Estimating perimeter using graph cuts

Abstract

AbstractWe investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For Ω ⊆D= (0, 1)dwithd≥ 2, we are givennrandom independent and identically distributed points onDwhose membership in Ω is known. We consider the sample as a random geometric graph with connection distance ε > 0. We estimate the perimeter of Ω (relative toD) by the, appropriately rescaled, graph cut between the vertices in Ω and the vertices inD∖ Ω. We obtain bias and variance estimates on the error, which are optimal in scaling with respect tonand ε. We consider two scaling regimes: the dense (when the average degree of the vertices goes to ∞) and the sparse one (when the degree goes to 0). In the dense regime, there is a crossover in the nature of the approximation at dimensiond= 5: we show that in low dimensionsd= 2, 3, 4 one can obtain confidence intervals for the approximation error, while in higher dimensions one can obtain only error estimates for testing the hypothesis that the perimeter is less than a given number.