Dictionary Learning of Symmetric Positive Definite Data Based on Riemannian Manifold Tangent Spaces and Local Homeomorphism

Symmetric Positive Definite (SPD) data are increasingly prevalent in dictionary learning recently. SPD data are the typical non-Euclidean data and cannot constitute a Euclidean space. Therefore, many dictionary learning algorithms cannot be directly adopted on SPD data. Reproducing Kernel Hilbert Spaces (RKHS) is now commonly used to deal with this problem. However, RKHS is an infinite-dimensional Hilbert space, rather than a Euclidean space, resulting in the inability of the dictionary learning to be directly used on SPD data. In this paper, we propose a novel dictionary learning algorithm for SPD data, which is based on the Riemannian Manifold Tangent Space (RMTS). Since RMTS is based on a finite-dimensional Hilbert space, i.e., Euclidean space, most machine learning algorithms developed on Euclidean space can be directly applied to RMTS. The introduction of RMTS provides a better mathematical platform for machine learning of non-Euclidean data. The proposed RMTS method first selects a point on a Riemannian manifold as an anchor point. It transforms the other points on the Riemannian manifold into tangent vectors of the geodesics between these points and the anchor point, with the tangent vectors at the anchor point. We set the length of the tangent vector equal to the length of the geodesic. As a result, such tangent vectors have an explicit geometric meaning, such as direction information, while the RKHS method may cause some geometric meaning to be lost in the original data during the mapping process. In addition, the proposed algorithm adds a regular term of local homomorphism between SPD data and its RMTS dictionary coding coefficients, so that the similarity of SPD data and its RMTS dictionary coding coefficients are as close as possible. Experimental results on four public datasets demonstrate that the proposed algorithm significantly outperforms five state-of-the-art algorithms. This work opens a new pathway towards SPD data dictionary learning methods.