Browsing by Subject "Core sets"

Now showing 1 - 6 of 6

Open Access
An algorithm and a core set result for the weighted euclidean one-center problem
(Institute for Operations Research and the Management Sciences (I N F O R M S), 2009) Kumar, P.; Yıldırım, A. E.
Given a set A of m points in n-dimensional space with corresponding positive weights, the weighted Euclidean one-center problem, which is a generalization of the minimum enclosing ball problem, involves the computation of a point c A n that minimizes the maximum weighted Euclidean distance from c A to each point in A In this paper, given ε > 0, we propose and analyze an algorithm that computes a (1 + ε)-approximate solution to the weighted Euclidean one-center problem. Our algorithm explicitly constructs a small subset X ⊆ A, called an ε-core set of A, for which the optimal solution of the corresponding weighted Euclidean one-center problem is a close approximation to that of A. In addition, we establish that \X\ depends only on ε and on the ratio of the smallest and largest weights, but is independent of the number of points m and the dimension n. This result subsumes and generalizes the previously known core set results for the minimum enclosing ball problem. Our algorithm computes a (1 + ε)-approximate solution to the weighted Euclidean one-center problem for A in O(mn\X\) arithmetic operations. Our computational results indicate that the size of the ε-core set computed by the algorithm is, in general, significantly smaller than the theoretical worst-case estimate, which contributes to the efficiency of the algorithm, especially for large-scale instances. We shed some light on the possible reasons for this discrepancy between the theoretical estimate and the practical performance.
Open Access
Computing minimum-volume enclosing axis-aligned ellipsoids
(Springer, 2008) Kumar, P.; Yıldırım, E. A.
Given a set of points S = {x1 ,..., xm}⊂ ℝn and ε>0, we propose and analyze an algorithm for the problem of computing a (1+ε)-approximation to the minimum-volume axis-aligned ellipsoid enclosing S. We establish that our algorithm is polynomial for fixed ε. 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 the minimum-volume axis-aligned ellipsoid enclosing X is a good approximation of the minimum-volume axis-aligned ellipsoid enclosing S. Our computational results indicate that the algorithm exhibits significantly better performance than the theoretical worst-case complexity estimate.
Open Access
A linearly convergent linear-time first-order algorithm for support vector classification with a core set result
(Institute for Operations Research and the Management Sciences (I N F O R M S), 2011) Kumar, P.; Yıldırım, E. A.
We present a simple first-order approximation algorithm for the support vector classification problem. Given a pair of linearly separable data sets and. ε (0,1), the proposed algorithm computes a separating hyperplane whose margin is within a factor of (1-ε) of that of the maximum-margin separating hyperplane. We discuss how our algorithm can be extended to nonlinearly separable and inseparable data sets. The running time of our algorithm is linear in the number of data points and in 1/ε. In particular, the number of support vectors computed by the algorithm is bounded above by O(ζ/ε. for all sufficiently small ε >, where ζ is the square of the ratio of the distances between the farthest and closest pairs of points in the two data sets. Furthermore, we establish that our algorithm exhibits linear convergence. Our computational experiments, presented in the online supplement, reveal that the proposed algorithm performs quite well on standard data sets in comparison with other first-order algorithms. We adopt the real number model of computation in our analysis.
Open Access
On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids
(Elsevier, 2007) Todd, M. J.; Yıldırım, E. A.
Given A {colon equals} { 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-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yi{dotless}ldi{dotless}ri{dotless}m, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yi{dotless}ldi{dotless}ri{dotless}m or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.
Open Access
On the minimum volume covering ellipsoid of ellipsoids
(Society for Industrial and Applied Mathematics, 2006) YIldırım, E. A.
Let S denote the convex hull of m full-dimensional ellipsoids in ℝn. Given ε > 0 and δ > 0, we study the problems of computing a (1 + ε)-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 Yildirim [J. Optim. Theory Appl., 126 (2005), pp. 1-21] that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in ℝn, which, in turn, is a modification of Khachiyan's algorithm [L. G. Khachiyan, Math. Oper. Res., 21 (1996), pp. 307-320]. Our algorithm can also compute a (1 + δ)n-rounding of 5. For fixed ε > 0 and δ > 0, we establish polynomial-time complexity results for the respective problems, each of which is linear in the number of ellipsoids m. In particular, our algorithm can approximate the minimum volume covering ellipsoid of S in asymptotically the same number of iterations as that required by the algorithm of Kumar and Yildirim to approximate the minimum volume covering ellipsoid of a set of m points. The main ingredient in our analysis is the extension of polynomial-time complexity of certain subroutines in the algorithm from a set of points to a set of ellipsoids. As a byproduct, our algorithm returns a finite "core" set χ ⊆ S with the property that the minimum volume covering ellipsoid of X provides a good approximation to the minimum volume covering ellipsoid of S. Furthermore, the size of the core set depends only on the dimension n and the approximation parameter ε, but not on the number of ellipsoids m. We also discuss the extent to which our algorithm can be used to compute an approximate minimum volume covering ellipsoid and an approximate n-rounding of the convex hull of other sets in ℝn. We adopt the real number model of computation in our analysis.
Open Access
Two algorithms for the minimum enclosing ball problem
(Society for Industrial and Applied Mathematics, 2008) Yıldırım, E. A.
Given A := {a1.....am} ⊂ ℝn and ε > 0, we propose and analyze two algorithms for the problem of computing a (1 + ε)-approximation to the radius of the minimum enclosing ball of A. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of the minimum enclosing ball problem. We establish that this algorithm converges in O(1/ε) iterations with an overall complexity bound of O(mn/ε) arithmetic operations. In addition, the algorithm returns a "core set" of size O(1/ε), which is independent of both m and n. The latter algorithm is obtained by incorporating "away" steps into the former one at each iteration and achieves the same asymptotic complexity bound as the first one. While the asymptotic bound on the size of the core set returned by the second algorithm also remains the same as the first one, the latter algorithm has the potential to compute even smaller core sets in practice, since, in contrast to the former one, it allows "dropping" points from the working core set at each iteration. Our analysis reveals that the leading terms in the asymptotic complexity analysis are reasonably small. In contrast to the first algorithm, we also establish that the second algorithm asymptotically exhibits linear convergence, which provides further insight into our computational results, indicating that the latter algorithm indeed terminates faster with smaller core sets in comparison with the first one. We also discuss how our algorithms can be extended to compute an approximation to the minimum enclosing ball of more general input sets without sacrificing the iteration complexity and the bound on the core set size. In particular, we establish the existence of a core set of size O(1/ε) for a much wider class of input sets. We adopt the real number model of computation in our analysis.