Variational limits of k-NN graph based functionals on data clouds
This work provides theoretical foundations for graph-based methods in machine learning, addressing convergence issues for researchers in statistical learning and data analysis.
The paper tackles the asymptotic behavior of data analysis procedures using k-NN graphs, specifically for balanced cut functionals, and proves that under certain scaling conditions, the graph-based solutions converge to continuum-level variational problems with probability one.
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. The 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.