Estimating perimeter using graph cuts

Abstract

We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For Ω ⊂ D = (0, 1) d , with d ≥ 2, we are given n random i.i.d. points on D whose 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 to D) by the, appropriately rescaled, graph cut between the vertices in Ω and the vertices in D\Ω. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to n and 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 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.Date: September 12, 2018.

A variational approach to the consistency of spectral clustering

“…The notion of Γ-convergence is recalled in Subsection 2.2. This notion of convergence is particularly suitable in order to study the convergence of minimizers of objective functionals on graphs as n → ∞, as it is discussed in [17]. The relevant continuum energy is the weighted Dirichlet energy G : …”

“…Recently the authors in [15], and together with Laurent, von Brecht and Bresson in [17], introduced a framework for showing the consistency of clustering algorithms based on minimizing an objective functional on graphs. In [17] they applied the technique to Cheeger and Ratio cuts.…”

“…In [17] they applied the technique to Cheeger and Ratio cuts. Here the framework of [15,17] is used to prove new results on consistency of spectral clustering, which establish the (almost) optimal rate at which the connectivity radius ε can be taken to 0 as n → ∞. We prove the convergence of the spectrum of the graph Laplacian towards the spectrum of a corresponding continuum operator.…”

Optimal Cheeger cuts and bisections of random geometric graphs

“…The almost sure convergence in (4.13) was proved in [18] for the case where ρ is constant on D and φ = φ U . In Remark 1.10 of [18] it is stated that the proof carries through to more general ρ and to all weight functions φ satisfying (2.6)-(2.8).…”

“…A significant recent theme in topological/geometrical data analysis and machine learning is the reconstruction of topological/geometrical properties of a continuous space such as a manifold from a random sample of points in that space via a graph, or more generally a simplicial complex, derived from the sample by connecting nearby points; see for example [8,12,15,18,21]. A prototypical graph of this type is the random geometric graph, where one connects every pair of points up to a specified distance r apart (we shall consider generalization of this to allow for weighted graphs).…”