Recognition on Grassmann manifolds via parametric modelling

Applicability of the standard Maximum Likelihood Estimation technique for utilising parametric models to real World applications has been a real hurdle due to the difficult calculation of normalising constant since decades. The aim of this paper is therefore to fill some gape by demonstrating a simple Maximum Likelihood Estimation (MLE) technique for classification on Grassmann manifolds sing the matrix-variate Bingham distributional model. The most challenging task in working with such high dimensional parametric models is the calculation of matrix based normalising constant. For calculating the values of normalising constants we propose some ad-hoc technique for approximating the values of normalising constants. These normalising constants are mostly represented by special functions, e.g., Hypergeometric functions, Confluent Hypergeometric functions and Bessel functions etc. We have approximated these special functions by Taylor series approximation and asymptotic series approximation methods, i.e by considering the first few terms of the series most important for calculating the numerical values of normalising constant. The calculated numerical value then boost the applicability of matrix-variate Bingham model for inferences via a simple Bayesian approach. Although these ad-hoc techniques does not have close-form solutions, but still they can produce very nice results for special cases of concentrated parameters.