A0531
Title: Riemannian functional regression and reproducing kernel tensor Hilbert spaces
Authors: Ke Yu - University of Oxford (United Kingdom) [presenting]
James Taylor - University of Oxford (United Kingdom)
Abstract: In many scientific fields, data arise in the form of smooth functions on Riemannian manifolds. Analyzing the relationship between a Riemannian functional response and a Riemannian functional predictor has become increasingly important. A Riemannian function-on-function regression model under the reproducing kernel tensor Hilbert space (RKTHS) framework is developed. As an extension of vector-valued reproducing kernel Hilbert spaces, the RKTHS we construct consists of functions taking values in tangent spaces along a curve on a manifold and is able to capture the intrinsic geometry of the manifold. The estimator of the regression coefficient achieves the optimal rate of convergence in mean prediction is proven. Moreover, a method is proposed to compare objects from different tensor Hilbert spaces based on Hilbert manifolds. Potential problems caused by nonnegative sectional curvatures of manifolds are also studied. Simulation studies demonstrate the numerical advantages of the RKTHS-based approach over the function-on-function regression based on Riemannian functional PCA. The proposed method is applied to tropical cyclone data to predict trajectories and brain imaging data of preterm and full-term infants in the Developing Human Connectome Project (dHCP) to study the linear relationship between homologous white matter fibre tracts in two hemispheres of the brain.
