Estimating perimeter using graph cuts

TL;DR

This paper proposes a method to estimate the perimeter of a set within a domain using graph cuts on a random geometric graph, providing optimal bias and variance estimates in different scaling regimes and dimensions.

Contribution

It introduces a new perimeter estimation technique based on graph cuts, with rigorous bias and variance analysis across dense and sparse regimes and various dimensions.

Findings

Optimal bias and variance estimates for perimeter approximation.

A phase transition at dimension d=5 affecting confidence interval feasibility.

Effective perimeter testing in higher dimensions with error bounds.

Abstract

We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For $\Omega \subset D = (0,1)^d$, with $d \geq 2$, we are given $n$ random i.i.d. points on $D$ whose membership in $\Omega$ is known. We consider the sample as a random geometric graph with connection distance $\varepsilon>0$. We estimate the perimeter of $\Omega$ (relative to $D$) by the, appropriately rescaled, graph cut between the vertices in $\Omega$ and the vertices in $D \backslash \Omega$. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to $n$ and $\varepsilon$. We consider two scaling regimes: the dense (when the average degree of the vertices goes to $\infty$) and the sparse one (when the degree goes to $0$). In the dense regime there is a crossover in the nature of approximation at dimension $d=5$: we show that in low dimensions $d=2,3,4$ one can obtain confidence intervals for the approximation error, while in higher dimensions one can only obtain error estimates for testing the hypothesis that the perimeter is less than a given number.