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...

This corollary extends work of Cao and Meyerhoff who had earlier shown that m003 and m004 were the smallest volume cusped manifolds. Also, the above list agrees with the SnapPea census of one-cusped manifolds produced by Jeff Weeks ([W]), whose initial members are conjectured to be an accurate list of small-volume cupsed manifolds. Let N be a closed hyperbolic 3-manifold with simple closed geod...

The MVE estimator is an important tool in robust regression and outlier detection in statistics. We develop fast and efficient algorithms for the MVE estimator problem and discuss how they can be implemented efficiently. The novelty of our approach stems from the recent developments in the first-order algorithms for solving the related Minimum Volume Enclosing Ellipsoid problem. Comparative com...

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a1, . . . , am ∈ R. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of m...

We consider the structure of Kr-free graphs with large minimum degree, and show that such graphs with minimum degree δ > (2r − 5)n/(2r − 3) are homomorphic to the join Kr−3 ∨ H where H is a triangle-free graph. In particular this allows us to generalize results from triangle-free graphs and show that Kr-free graphs with such minimum degree have chromatic number at most r+ 1. We also consider th...

This paper develops theory and algorithms concerning a new metric for clustering data. The metric minimizes the total volume of clusters, where the volume of a cluster is defined as the volume of the minimum volume ellipsoid (MVE) enclosing all data points in the cluster. This metric is scale-invariant, that is, the optimal clusters are invariant under an affine transformation of the data space...

We perform collision detection between static rigid objects using a bounding volume hierarchy which consists of an oriented bounding box (OBB) tree enhanced with bounding spheres. This approach combines the compactness of OBBs and the simplicity of spheres. The majority of distant objects are separated using the simpler sphere tests. The remaining objects are in close proximity, where some sepa...