Riemannian Optimization for Non-Centered Mixture of Scaled Gaussian Distributions

This article studies the statistical model of the non-centered mixture of scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information geometry associated with this distribution, we derive a Riemannian gradient descent algorithm. This algorithm is leveraged for two minimization problems. The first 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 second minimization problem. The latter is the computation of centers of mass of several NC-MSGs. Numerical experiments show the good performance and the speed of the Riemannian gradient descent on the two problems. Finally, a
Nearest centroïd 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 and robustness to rigid transformations of the test set.