Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices
This work addresses the challenge of accurately modeling SPD matrices in machine learning for applications like image processing and medical imaging, representing an incremental improvement by adapting existing kernel methods to a specific geometric context.
The paper tackled the problem of applying kernel methods to symmetric positive definite (SPD) matrices by introducing a family of provable positive definite kernels that encode the Riemannian manifold geometry, enabling the extension of algorithms like SVM and kernel PCA to this manifold and demonstrating benefits in tasks such as pedestrian detection and DTI segmentation.
Symmetric Positive Definite (SPD) matrices have become popular to encode image information. Accounting for the geometry of the Riemannian manifold of SPD matrices has proven key to the success of many algorithms. However, most existing methods only approximate the true shape of the manifold locally by its tangent plane. In this paper, inspired by kernel methods, we propose to map SPD matrices to a high dimensional Hilbert space where Euclidean geometry applies. To encode the geometry of the manifold in the mapping, we introduce a family of provably positive definite kernels on the Riemannian manifold of SPD matrices. These kernels are derived from the Gaussian ker- nel, but exploit different metrics on the manifold. This lets us extend kernel-based algorithms developed for Euclidean spaces, such as SVM and kernel PCA, to the Riemannian manifold of SPD matrices. We demonstrate the benefits of our approach on the problems of pedestrian detection, ob- ject categorization, texture analysis, 2D motion segmentation and Diffusion Tensor Imaging (DTI) segmentation.