that the proposed method is efficient and more accurate. Our method uses asymptotic and Taylor series approximations, with good accuracy estimate.