  This paper studies the statistical model of the non-centered mixture of
scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information
geometry associated to this distribution, we derive a Riemannian gradient
descent algorithm. This algorithm is leveraged for two minimization problems.
The first one is the minimization of a regularized negative log-likelihood
(NLL). The latter makes the trade-off between a white Gaussian distribution and
the NC-MSG. Conditions on the regularization are given so that the existence of
a minimum to this problem is guaranteed without assumptions on the samples.
Then, the Kullback-Leibler (KL) divergence between two NC-MSG is derived. This
divergence enables us to define a minimization problem to compute centers of
mass of several NC-MSGs. The proposed Riemannian gradient descent algorithm is
leveraged to solve this second minimization problem. Numerical experiments show
the good performance and the speed of the Riemannian gradient descent on the
two problems. Finally, a Nearest centroid classifier is implemented leveraging
the KL divergence and its associated center of mass. Applied on the large scale
dataset Breizhcrops, this classifier shows good accuracies as well as
robustness to rigid transformations of the test set.
