A Graph-based approach to derive the geodesic distance on Statistical manifolds: Application to Multimedia Information Retrieval

Abstract

In this paper, we leverage the properties of non-Euclidean Geometry to define the Geodesic distance (GD) on the space of statistical manifolds. The Geodesic distance is a real and intuitive similarity measure that is a good alternative to the purely statistical and extensively used Kullback-Leibler divergence (KLD). Despite the effectiveness of the GD, a closed-form does not exist for many manifolds, since the geodesic equations are hard to solve. This explains that the major studies have been content to use numerical approximations. Nevertheless, most of those do not take account of the manifold properties, which leads to a loss of information and thus to low performances. We propose an approximation of the Geodesic distance through a graph-based method. This latter permits to well represent the structure of the statistical manifold, and respects its geometrical properties. Our main aim is to compare the graph-based approximation to the state of the art approximations. Thus, the proposed approach is evaluated for two statistical manifolds, namely the Weibull manifold and the Gamma manifold, considering the Content-Based Texture Retrieval application on different databases.