Results 1  10
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15
Determinant maximization with linear matrix inequality constraints
 SIAM Journal on Matrix Analysis and Applications
, 1998
"... constraints ..."
MinimumVolume Enclosing Ellipsoids and Core Sets
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 2005
"... We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly improved ..."
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Cited by 25 (4 self)
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We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly improved complexity bound of O(nd (0, 1). As a byproduct, our algorithm returns a core set with the property that the minimum volume enclosing ellipsoid of provides a good approximation to that of S.
Improved feasible solution algorithms for high breakdown estimation
 Computational Statistics and Data Analysis
, 1999
"... High breakdown estimation allows one to get reasonable estimates of the parameters from a sample of data even if that sample is contaminated by large numbers of awkwardly placed outliers. Two particular application areas in which this is of interest are multiple linear regression, and estimation of ..."
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Cited by 13 (4 self)
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High breakdown estimation allows one to get reasonable estimates of the parameters from a sample of data even if that sample is contaminated by large numbers of awkwardly placed outliers. Two particular application areas in which this is of interest are multiple linear regression, and estimation of the location vector and scatter matrix of multivariate data. Standard high breakdown criteria for the regression problem are the least median of squares (LMS) and least trimmed squares (LTS); those for the multivariate location/scatter problem are the minimum volume ellipsoid (MVE) and minimum covariance determinant (MCD). All of these present daunting computational problems. The ‘feasible solution algorithms ’ for these criteria have been shown to have excellent performance for textbook sized problems, but their performance on much larger data sets is less impressive. This paper points out a computationally cheaper feasibility condition for LTS, MVE and MCD, and shows how the combination of the criteria leads to improved performance on large data sets. Algorithms incorporating these improvements are available from the author’s Web site.
An InteriorPoint Algorithm for the MaximumVolume Ellipsoid Problem
 RICE UNIVERSITY
, 1999
"... In this report, we consider the problem of finding the maximumvolume ellipsoid inscribing a given fulldimensional polytope in R^n defined by a finite set of affine inequalities. We present several formulations for the problem that may serve as algorithmic frameworks for applying interiorpoint met ..."
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Cited by 10 (1 self)
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In this report, we consider the problem of finding the maximumvolume ellipsoid inscribing a given fulldimensional polytope in R^n defined by a finite set of affine inequalities. We present several formulations for the problem that may serve as algorithmic frameworks for applying interiorpoint methods. We propose a practical interiorpoint algorithm based on one of the formulations and present preliminary numerical results.
Exact Primitives for Smallest Enclosing Ellipses
 In Proc. 13th Annu. ACM Symp. on Computational Geometry
, 1997
"... The problem of finding the unique closed ellipsoid of smallest volume enclosing an npoint set P in dspace (known as the LoewnerJohn ellipsoid of P) is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is fixed. The problemspecific parts of these ..."
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Cited by 9 (2 self)
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The problem of finding the unique closed ellipsoid of smallest volume enclosing an npoint set P in dspace (known as the LoewnerJohn ellipsoid of P) is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is fixed. The problemspecific parts of these methods are encapsulated in primitive operations that deal with subproblems of constant size. We derive explicit formulae for the primitive operations of Welzl's randomized method in dimension d=2. Compared to previous ones, these formulae are simpler and faster to evaluate, and they only contain rational expressions, allowing for an exact solution.
Quantile Approximation for Robust Statistical Estimation and kEnclosing Problems
, 2000
"... is concerned with finding the smallest shape of some type that encloses all the points of P . Wellknown instances of this problem include finding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points ..."
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Cited by 6 (1 self)
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is concerned with finding the smallest shape of some type that encloses all the points of P . Wellknown instances of this problem include finding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points in R , find the smallest shape in question that contains at least k points or a certain quantile of the data. This type of problem is known as a kenclosing problem. We present a simple algorithmic framework for computing quantile approximations for the minimum strip, ellipsoid, and annulus containing a given quantile of the points. The algorithms run in O(n log n) time.
Doptimality for minimum volume ellipsoid with outliers
 In Proceedings of the Seventh International Conference on Signal/Image Processing and Pattern Recognition, (UkrOBRAZ’2004
, 2004
"... A family of oneclass classification methods is extended by the determinant maximization novelty detection (DMND) model based on the Doptimum experimental design approach for the ellipsoid estimation. Similar to the oneclass classification methods based on the support vector machine or the socall ..."
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Cited by 5 (3 self)
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A family of oneclass classification methods is extended by the determinant maximization novelty detection (DMND) model based on the Doptimum experimental design approach for the ellipsoid estimation. Similar to the oneclass classification methods based on the support vector machine or the socalled support vector data description (SVDD) approach, DMND is a method that fits a geometrical object around the training data. However, in contrast to SVDD, DMND finds the hyperellipsoid of the smallest volume covering the target objects that can contain outliers by maximizing the determinant of an information matrix. Simulation results are presented for the case when training data are contaminated by compactly located outliers. 1.
Connections Between SemiInfinite and Semidefinite Programming
"... We consider convex optimization problems with linear matrix inequality (LMI) constraints, i.e., constraints of the form F (x) =F0+x1F1+ +xmFm 0; (1.1) where the matrices Fi = F T ..."
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Cited by 5 (2 self)
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We consider convex optimization problems with linear matrix inequality (LMI) constraints, i.e., constraints of the form F (x) =F0+x1F1+ +xmFm 0; (1.1) where the matrices Fi = F T
Smallest Enclosing Ellipses  Fast and Exact
, 1997
"... The problem of finding the smallest enclosing ellipsoid of an npoint set P in dspace is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is fixed. The problemspecific parts of these methods are encapsulated in primitive operations that deal wit ..."
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Cited by 4 (1 self)
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The problem of finding the smallest enclosing ellipsoid of an npoint set P in dspace is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is fixed. The problemspecific parts of these methods are encapsulated in primitive operations that deal with subproblems of constant size. We derive explicit formulae for the primitive operations of Welzl's randomized method [22] in dimension d = 2. Compared to previous ones, these formulae are simpler and faster to evaluate, and they only contain rational expressions, allowing for an exact solution. * supported by the ESPRIT IV LTR Project No. 21957 (CGAL), a preliminary version appeared in Proc. 13th Annu. ACM Symp. on Computational Geometry (SoCG), 1997 [8] y Institut fur Theoretische Informatik, ETH Zurich, Haldeneggsteig 4, CH8092 Zurich, Switzerland, email: gaertner@inf.ethz.ch z Institut fur Informatik, Freie Universitat Berlin, Takustr. 9, D14195 Berlin, Germany, email: sven@inf....
A note on Approximate Minimum Volume Enclosing Ellipsoid of Ellipsoids
"... We study the problem of computing the Minimum Volume Enclosing Ellipsoid (MVEE) containing a given set of ellipsoids S = {E1, E2,..., En} ⊆ Rd. We show how to efficiently compute a small set X ⊆ S of size at most α = X  = O ( d2) whose minimum volume ..."
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We study the problem of computing the Minimum Volume Enclosing Ellipsoid (MVEE) containing a given set of ellipsoids S = {E1, E2,..., En} ⊆ Rd. We show how to efficiently compute a small set X ⊆ S of size at most α = X  = O ( d2) whose minimum volume