Variational Limits of $k$-NN Graph-Based Functionals on Data Clouds

This paper studies the large sample asymptotics of data analysis procedures based on the optimization of functionals defined on $k$-NN graphs on point clouds. This paper is framed in the context of minimization of balanced cut functionals, but our techniques, ideas, and results can be adapted to other functionals of relevance. We rigorously show that provided the number of neighbors in the graph $k:=k_n$ scales with the number of points in the cloud as $n \gg k_n \gg \log(n)$, then with probability one the solution to the graph cut optimization problem converges towards the solution of an analogue variational problem at the continuum level.