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

TL;DR

This paper explores the relationship between the Cheeger constant of a domain and the conductance of neighborhood graphs built from random samples, establishing consistency and convergence results as sample size grows.

Contribution

It introduces a normalized graph cut approach linking discrete graph properties to continuous Cheeger sets, with proofs of consistency and convergence.

Findings

Normalized conductance converges to the Cheeger constant as sample size increases.

Minimizing subsets of the graph converge to Cheeger sets of the domain.

Provides theoretical foundations for graph-based clustering methods.

Abstract

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.