On Kernel density estimation on Product Riemannian manifolds

Estimating the kernel density function of a random vector taking values on Riemannian manifolds are considered. More Precisely, we consider the problem of the estimation of the probability density of n i.i.d. random objects on the product of two compact Riemannian manifolds without boundary. The proposed methodology adapts the Pelletier’s approach which is the kernel density estimation on non Euclidean setting. Under sufficient regularity assumptions on the underlying density, L2 convergence rates are obtained. Riemannian products are the most natural and fruitful generalization of Cartesian products. These manifold models are of interest to many fields of applied mathematics, natural sciences and technology. Let us recall here as examples, the use of these products in the domain of computer vision and in biomedical sciences. In this note, we construct an estimator of the density on a product manifold from one of the estimators existing in the literature. Among the estimation methods, we mainly cite a method based on Fourier developments proposed in (Hendriks, 1990). Another method based on geodesic distance is proposed later by B. Pelletier in (Pelletier, 2005). This method used in the present work, is a generalization of the kernel method in the affine case. Another method recently developed by Kim et al. in (Kim, 2013) use the concept of exponential map to define a kernel estimator of the density on a Riemannian manifold. The aim in the present work, is to show that Pelletier's method is the best suited to the case of the Riemannian product.