Density estimation on Stiefel manifolds using matrix-Fisher model

Our main focus in this paper is on the matrix-variate Fisher distribution for the product case of Stiefel manifolds and perform density estimation or classification via straightforward way of Maximum Likelihood Estimation (MLE) of parameter. The novelty of our proposed method is its strict dependency on normalisation constant appearing in parametric models, i.e., we have implemented our proposed matrix Fisher density function for classification with normalisation constant included in a more general context for Stiefel manifolds. An accurate way of calculating the log-likehood function of matrix based normalising constant and its practicability to matrix variate parametric modelling has been a big hurdle and is treated in this paper for classification on Stiefel manifolds. Instead of ad-hoc approximation of normalisation constant we have considered the method of Saddle Point Approximation (SPA). With the inclusion of calculated normalising constant with matrix-variate Fisher parametric model, the direct MLE with simple Bayesian approach for numerical experiments is employed for classification example using Synthetic and real World dataset, with promising accuracy.