Kernel and wavelet density estimators on manifolds and more general metric spaces

TL;DR

This paper develops kernel and wavelet density estimators for data on manifolds and metric spaces, establishing convergence rates similar to classical real-valued cases.

Contribution

It introduces and analyzes kernel and wavelet density estimators on complex spaces like manifolds, extending classical methods to more general settings.

Findings

Convergence rates comparable to classical real-valued cases.

Development of localized spectral kernels and wavelet systems.

Establishment of Besov regularity spaces on general metric spaces.

Abstract

We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but also sufficiently rich in allowing the development of smooth functional calculus with well localized spectral kernels, Besov regularity spaces, and wavelet type systems. Kernel and both linear and nonlinear wavelet density estimators are introduced and studied. Convergence rates for these estimators are established, which are analogous to the existing results in the classical setting of real-valued variables.