Nonlinear regression on riemannian manifolds and its applications to neuro-image analysis

Regression in its most common form where independent and dependent variables are in Rn is a ubiquitous tool in Sciences and Engineering. Recent advances in Medical Imaging has lead to a wide spread availability of manifoldvalued data leading to problems where the independent variables are manifoldvalued and dependent are real-valued or vice-versa. The most common method of regression on a manifold is the geodesic regression, which is the counterpart of linear regression in Euclidean space. Often, the relation between the variables is highly complex, and existing most commonly used geodesic regression can prove to be inaccurate. Thus, it is necessary to resort to a non-linear model for regression. In this work we present a novel Kernel based non-linear regression method when the mapping to be estimated is either from M → ℝn or ℝn → M, where M is a Riemannian manifold. A key advantage of this approach is that there is no requirement for the manifold-valued data to necessarily inherit an ordering from the data in ℝn.We present several synthetic and real data experiments along with comparisons to the state-of-the-art geodesic regression method in literature and thus validating the effectiveness of the proposed algorithm.