constraints

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 first-order 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.

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 text-book 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.

In this report, we consider the problem of finding the maximumvolume ellipsoid inscribing a given full-dimensional polytope in ! n defined by a finite set of affine inequalities. We present several formulations for the problem that may serve as algorithmic frameworks for applying interior-point methods. We propose a practical interior-point algorithm based on one of the formulations and present preliminary numerical results. 1

The problem of finding the unique closed ellipsoid of smallest volume enclosing an n-point set P in d-space (known as the Loewner-John 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 problem-specific 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.

is concerned with finding the smallest shape of some type that encloses all the points of P . Well-known 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 k-enclosing 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.

The problem of finding the smallest enclosing ellipsoid of an n-point set P in d-space is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is fixed. The problem-specific 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, CH-8092 Zurich, Switzerland, e-mail: gaertner@inf.ethz.ch z Institut fur Informatik, Freie Universitat Berlin, Takustr. 9, D-14195 Berlin, Germany, e-mail: sven@inf....

A family of one-class classification methods is extended by the determinant maximization novelty detection (DMND) model based on the D-optimum experimental design approach for the ellipsoid estimation. Similar to the one-class classification methods based on the support vector machine or the so-called 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.

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

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