parts. First, we study the geometry of a continuum analogue Laplacian embedding using techniques from PDEs and the calculus of variations. In the second part, we compare the two embeddings, discrete and continuum. For the second part we use recent results quantifying the approximation error of the spectra of continuum Laplacian operators with the spectra of graph Laplacians. It is only in the second step that we study statistical properties of random objects, and the first part of the paper is, to some extent, of independent interest.

Overall, our ideas, definitions, and results provide new theoretical insights on spectral clustering. They further justify the use of spectral clustering beyond the trivial setting where a manifold is disconnected. In particular, we show that when the ground-truth measure $\nu$ has well-separated components, the geometry of the embedded data set $F_n({\mathcal{M}}_n)$ is simple enough so that an algorithm like $k$-means is successful in correctly identifying the coarse structure of the data set.

Overview of main results
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Let us make our setup more precise. We assume that ${\mathcal{M}}$ is a connected manifold and that $\nu$ is a mixture model with $N$ components supported on ${\mathcal{M}}$ (see Section \[Set-up\] for detailed assumptions on this 