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.

In modern statistics, the robust estimation of parameters of a regression hyperplane is a central problem, i. e., an estimation that is not or only slightly affected by outliers in the data. In this paper we will consider the least median of squares (LMS) estimator. For n points in d dimensions we describe a randomized algorithm for LMS running in O � n d � time and O(n) space, for d fixed, and in time O � d 3 · (2n) d � and O(dn) space, for arbitrary d.

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