Optimal Cheeger cuts and bisections of random geometric graphs

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Abstract

Let \(d \geq 2\). The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on \(n\) random points in a \(d\)-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of \(n\)) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large \(n\) to an analogous Cheeger-type constant of the domain. Previously, Garc\'ia Trillos {\em et al.} had shown this for \(d \geq 3\) but had required an extra condition on the distance parameter when \(d=2\).

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A Tutorial on Spectral Clustering

Ulrike von Luxburg (2007)

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.