Nonparametric regression estimation on closed Riemannian manifolds

The nonparametric estimation of the regression function of a real-valued random variable Y on a random object X valued in a closed Riemannian manifold M is considered. A regression estimator which generalizes kernel regression estimators on Euclidean sample spaces is introduced. Under classical assumptions on the kernel and the bandwidth sequence, the asymptotic bias and variance are obtained, and the estimator is shown to converge at the same L2-rate as kernel regression estimators on Euclidean spaces. Index Terms — Nonparametric regression, Kernel regression, Riemannian manifolds, L2-convergence.