  A Stiefel manifold of the compact type is often encountered in many fields of
Engineering including, signal and image processing, machine learning, numerical
optimization and others. The Stiefel manifold is a Riemannian homogeneous space
but not a symmetric space. In previous work, researchers have defined
probability distributions on symmetric spaces and performed statistical
analysis of data residing in these spaces. In this paper, we present original
work involving definition of Gaussian distributions on a homogeneous space and
show that the maximum-likelihood estimate of the location parameter of a
Gaussian distribution on the homogeneous space yields the Fr\'echet mean (FM)
of the samples drawn from this distribution. Further, we present an algorithm
to sample from the Gaussian distribution on the Stiefel manifold and
recursively compute the FM of these samples. We also prove the weak consistency
of this recursive FM estimator. Several synthetic and real data experiments are
then presented, demonstrating the superior computational performance of this
estimator over the gradient descent based non-recursive counter part as well as
the stochastic gradient descent based method prevalent in literature.
