In this paper, we develop an approach to exploiting kernel methods with manifold-valued data. In many computer vision problems, the data can be naturally represented as points on a Riemannian manifold. Due to the non-Euclidean geometry of Riemannian manifolds, usual Euclidean computer vision and machine learning algorithms yield inferior results on such data. In this paper, we define Gaussian radial basis function (RBF)-based positive definite kernels on manifolds that permit us to embed a given manifold with a corresponding metric in a high dimensional reproducing kernel Hilbert space. These kernels make it possible to utilize algorithms developed for linear spaces on nonlinear manifold-valued data. Since the Gaussian RBF defined with any given metric is not always positive definite, we present a unified framework for analyzing the positive definiteness of the Gaussian RBF on a generic metric space. We then use the proposed framework to identify positive definite kernels on two specific manifolds commonly encountered in computer vision: the Riemannian manifold of symmetric positive definite matrices and the Grassmann manifold, i.e., the Riemannian manifold of linear subspaces of a Euclidean space. We show that many popular algorithms designed for Euclidean spaces, such as support vector machines, discriminant analysis and principal component analysis can be generalized to Riemannian manifolds with the help of such positive definite Gaussian kernels.
In computer vision applications, features often lie on Riemannian manifolds with known geometry. Popular learning algorithms such as discriminant analysis, partial least squares, support vector machines, etc., are not directly applicable to such features due to the non-Euclidean nature of the underlying spaces. Hence, classification is often performed in an extrinsic manner by mapping the manifolds to Euclidean spaces using kernels. However, for kernel based approaches, poor choice of kernel often results in reduced performance. In this paper, we address the issue of kernel selection for the classification of features that lie on Riemannian manifolds using the kernel learning approach. We propose two criteria for jointly learning the kernel and the classifier using a single optimization problem. Specifically, for the SVM classifier, we formulate the problem of learning a good kernel-classifier combination as a convex optimization problem and solve it efficiently following the multiple kernel learning approach. Experimental results on image set-based classification and activity recognition clearly demonstrate the superiority of the proposed approach over existing methods for classification of manifold features. The original mean shift algorithm [1] on Euclidean spaces (MS) was extended in [2] to operate on general Riemannian manifolds. This extension is extrinsic (Ext-MS) since the mode seeking is performed on the tangent spaces [3], where the underlying curvature is not fully considered (tangent spaces are only valid in a small neighborhood). In [3] was proposed an intrinsic mean shift designed to operate on two particular Riemannian manifolds (IntGS-MS), i.e. Grassmann and Stiefel manifolds (using manifold-dedicated density kernels). It is then natural to ask whether mean shift could be intrinsically extended to work on a large class of manifolds. We propose a novel paradigm to intrinsically reformulate the mean shift on general Riemannian manifolds. This is accomplished by embedding the Riemannian manifold into a Reproducing Kernel Hilbert Space (RKHS) by using a general and mathematically well-founded Riemannian kernel function, i.e. heat kernel [5]. The key issue is that when the data is implicitly mapped to the Hilbert space, the curvature of the manifold is taken into account (i.e. exploits the underlying information of the data). The inherent optimization is then performed on the embedded space. Theoretic analysis and experimental results demonstrate the promise and effectiveness of this novel paradigm.