\rho$. One way to obtain ${\mathbf{x}}$ is to first choose a random number $\bm{k} \in \{1, \dots, N\}$ where $\operatorname{\mathbb{P}}(\bm{k}= k) = w_k$ and then sample ${\mathbf{x}}$ from $\rho_{\bm{k}}$. Conversely given the value ${\mathbf{x}}=x$, we can check that: $$\operatorname{\mathbb{P}}(\bm{k}= k | {\mathbf{x}}=x) = \frac{w_k \rho_k(x)}{\rho(x)}.$$ In that order of ideas, $q_k(x)$ is simply the square root of the likelihood of ${\mathbf{x}}$ being sampled from $\rho_k$.

Geometric structure of Laplacian embeddings
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### The continuum setting {#sec:continuum}

In what follows we describe the geometry of the ground-truth spectral embedding (i.e. the measure clustering embedding in the continuum). First, let $u_1, \dots, u_N$ be the first $N$ orthonormal (with respect to $\langle \cdot  , \cdot \rangle_{\rho}$) eigenfunctions of $\Delta_\rho$ corresponding to its $N$ smallest eigenvalues. Define the Laplacian embedding $$F: x \in {\mathcal{M}}\longmapsto \left( \begin{matrix} u_{1}(x) \\ \vdots \\ u_{N}(x)  \end{matrix} \right)
  \in {\mathbb{R}}^N. 
\label{ContEmbedding}$$

The push-forward of the measure $\nu$ by $F$, denoted $\mu:= F_{\sharp } \nu $, is a measure on ${\mathbb{R}}^N$ which describes the distribution of points originally in ${\mathcal{M}}$ after transformation by the Laplacian map $F$. Our first main result asserts that if the mixture model is well-separated, then the measure $\mu$ has an orthogonal cone structure; this 