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سند علمی

Approximate Minimum Volume Enclosing Ellipsoids Using Core Sets

Piyush Kumar, E. Alper Yıldırım

چکیده

We study the problem of computing the minimum volume enclosing ellipsoid containing a given point set S = {p1, p2, . . . , pn} ⊆ R. Using “core sets” and a column generation approach, we develop a (1 + )-approximation algorithm. We prove the existence of a core set X ⊆ S of size at most |X| = α = O ( d ( log d + 1 )) . We describe an algorithm that computes the set X and a (1 + )-approximation to the minimum volume enclosing ellipsoid of S in O(ndα+ α log α ) operations by using Khachiyan’s algorithm to solve each subproblem. This result is an improvement over the previously known algorithms especially for input sets with n d and reasonably small values of . We also discuss extensions to the cases in which the input set consists of balls or ellipsoids.

اسناد مشابه

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    Abstract. We study the problem of computing a (1 + )-approximation to the minimum volume enclosing ellipsoid of a given point set S = {p1, p2, . . . , pn} ⊆ Rd. Based on a simple, initial volume approximation method, we propose a modification of Khachiyan’s first-order algorithm.

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    We study the problem of computing a (1 + )-approximation to the minimum volume enclosing ellipsoid of a given point set S = {p, p, . . . , p} ⊆ R. Based on a simple, initial volume approximation method, we propose a modification of Khachiyan’s first-order algorithm. Our analysis

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    Abstract For a given multidimensional data set (point cloud), we investigate methods for computing the minimum-volume enclosing ellipsoid (MVEE), which provides an efficient representation of that is useful in many applications, including analysis, optimal design, and computation

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