The Normalized Graph Cut and Cheeger Constant: from Discrete to Continuous

Ery Arias-Castro (1) , Bruno Pelletier (2) , Pierre Pudlo (3)

1 MATH - UC San Diego - Department of Mathematics [Univ California San Diego]
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Let M be a bounded domain of a Euclidian space with smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.

Mots clés: Cheeger isoperimetric constant of a manifold; conductance of a graph; neighborhood graph; spectral clustering; U-processes; empirical processes

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Ery Arias-Castro, Bruno Pelletier, Pierre Pudlo. The Normalized Graph Cut and Cheeger Constant: from Discrete to Continuous. Advances in Applied Probability, 2012, 44 (4), pp.907-937. ⟨10.1239/aap/1354716583⟩. ⟨hal-00473264v3⟩