Kernel smoothing on manifolds
This work addresses the challenge of nonparametric estimation on manifolds, which is incremental as it extends existing kernel smoothing theory to manifold settings.
The authors tackled the problem of kernel smoothing on unknown compact manifolds without boundary, deriving finite sample bounds and establishing asymptotic normality for kernel smoothing and its derivatives, with applications to kernel density estimation, kernel regression, and the heat kernel signature.
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.