limit Laplacian embedding $F$) which forms the basis of the measure clustering procedure. In particular, we identify conditions on the ground-truth model that ensure that the measure $F_\sharp \nu$ has an orthogonal cone structure. This is the content of our first key result Theorem \[mainTheorem\] which is informally stated below.

\[thm:well-separated-gives-cone-structure-continuum\] If the mixture $\nu$ has well-separated components (see Definition \[def:well-separated-mixture\]), then $F_\sharp\nu$, the push forward of $\nu$ through the Laplacian embedding $F$, has an orthogonal cone structure (see Definition \[def:contcones\]).

Let us now turn our attention to the geometry of the embedded point cloud $F_n( {\mathcal{M}}_n)$. As mentioned above, when constructing the graph $G$ from the data ${\mathcal{M}}_n$, one often scales ${\varepsilon}$ with $n$ in such a way that when $n \rightarrow \infty$, ${\varepsilon}\to 0$ (i.e. the weight matrix ${\mathbf{W}}_n$ localizes) at a slow enough rate. Analyzing this regime is motivated by the desire to balance between graph connectivity and computational efficiency. Recent results in the literature provide explicit rates of convergence for the spectra of graph Laplacians towards the spectrum of an elliptic differential operator (see and Section \[App:A\] below) in such regime. Moreover, the convergence of the eigenvectors is with respect to a strong enough topology, implying 