structure provided the overlapping parameter of the mixture model $\mathcal{S}$ is small enough (see Proposition \[prop:muQcone\]). In Section \[sec:Conic\] we show that orthogonal cone structures are stable under Wasserstein perturbations. Theorem \[mainTheorem\] is then proved by showing that the measures $F^Q_{\sharp} \nu$ and $F_{\sharp} \nu$ are close to each other in the Wasserstein sense modulo an orthogonal transformation. To achieve this we will essentially show that $q_1, \dots, q_N$ are close to spanning the subspace generated by the first $N$ eigenfunctions of $\Delta_\rho$. This statement is trivially true in the extreme case when ${\mathcal{M}}$ has $N$ connected components and the $\rho_k dx$ are the uniform distributions of the connected components. Indeed, in that case, the first $N$ eigenvalues of $\Delta_\rho$ are zero, and the first $N$ eigenfunctions of $\Delta_\rho$ are rescaled versions of the indicator functions of the components. In other words, the span of the $\rho_k$ is exactly equal to the span of the first $N$ eigenfunctions of $\Delta_\rho$, and in particular up to a rotation we have $F^Q=F  $.

### The discrete setting {#sec:fromconttodiscr}

Having discussed spectral clustering at the continuum level, we now move to the discrete setting. Let ${\mathcal{M}}_n := \left\{ {\mathbf{x}}_1, \dots, {\mathbf{x}}_n \right\}$ be 