i.i.d. samples from the probability distribution $d \nu = \rho(x) dx$. We denote by $\nu_n$ the empirical measure associated to the samples ${\mathbf{x}}_1,\dots, {\mathbf{x}}_n$, $$\nu_n := \frac{1}{n}\sum_{i=1}^{n}\delta_{{\mathbf{x}}_i}.
\label{Empirical}$$ We denote by $L^2(\nu_n)$ the space of functions $u_n : {\mathcal{M}}_n \rightarrow {\mathbb{R}}$ and identify $u_n \in L^2(\nu_n)$ with a column vector $(u_n({\mathbf{x}}_1) , \dots, u_n({\mathbf{x}}_n))^T $ in ${\mathbb{R}}^n$.

We associate a weighted graph structure to ${\mathcal{M}}_n$ as follows. Let $\eta: [0,\infty) \rightarrow [0,\infty)$ be a non-increasing Lipschitz function with compact support and let $$\alpha_\eta:= \int_{0}^{\infty}\eta(r)r^{m+1}dr, 
\label{alphaeta}$$ where we recall $m\leq d$ is the intrinsic dimension of the manifold ${\mathcal{M}}$ (typically we think of $m \ll d$, but this is unimportant for the purposes of the results discussed here). We assume that the kernel has been normalized so that $$\int_{{\mathbb{R}}^m}\eta(|x|)dx =1.$$ For a given ${\varepsilon}>0$ we define the weight matrix ${\mathbf{W}}_n$ with entries $({\mathbf{W}}_n)_{ij}$, $$({\mathbf{W}}_n)_{ij} := \frac{2 \eta_{{\varepsilon}}(\lvert {\mathbf{x}}_i-{\mathbf{x}}_j \rvert)}{ \alpha_\eta{\varepsilon}^2 (d_{\varepsilon}({\mathbf{x}}_i))^{1/2} \cdot (d_{\varepsilon}({\mathbf{x}}_j))^{1/2}}, 
\label{KernelizedWeights}$$ measuring how similar points ${\mathbf{x}}_i$ and ${\mathbf{x}}_j$ are. Here, $|\cdot|$ denotes the Euclidean distance in the ambient space ${\mathbb{R}}^d$, $\eta_{\varepsilon}$ is defined as $$\eta_{\varepsilon}(r):= \frac{1}{{\varepsilon}^m}\eta\left(\frac{r}{ {\varepsilon}}\right),$$ and $$d_{\varepsilon}(y):= \sum_{j=1}^{n} \eta_{\varepsilon}(|y- {\mathbf{x}}_j|)\qquad y\in{\mathcal{M}}\,.$$ We notice that the weights in have been appropriately rescaled so that the eigenvalues of 