Optimal Cheeger cuts and bisections of random geometric graphs

Abstract

Let d ≥ 2. The Cheeger constant of a graph is the minimum surface-tovolume 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, García Trillos et al. had shown this for d ≥ 3 but had required an extra condition on the distance parameter when d = 2.

Mumford-Shah functionals on graphs and their asymptotics

“…In particular the approach outlined at the beginning of Subsection 4.2 would allow one to obtain optimal estimates for total variation, Laplacian, andf p-Laplacian functionals considered in [GTS16], [GTS18] and [ST19] respectively. We note that for the graph total variation optimal estimates in 2D were recently obtained by Müller and Penrose [MP18].…”

Mumford–Shah functionals on graphs and their asymptotics

“…In particular the approach outlined at the beginning of section 4.2 would allow one to obtain optimal estimates for total variation, Laplacian, and p-Laplacian functionals considered in [GTS16, GTS18,ST19] respectively. We note that for the graph total variation optimal estimates in 2D were recently obtained by Müller and Penrose [MP18]. The approach here is simpler, but does use the insight of Müller and Penrose that binning at an intermediate scale can be advantageous.…”