Probability Distribution-Based Dimensionality Reduction on Riemannian Manifold of SPD Matrices

Abstract

Representing images and videos with Symmetric Positive Definite (SPD) matrices and utilizing the intrinsic Riemannian geometry of the resulting manifold has proved successful in many computer vision tasks. Since SPD matrices lie in a nonlinear space known as a Riemannian manifold, researchers have recently shown a growing interest in learning discriminative SPD matrices with appropriate Riemannian metrics. However, the computational complexity of analyzing high-dimensional SPD matrices is nonnegligible in practical applications. Inspired by the theory of nonparametric estimation, we propose a probability distribution-based approach to overcome this drawback by learning a mapping from the manifold of high-dimensional SPD matrices to a manifold of lower-dimension, which can be expressed as an optimization problem on the Grassmann manifold. Specifically, we perform the dimensionality reduction for high-dimensional SPD matrices with popular Riemannian metrics and an affinity matrix constructed using an estimated probability distribution function (PDF) to achieve maximum class separability. The evaluation of several classification tasks shows the competitiveness of the proposed approach compared with state-of-the-art methods.

Keywords

• Probability distribution
• Riemannian geometry
• SPD matrices