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

Part of: Nonparametric inference

Published online by Cambridge University Press:  04 January 2016

Ery Arias-Castro ,
Bruno Pelletier  and
Pierre Pudlo
Ery Arias-Castro*
Affiliation:
University of California
Bruno Pelletier*
Affiliation:
Université Rennes II
Pierre Pudlo*
Affiliation:
Université Montpellier II
*
∗ Postal address: Department of Mathematics, University of California, San Diego, USA.
∗∗ Postal address: Département de Mathématiques, IRMAR – UMR CNRS 6625, Université Rennes II, France. Email address: [email protected]
∗∗∗ Postal address: Département de Mathématiques, I3M – UMR CNRS 5149, Université Montpellier II, France.
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Abstract

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Let M be a bounded domain of with a 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.

Keywords

Cheeger isoperimetric constant of a manifoldconductance of a graph; neighborhood graphspectral clusteringU-processempirical process

MSC classification

Primary: 62G05: Estimation 62G20: Asymptotic properties
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 907 - 937
Copyright
© Applied Probability Trust 

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