Approximate Minimum Volume Enclosing Ellipsoids Using Core Sets
نویسندگان
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.
منابع مشابه
On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids
Given A := {a1, . . . , am} ⊂ Rd whose affine hull is Rd, we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan’s barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two...
متن کاملMinimum Volume Enclosing Ellipsoids and Core Sets
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. Our analysis leads to a slightly improved complexity bound of O(nd3/ ) operations for ∈ (0, 1). As a byproduct, our ...
متن کاملComputing Minimum-Volume Enclosing Axis-Aligned Ellipsoids
Given a set of points S = {x1, . . . , xm} ⊂ R and > 0, we propose and analyze an algorithm for the problem of computing a (1 + )-approximation to the minimum-volume axis-aligned ellipsoid enclosing S . We establish that our algorithm is polynomial for fixed . In addition, the algorithm returns a small core set X ⊆ S , whose size is independent of the number of points m, with the property that ...
متن کاملMinimum-Volume Enclosing Ellipsoids and Core Sets1
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 leads to a slightly improved complexity bound of O(nd/ ) operations for ∈ (0, 1). As a byproduct, our algorithm retur...
متن کاملOn the Minimum Volume Covering Ellipsoid of Ellipsoids
Let S denote the convex hull of m full-dimensional ellipsoids in Rn. Given > 0 and δ > 0, we study the problems of computing a (1 + )-approximation to the minimum volume covering ellipsoid of S and a (1 + δ)n-rounding of S. We extend the first-order algorithm of Kumar and Yıldırım that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in Rn, which, in ...
متن کامل