Variational limits of k-NN graph based functionals on data clouds

We consider i.i.d. samples $x_1, \dots, x_n$ from a measure $\nu$ with density supported on a bounded Euclidean domain $D \subseteq R^d$ where $d\geq3$. A graph on the point cloud is obtained by connecting two points if one of them is among the $k$-nearest neighbors of the other. Our goal is to study consistency of graph based procedures to clustering, classification and dimensionality reduction by studying the variational convergence of the graph total variation associated to such $k$-NN graph. We prove that provided $k:=k_n$ scales like $n \gg k_n \gg \log(n)$, then the $\Gamma$-convergence of the graph total variation towards an appropriate weighted total variation is guaranteed.