Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels

TL;DR

This paper introduces Gaussian RBF kernels on Riemannian manifolds, enabling the use of linear algorithms for manifold-valued data in computer vision, with a focus on positive definiteness and specific manifolds.

Contribution

It develops a unified framework for positive definite Gaussian kernels on Riemannian manifolds and applies it to key manifolds in computer vision.

Findings

Gaussian RBF kernels can be positive definite on certain manifolds.

Support vector machines and PCA can be extended to Riemannian manifolds.

Framework applies to symmetric positive definite matrices and Grassmann manifolds.

Abstract

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.