Nonparametric recursive regression estimation on Riemannian Manifolds

The considerations of this paper are restricted to random variables with values on Riemannian manifolds M and hence we propose a geometric framework to estimate their recursive regression function. Suppose we are given observations (X_i,Y_i)_i=1⋯n, where X_i∈M and Y_i∈R. In this work we define and study a new estimator of the regression function on Riemannian Manifold M. Precisely, we employ a recursive version of the Nadaraya–Watson estimator on Riemannian Manifolds. Under some assumptions in Riemannian Manifolds data analysis, we study the properties of a recursive family kernels regression. The bias, variance are computed explicitly.