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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.05633 |
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| _version_ | 1866912264682995712 |
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| author | Akwei, Bernard Rogers, Luke Teplyaev, Alexander |
| author_facet | Akwei, Bernard Rogers, Luke Teplyaev, Alexander |
| contents | 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 $n\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distributional Convergence of the Empirical Laplacians with Integral Kernels on Domains with Boundaries Akwei, Bernard Rogers, Luke Teplyaev, Alexander Functional Analysis Probability 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 $n\to\infty$. |
| title | Distributional Convergence of the Empirical Laplacians with Integral Kernels on Domains with Boundaries |
| topic | Functional Analysis Probability |
| url | https://arxiv.org/abs/2503.05633 |