where $m$ is the intrinsic dimension of the manifold ${\mathcal{M}}$ and $p_m >0$ is an appropriate constant depending on $m$. Nevertheless, the spectral convergence was asymptotic and no rate was provided until [@Shi2015]. In [@BIK], the authors took a somewhat similar approach to [@GTSSpectralClustering], and in a non-probabilistic setting, conducted a careful analysis which ultimately allowed to provide very precise rates of convergence for the spectrum of graph Laplacians. The work [@Moritz] extended this analysis to cover the relevant probabilistic setting where point clouds are samples from a ground truth measure supported on a manifold. The rates in [@Moritz] are, to the best of our knowledge, state of the art. Many of the results and ideas introduced in [@Moritz] will be revisited in Section \[App:A\]; as a matter of fact, the probabilistic estimates that we need for this paper are proved in a very similar fashion to [@Moritz] after making some slight modifications that we will discuss later on.

All of the publications listed above focus on spectral convergence of graph Laplacians, but do not study the geometry of Laplacian embeddings themselves (i.e. what is the geometry of the embedded data set, and how such geometry may affect the subsequent 