the graph Laplacian $\Delta_n$ defined below approximate those of $\Delta_\rho$. We would also like to notice that $({\mathbf{W}}_{n})_{ij}$ can be written as $$({\mathbf{W}}_n)_{ij}=\frac{2 \eta_{{\varepsilon}}(\lvert {\mathbf{x}}_i-{\mathbf{x}}_j \rvert)}{  \alpha_\eta n {\varepsilon}^2 (d_{\varepsilon}({\mathbf{x}}_i)/n)^{1/2} \cdot (d_{\varepsilon}({\mathbf{x}}_j)/n)^{1/2}}     ,$$ and that $d_{\varepsilon}(\cdot) /n$ is nothing but a kernel density estimator for the density $\rho$ due to the fact that $\eta$ has been assumed to be normalized.

The graph Laplacian operator $\Delta_n: L^2(\nu_n)\rightarrow L^2(\nu_n)$ is then defined as the matrix $$\Delta_n := {\mathbf{D}}_n- {\mathbf{W}}_n,$$ where ${\mathbf{D}}_n$ is the degree matrix associated to the weight matrix ${\mathbf{W}}_n$, that is, ${\mathbf{D}}_n$ is a diagonal matrix whose diagonal entries $({\mathbf{D}}_n)_{ii}$ are given by $({\mathbf{D}}_n)_{ii}= \sum_{j=1}^{n}({\mathbf{W}}_n)_{ij}$. Notice that here we have suppressed the dependence of $\Delta_n$ on ${\varepsilon}$ for notational convenience.

The spectrum of $\Delta_n$ induces a graph Laplacian embedding $$F_n : {\mathcal{M}}_n \rightarrow {\mathbb{R}}^N$$ $$F_n: {\mathbf{x}}_i \longmapsto \left( \begin{matrix} u_{n,1}({\mathbf{x}}_i) \\ \vdots \\ u_{n,N}({\mathbf{x}}_i)  \end{matrix} \right),
\label{DiscreteEmbedding}$$ where $u_{n,1}, \dots, u_{n,N}$ are the first $N$ eigenvectors of $\Delta_n$.

The normalization terms appearing in induce the *kernelized* graph Laplacian $\Delta_n$. There are other possible normalizations that one can consider such as the ones in the parametric family introduced in [@Coifman1]. The effect of the different normalizations can be observed asymptotically (as 