[@Rohe]. In the geometric graph setting (our setting) [@vonLuxburg] was the first to rigorously establish a statistical consistency result for graph-based spectral embeddings when the number of data points $n$ goes to infinity and the connectivity ${\varepsilon}>0$ remains constant. Up until then most papers focused on proving pointwise convergence estimates (see [@bel_niy_LB; @HeinvonLuxburgAudibert] and references within). The work [@vonLuxburg] however, did not address the question of determining scalings for ${\varepsilon}:={\varepsilon}(n) \rightarrow 0$ that implied spectral convergence towards differential operators in the continuum limit. Moreover, it was still a pending task to analyze the entire spectral clustering algorithm, where in addition to the embedding step, there is an extra clustering step (typically $k$-means).

In [@GarciaTrillosSlepcev2015], the authors introduced a framework using notions from optimal transport and the calculus of variations to study large data limits of optimization problems defined on random geometric graphs. Many large data limits of relevant functionals for machine learning have been studied using similar approaches [@SlepcevThorpe; @StuartSlepcevThorpeDunlop; @GarciaMurray; @GarciaSanzAlonso]. In particular, the optimal transport framework was used in [@GarciaTrillosSlepcev2015] to analyze the entire spectral clustering procedure (embedding step and $k$-means step) and the authors were able to show statistical consistency for scalings $ \log(n)^{p_m}/n^{1/m} \ll {\varepsilon}\ll 1$ 