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

Quantitative stability for minimizing Yamabe metrics

“…In a slightly different direction, quantitative stability estimates for critical points have been addressed for the isoperimetric inequality on Euclidean space [23,38] and for the Sobolev inequality [22,28]. Quantitative stability estimates have wideranging applications to contexts including characterization of minimizers in variational problems [21], rates of convergence of PDE [16], regularity of interfaces in free boundary problems [2], and even data science [35]. Apart from [18], all of these results make crucial use of the explicit form of minimizers and critical points or of the symmetries of the ambient space.…”

“…It is clear that F (0) = 0 since N (0) = 0. To see that ∇F (0) = 0, we appeal to (38) with ϕ = 0 and see that it suffices to show that π K ⊥ ∇N −1 (0)[η] = 0 for any η ∈ K. And indeed, by (35), we see that ∇ B N (v) maps K to K and that ∇ B N (v)| K = (∇N −1 (0)) −1 | K = Id. Thus ∇F (0) = 0.…”

Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning

“…An interesting consequence of our Gamma-convergence result is the convergence of adversarial training (1.1) as ε → 0 to a solution of the problem with ε = 0 with minimal perimeter. Furthermore, the primary result in the continuum setting can be used to recover a Gamma-convergence result for graph discretizations of the nonlocal perimeter; a setting that is especially relevant in the context of graph-based machine learning [33,34,36]. Our approach for this discrete to continuum convergence is in the spirit of these works and relies on T L p (transport L p ) spaces which were introduced for the purpose of proving Gammaconvergence of a graph total variation by García Trillos and Slepčev in [34], but have also been used to prove quantitative convergence statements for graph problems, see, e.g., the works [14,15] by Calder et al…”

Rates of convergence for regression with the graph poly-Laplacian

“…This has motivated theoretical works like [ 67 ] which study the convergence of graph total variation to a continuum weighted total variation (the same paper proposed a topology to study the convergence that didn’t require regularity—in particular pointwise evaluation—of the continuum function). Total variation functionals are also widely used for clustering and segmentation such as in graph cut methods, for example ratio or Cheeger cuts [ 68 , 69 ], graph modularity clustering [ 70 , 71 ], and Ginzburg–Landau segmentation [ 72 – 74 ].…”