$$\langle u, v \rangle_{\varrho} := \int_{\mathcal{M}}u(x) v(x) \varrho(x) dx.$$ Throughout the article we routinely write $L^2(\varrho)$ instead of $L^2({\mathcal{M}}, \varrho)$ when the domain of the function space is clear from the context. In section \[App:A\] we modify our notation slightly and use $L^2(\mu)$ to denote $L^2({\mathcal{M}}, \varrho)$ given the measure $\mu = \varrho dx$ and use $\langle\cdot, \cdot\rangle_{L^2(\mu)}$ to denote $\langle \cdot, \cdot \rangle_\rho$. The reason for choosing the specific form for the operator $\Delta_\rho$ is justified by its close connection with the kernelized graph Laplacian that we will introduce and discuss in Sections \[sec:fromconttodiscr\] and \[sec:differentweights\] below.

We will make the following assumption.

\[Assump1\] We assume that the mixture model $\nu$ satisfies the following: $\rho_1, \dots, \rho_N
\in C^1({\mathcal{M}})$ with $\int_{\mathcal{M}}\rho_k(x)\,dx=1$ for all $k$ are such that the operators $\Delta_{\rho_k}$ and $\Delta_{\rho}$ have a discrete point spectrum with an associated orthonormal basis of eigenfunctions for $L^2(\rho_k)$ and $L^2(\rho)$ respectively.

Well-separated mixture models {#sec:wellsepa}
-----------------------------

Let us now introduce the notion of a well-separated mixture model. This is a notion defined in terms of three quantities associated to the model $\nu$ that we refer to as *overlapping*, *coupling*, and *indivisibility* parameters.

For $k =1, \dots, N$, define the functions $$\label{qk-definition}
 