Riemannian Generalized Gaussian Distributions on the Space of SPD Matrices for Image Classification

Abstract

The space of symmetric positive definite (SPD) matrices, denoted as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{m}$ </tex-math></inline-formula>, plays a crucial role in various domains, including computer vision, medical imaging, and signal processing. Its significance lies in its capacity to represent the underlying structure in nonlinear data using its Riemannian geometry. Nevertheless, a notable gap exists in the absence of statistical distributions capable of characterizing the statistical properties of data within this space. This paper proposes a new Riemannian Generalized Gaussian distribution (RGGD) on that space. The major contributions of this paper are, first of all, providing the exact expression of the probability density function (PDF) of the RGGD model, as well as an exact expression of the normalizing factor. Furthermore, an estimation of parameters is given using the maximum likelihood of this distribution. The second contribution involves exploiting the second-order statistics of feature maps derived from the first layers of deep convolutional neural networks (DCNNs) through the RGGD stochastic model in an image classification framework. Experiments were carried out on four well-known datasets, and the results demonstrate the efficiency and competitiveness of the proposed model.