Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels


In this paper, we have a tendency to develop an approach to exploiting kernel methods with manifold-valued knowledge. In many laptop vision issues, the data can be naturally represented as points on a Riemannian manifold. Because of the non-Euclidean geometry of Riemannian manifolds, usual Euclidean laptop vision and Machine Learning algorithms yield inferior results on such data. In this paper, we outline Gaussian radial basis operate (RBF)-based mostly positive definite kernels on manifolds that let us to embed a given manifold with a corresponding metric in a very high dimensional reproducing kernel Hilbert house. These kernels create it attainable to utilize algorithms developed for linear areas on nonlinear manifold-valued data. Since the Gaussian RBF defined with any given metric isn't invariably positive definite, we present a unified framework for analyzing the positive definiteness of the Gaussian RBF on a generic metric space. We tend to then use the proposed framework to spot positive definite kernels on two specific manifolds commonly encountered in computer vision: the Riemannian manifold of symmetric positive definite matrices and therefore the Grassmann manifold, i.e., the Riemannian manifold of linear subspaces of a Euclidean area. We tend to show that many widespread algorithms designed for Euclidean areas, like support vector machines, discriminant analysis and principal component analysis can be generalized to Riemannian manifolds with the help of such positive definite Gaussian kernels.

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