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 recent years, several optimization algorithms have been proposed for Riemannian manifolds. In particular, LogitBoost for classification on Riemannian manifolds was introduced in @cite_28 . This algorithm has the drawbacks of approximating the manifold by tangent spaces and not scaling with the number of training samples due to the heavy use of exponential logarithmic maps to transit between the manifold and the tangent space, as well as of gradient descent based Karcher mean calculation. Here, our positive definite kernels enable us to use more efficient and accurate classification algorithms on manifolds without requiring tangent space approximations. Furthermore, as shown in @cite_5 @cite_56 , extending existing kernel-free, manifold-based binary classifiers to the multi-class case is not straightforward. In contrast, the kernel-based classifiers on manifolds described in this paper can readily be used in multi-class scenarios.
@cite_28 @cite_5 @cite_56
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We present a new algorithm to detect pedestrian in still images utilizing covariance matrices as object descriptors. Since the descriptors do not form a vector space, well known machine learning techniques are not well suited to learn the classifiers. The space of d-dimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold. The main contribution of the paper is a novel approach for classifying points lying on a connected Riemannian manifold using the geometry of the space. The algorithm is tested on INRIA and DaimlerChrysler pedestrian datasets where superior detection rates are observed over the previous approaches. In video surveillance, classification of visual data can be very hard, due to the scarce resolution and the noise characterizing the sensors' data. In this paper, we propose a novel feature, the ARray of COvariances (ARCO), and a multi-class classification framework operating on Riemannian manifolds. ARCO is composed by a structure of covariance matrices of image features, able to extract information from data at prohibitive low resolutions. The proposed classification framework consists in instantiating a new multi-class boosting method, working on the manifold Symd+ of symmetric positive definite d × d (covariance) matrices. As practical applications, we consider different surveillance tasks, such as head pose classification and pedestrian detection, providing novel state-of-the-art performances on standard datasets.