From Graph Cuts to Isoperimetric Inequalities: Convergence Rates of Cheeger Cuts on Data Clouds

Abstract In this work we study statistical properties of graph-based clustering algorithms that rely on the optimization balanced graph cuts, main example being Cheeger cuts. We consider proximity graphs built from data sampled an underlying distribution supported a generic smooth compact manifold $${\mathcal {M}}$$ M . setting, obtain high probability convergence rates for both constant and associated cuts towards their continuum counterparts. The key technical tools are careful estimates interpolation operators which lift empirical to continuum, as well stability isoperimetric problems. To best our knowledge quantitative obtained here first kind.