Distributional Convergence of the Empirical Laplacians with Integral Kernels on Domains with Boundaries
Bernard Akwei, Luke Rogers, Alexander Teplyaev
TL;DR
The paper analyzes when and how the empirical graph Laplacians built from i.i.d. samples with kernel K converge to a Laplacian-type operator on a domain, including manifolds with boundary. It introduces D_{ε,n} and a kernel-induced operator Δ_K, proving a central limit theorem for the deviation between the empirical and averaging operators and establishing convergence to Δ_K under weakened smoothness (f ∈ C^{2,θ}) and non-uniform sampling density g. It also demonstrates asymptotically vanishing correlations of the fluctuations across distinct points, and extends the theory to domains with boundary, defining a boundary-aware Laplacian Δ_{K, A(p)} that accounts for the local geometry of the boundary. The results generalize prior work to more general kernels, lower regularity, non-uniform sampling, and boundary effects, providing a more robust foundation for Laplacian-based dimension reduction in real-world, non-smooth data settings.
Abstract
Motivated by the problem of understanding theoretical bounds for the performance of the Belkin-Niyogi Laplacian eigencoordinate approach to dimension reduction in machine learning problems, we consider the convergence of random graph Laplacian operators to a Laplacian-type operator on a manifold. For $\{X_j\}$ i.i.d.\ random variables taking values in $\mathbb{R}^d$ and $K$ a kernel with suitable integrability we define random graph Laplacians \begin{equation*} D_{ε,n}f(p)=\frac{1}{nε^{d+2}}\sum_{j=1}^nK\left(\frac{p-X_j}ε\right)(f(X_j)-f(p)) \end{equation*} and study their convergence as $ε=ε_n\to0$ and $n\to\infty$ to a second order elliptic operator of the form \begin{align*} Δ_K f(p) &= \sum_{i,j=1}^d\frac{\partial f}{\partial x_i}(p)\frac{\partial g}{\partial x_j}(p)\int_{\mathbb{R}^d}K(-t)t_it_jdλ(t)\\ &\quad +\frac{g(p)}{2}\sum_{i,j=1}^d\frac{\partial^2f}{\partial x_i\partial x_j}(p)\int_{\mathbb{R}^d}K(-t)t_it_jdλ(t). \end{align*} Our results provide conditions that guarantee that $D_{ε_n,n}f(p)-Δ_Kf(p)$ converges to zero in probability as $n\to\infty$ and can be rescaled by $\sqrt{nε_n^{d+2}}$ to satisfy a central limit theorem. They generalize the work of Giné--Koltchinskii~\cite{gine2006empirical} and Belkin--Niyogi~\cite{belkin2008towards} to allow manifolds with boundary and a wider choice of kernels $K$, and to prove convergence under weaker smoothness assumptions and a correspondingly more precise choice of conditions on the asymptotics of $ε_n$ as $n\to\infty$.