An Information Geometry-Based Distance Between High-Dimensional Covariances for Scalable Classification

Modeling images/videos with covariance matrices has attracted increasing attentions in various vision tasks, especially in visual classification. For covariances-based visual classification, measuring the distances between covariances is one of the key issues and has been studied for decades. Since the space of covariances is a Riemannian manifold, the geometrical structure of covariances should be favorably considered when designing distance metrics. Although this problem has been widely studied, designing an effective and efficient metric between high-dimensional covariances (HDCOV) for scalable classification is still an open problem. In this paper, we present an information geometry-based distance (IGBD) to tackle this challenge from the perspective of information geometry. Our idea is based on the fact that each covariance can be viewed as a zero-mean Gaussian distribution, and thus the distances between covariances are measured by those between the corresponding Gaussian distributions. The core of our method is to project each distribution, in the form of a set of random samples, to a vector on the tangent space of a common, known distribution on the statistical manifold, based on Fisher information metric and maximum likelihood method. On the tangent space, the Euclidean norm can be used to measure the distances between those sets of projection vectors (or equivalently distributions). The proposed IGBD for HDCOV is computationally efficient and easily combined with a linear support vector machine, suitable for scalable visual classification. The experiments are conducted on various kinds and sizes of benchmarks, and results show the proposed method is efficient and the combination of HDCOV can achieve very competitive performance.