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Asymptotic normality in density support estimation
 Electron. J. Probab
"... c i E l e c t r o n J o u r n a l o f ..."
The Normalized Graph Cut and Cheeger Constant: from Discrete to Continuous
, 2012
"... Let M be a bounded domain of W ' with a smooth boundary. We relate the Cheeger eonstant of M and the eonductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normal ..."
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Let M be a bounded domain of W ' with a smooth boundary. We relate the Cheeger eonstant of M and the eonductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.
On nonparametric estimation of boundary measures
"... Abstract: The measure of the boundary ∂G of a compact body G ⊂ [0, 1]d can be expressed in terms of the Minkowski content defined by L0(G) = lim →0 µ (B(∂G, )) ..."
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Abstract: The measure of the boundary ∂G of a compact body G ⊂ [0, 1]d can be expressed in terms of the Minkowski content defined by L0(G) = lim →0 µ (B(∂G, ))
Testing Surface Area
, 2013
"... We consider the problem of estimating the surface area of an unknown ndimensional set F given membership oracle access. In contrast to previous work, we do not assume that F is convex, and in fact make no assumptions at all about F. By necessity this means that we work in the property testing model ..."
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We consider the problem of estimating the surface area of an unknown ndimensional set F given membership oracle access. In contrast to previous work, we do not assume that F is convex, and in fact make no assumptions at all about F. By necessity this means that we work in the property testing model; we seek an algorithm which, given parameters A and ɛ, satisfies: • if surf(F) ≤ A then the algorithm accepts (whp); • if F is not ɛclose to some set G with surf(G) ≤ κA, then the algorithm rejects (whp). We call κ ≥ 1 the “approximation factor ” of the testing algorithm. The n = 1 case (in which “surf(F) = 2m ” means F is a disjoint union of m intervals) was introduced by Kearns and Ron [KR98], who solved the problem with κ = 1/ɛ and O(1/ɛ) oracle queries. Later, Balcan et al. [BBBY12] solved it with with κ = 1 and O(1/ɛ4) queries. We give the first result for higher dimensions n. Perhaps surprisingly, our algorithm completely evades the “curse of dimensionality”: for any n and any κ> 4 π ≈ 1.27 we give a test that uses O(1/ɛ)
Asymptotic Normality in Density Support Estimation
"... Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as the union of balls centered at the Xi and with common radius rn. Using tools from Riemannian geometry, and u ..."
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Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as the union of balls centered at the Xi and with common radius rn. Using tools from Riemannian geometry, and under mild assumptions on f and the sequence (rn), we prove a central limit theorem for λ(Sn∆Sf), where λ denotes the Lebesgue measure on R d and ∆ the symmetric difference operation.
A Weighted kNearest Neighbor Density Estimate for Geometric Inference
"... Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the knearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the light ..."
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Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the knearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong approximation result is obtained and the choice of the optimal set of weights is discussed. In particular, the classical knearest neighbor estimate is not optimal in a sense described in the manuscript. The proposed method has been implemented to recover level sets in both simulated and reallife data. Index Terms — Geometric inference, level sets, density estimation, knearest neighbor estimate, weighted estimate, consistency, rates of convergence, central limit theorem, strong approximation.
© Institute of Mathematical Statistics, 2011 NONPARAMETRIC ESTIMATION OF SURFACE INTEGRALS
"... The estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics, with powerful applications to physics and image analysis, among other fields. Provided that one can determine whether random shots hit the body, Cuevas et al. [Ann. Statist. ..."
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The estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics, with powerful applications to physics and image analysis, among other fields. Provided that one can determine whether random shots hit the body, Cuevas et al. [Ann. Statist. 35 (2007) 1031–1051] estimate the boundary measure (the boundary length for planar sets and the surface area for 3dimensional objects) via the consideration of shots at a box containing the body. The statistics considered by these authors, as well as those in subsequent papers, are based on the estimation of Minkowski content and depend on a smoothing parameter which must be carefully chosen. For the same sampling scheme, we introduce a new approach which bypasses this issue, providing strongly consistent estimators of both the boundary measure and the surface integrals of scalar functions, provided one can collect the function values at the sample points. Examples arise in experiments in which the density of the body can be measured by physical properties of the impacts, or in situations where such quantities as temperature and humidity are observed by randomly distributed sensors. Our method is based on random Delaunay triangulations and involves a simple procedure for surface reconstruction from a dense cloud of points inside and outside the body. We obtain basic asymptotics of the estimator, perform simulations and discuss, via Google Earth’s data, an application to the image analysis of the Aral Sea coast and its cliffs. 1. Introduction. The