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Learning minimum volume sets
 J. Machine Learning Res
, 2006
"... Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence region ..."
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Cited by 41 (9 self)
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Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure µ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency and an oracle inequality. Estimators based on histograms and dyadic partitions illustrate the proposed rules. 1
Minimax optimal level set estimation
 in Proc. SPIE, Wavelets XI, 31 July  4
, 2005
"... Abstract — This paper describes a new methodology and associated theoretical analysis for rapid and accurate extraction of level sets of a multivariate function from noisy data. The identification of the boundaries of such sets is an important theoretical problem with applications for digital elevat ..."
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Cited by 21 (5 self)
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Abstract — This paper describes a new methodology and associated theoretical analysis for rapid and accurate extraction of level sets of a multivariate function from noisy data. The identification of the boundaries of such sets is an important theoretical problem with applications for digital elevation maps, medical imaging, and pattern recognition. This problem is significantly different from classical segmentation because level set boundaries may not correspond to singularities or edges in the underlying function; as a result, segmentation methods which rely upon detecting boundaries would be potentially ineffective in this regime. This issue is addressed in this paper through a novel error metric sensitive to both the error in the location of the level set estimate and the deviation of the function from the critical level. Hoeffding’s inequality is used to derive a novel regularization
Exact Rates in Density Support Estimation
"... Let f be an unknown multivariate probability density with compact support Sf. Given n independent observations X1,...,Xn drawn from f, this paper is devoted to the study of the estimator Ŝn of Sf defined as unions of balls centered at the Xi and of common radius rn. measure the proximity between S ..."
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Cited by 17 (1 self)
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Let f be an unknown multivariate probability density with compact support Sf. Given n independent observations X1,...,Xn drawn from f, this paper is devoted to the study of the estimator Ŝn of Sf defined as unions of balls centered at the Xi and of common radius rn. measure the proximity between Ŝn and Sf, we employ a general criterion dg, based on some function g, which encompasses many statistical situations of interest. Under mild assumptions on the sequence (rn) and some analytic conditions on f and g, the exact rates of convergence of dg(Ŝn, Sf) are obtained using tools from Riemannian geometry. The conditions on the radius sequence are found to be sharp and consequences of the results are discussed from a statistical perspective.
Adaptation to lowest density regions with application to support recovery
, 2014
"... Adaptation to lowest density regions with ..."
Estimation of the support of the density and its boundary using Random Polyhedron
, 2013
"... WeconsiderrandomsamplesinR d drawnfromanunknowndensity. When the support is assumed to be convex and with sharp boundary, the convex hull is an estimator of the support that converges to S with a rate of n −2/(d+1). When the boundary of the support is sharp but the support is no longer assumed to be ..."
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Cited by 1 (0 self)
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WeconsiderrandomsamplesinR d drawnfromanunknowndensity. When the support is assumed to be convex and with sharp boundary, the convex hull is an estimator of the support that converges to S with a rate of n −2/(d+1). When the boundary of the support is sharp but the support is no longer assumed to be convex, the usual support estimators convergeswith a rate of n −1/d or (ln(n)/n) −1/d. This paper is devoted to presenting some new estimators of the support of the density, which are based on some local convexity criteria and converge to S with a rate of (n/lnn) −2/(d+1) (and their boundary converges toward ∂S with the same rate) when the support is assumed to have a sharp C 2 boundary. The convergence rate is also given when the sharpness hypothesis is relaxed (and it is close to the optimal rate when the dimension is two). key words: Delaunay complex, polyhedron, support estimation, topological data analysis, geometric inference.
support and
, 2012
"... Using the k−nearest neighbor restricted Delaunay polyhedron to estimate the density ..."
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Using the k−nearest neighbor restricted Delaunay polyhedron to estimate the density
Local Convex Hull support and boundary estimation
, 2014
"... In this paper we study a new estimator for the support of a multivariate density. It is defined as a union of convexhulls of observations contained in small balls. We study the asymptotic behavior of this “local convex hull ” as an estimator of the support and the asymptotic behaviors of its boun ..."
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In this paper we study a new estimator for the support of a multivariate density. It is defined as a union of convexhulls of observations contained in small balls. We study the asymptotic behavior of this “local convex hull ” as an estimator of the support and the asymptotic behaviors of its boundary as an estimator of the boundary of the support. We analyze as well its ”topologypreserving ” properties.
Journal of Machine Learning Research x (2006) xx Submitted 9/05; Published xx/xx Learning Minimum Volume Sets
"... Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence region ..."
Abstract
 Add to MetaCart
Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure µ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency, an oracle inequality, and rates of convergence. The proposed estimators are illustrated with histogram and decision tree set estimation rules.