Kernel smoothing on manifolds

TL;DR

This paper develops finite sample bounds and asymptotic normality results for kernel smoothing methods on unknown manifolds, covering density estimation, regression, and heat kernel signatures, with links to graph Laplacians.

Contribution

It provides the first finite sample and asymptotic analysis of kernel smoothing on unknown manifolds, extending classical methods to geometric data.

Findings

Finite sample bounds for kernel smoothing on manifolds

Asymptotic normality established via Berry-Esseen bounds

Connections made between kernel methods and graph Laplacians

Abstract

Under the assumption that data lie on a compact (unknown) manifold without boundary, we derive finite sample bounds for kernel smoothing and its (first and second) derivatives, and we establish asymptotic normality through Berry-Esseen type bounds. Special cases include kernel density estimation, kernel regression and the heat kernel signature. Connections to the graph Laplacian are also discussed.