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In this work we study statistical properties of graph-based clusteringalgorithms that rely on the optimization of balanced graph cuts, the mainexample being the optimization of Cheeger cuts. We consider proximity graphsbuilt from data sampled from an underlying distribution supported on a genericsmooth compact manifold $M$. In this setting, we obtain high probabilityconvergence rates for both the Cheeger constant and the associated Cheeger cutstowards their continuum counterparts. The key technical tools are carefulestimates of interpolation operators which lift empirical Cheeger cuts to thecontinuum, as well as continuum stability estimates for isoperimetric problems.To our knowledge the quantitative estimates obtained here are the first oftheir kind.

From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds