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Kernel smoothing on manifolds

Eunseong Bae, Wolfgang Polonik

TL;DR

The paper develops a unified framework for kernel smoothing on manifolds, deriving finite-sample uniform bounds and Berry–Esseen type bounds that yield standard rates of uniform consistency and asymptotic normality for estimators of functions and their derivatives on a compact Riemannian manifold. By linking ambient-distance kernel smoothing to intrinsic operators, it connects kernel smoothing with Laplace–Beltrami operator estimation via graph Laplacians and to heat-kernel signatures, providing precise bias and variance decompositions. The results are specialized to kernel density estimation, kernel regression, and the estimation of heat-kernel signatures, with multi-point multivariate normality results and derivative extensions, offering rigorous guarantees for non-Euclidean data analysis and manifold learning tasks. The work also discusses extensions to graph-based approximations, spectral methods, and practical bandwidth choices, highlighting implications for diffusion geometry and shape analysis in high dimensions.

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.

Kernel smoothing on manifolds

TL;DR

The paper develops a unified framework for kernel smoothing on manifolds, deriving finite-sample uniform bounds and Berry–Esseen type bounds that yield standard rates of uniform consistency and asymptotic normality for estimators of functions and their derivatives on a compact Riemannian manifold. By linking ambient-distance kernel smoothing to intrinsic operators, it connects kernel smoothing with Laplace–Beltrami operator estimation via graph Laplacians and to heat-kernel signatures, providing precise bias and variance decompositions. The results are specialized to kernel density estimation, kernel regression, and the estimation of heat-kernel signatures, with multi-point multivariate normality results and derivative extensions, offering rigorous guarantees for non-Euclidean data analysis and manifold learning tasks. The work also discusses extensions to graph-based approximations, spectral methods, and practical bandwidth choices, highlighting implications for diffusion geometry and shape analysis in high dimensions.

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.
Paper Structure (31 sections, 30 theorems, 182 equations, 4 figures)

This paper contains 31 sections, 30 theorems, 182 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform consistency of kernel smoothing on manifolds
  4. Controlling the bias
  5. Controlling the stochastic part
  6. Uniform consistency
  7. Asymptotic normality and Berry-Esseen bounds
  8. Relation to the estimation of the Laplace-Beltrami operator
  9. Derivatives of kernel smoothing on manifolds
  10. Applications
  11. Kernel density estimation on manifolds
  12. Kernel regression on manifolds
  13. Estimation of heat kernel signature on manifolds
  14. Discussion
  15. Simulation study
  16. ...and 16 more sections

Key Result

Theorem 3.5

Suppose $f \in C^{2,\alpha}(\mathbb{M})$ with $\alpha \in (0,1]$. Then, there exist $\varepsilon_0,C_0>0$ such that if $\varepsilon < \varepsilon_0$, we have where $c(x)$ depends only on the curvature of $\mathbb{M}$ at $x$. The constants $\varepsilon_0$ and $C_0$ are $\mathbb{M}$-dependent, and they also depend on the $C^{2,\alpha}$-norms of $f$ and $\rho$. For unif:bias-res2, $\varepsilon_0$ an

Figures (4)

  • Figure 1: Empirical distribution of \ref{['formula: simul1']} over 300 simulated datasets from the circle case and the corresponding theoretic asymptotic density (red line) at two points (5,0) (top) and (0,5) (bottom).
  • Figure 2: Empirical distribution of (\ref{['formula: simul2']}) over 300 simulated datasets from the circle case and the corresponding theoretic asymptotic density (red line) at two points (5,0) (top) and (0,5) (bottom).
  • Figure 3: Empirical distribution of \ref{['formula: simul1']} over 300 simulated datasets from the torus case and the corresponding theoretic asymptotic density (red line) at two points (1/6,0,0) (top) and (0,-1/2,1/3) (bottom).
  • Figure 4: Empirical distribution of (\ref{['formula: simul2']}) over 300 simulated datasets from the torus case and the corresponding theoretic asymptotic density (red line) at two points (1/6,0,0) (top) and (0,-1/2,1/3) (bottom).

Theorems & Definitions (62)

  • Definition 3.4: $\mathbb{M}$-dependence
  • Theorem 3.5
  • Theorem 3.6
  • Corollary 3.7
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Theorem 5.1
  • ...and 52 more