Statistical Learning over Manifolds and its applications in Computer Vision

The aim of this thesis is to develop a consistent general parametric modeling framework based on analytic manifolds. In practice, the structures of most real-world data such as images and videos have a very complex nature, and that is why it is assumed that instead of linear vector space (Euclidean space), such data may actually lie in nonlinear differentiable manifolds (Riemannian space). This assumption of differentiable manifolds suggests that the nonlinear manifolds of high dimensional structures are intrinsically low dimensional spaces. In other words, such nonlinear manifold structures of data are locally homeomorphic to Euclidean space of comparatively very low dimensionality.

Based on the manifolds assumption, in this thesis a novel parametric classification method on Stiefel and Grassmann manifolds is presented. In order to process the manifold-valued data, matrix-variate Fisher, Angular Central Gaussian (ACG) and Bingham distributions of most general settings were considered. The proposed method exploits these distributional models with the inclusion of normalizing constants that naturally appear within them. To calculate the normalizing constants of the prescribed models, the method of asymptotic series approximation, Taylor series approximation, Laplace's approximation and the method of Saddle-Point Approximation (SPA) to a new setting have been extended. Furthermore, the standard theory Maximum Likelihood Estimation (MLE) have been employed to evaluate the involved parameters in the
used probability density functions.

The validity and performance of the proposed generalized Bayesian framework based on normalizing constants being calculated by several different approaches have been studied for one set of synthetic data for Stiefel manifolds case and for 36 real world visual classification databases in the context of Grassmann manifolds. The new approach has been compared against existing approaches in the current literature and it has been confirmed that this new generalised Bayesian parametric modeling approach measurably improved the classification accuracy.