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.

翻译：在数据位于无边界紧致（未知）流形的假设下，我们推导了核平滑及其（一阶和二阶）导数的有限样本界，并通过Berry-Esseen型界建立了渐近正态性。特例包括核密度估计、核回归与热核特征。文中亦讨论了与图拉普拉斯算子的关联。