Quantile approximation for robust statistical estimation

Given a set P of n points in R d , a fundamental problem in computational geometry is concerned with nding the smallest enclosing \range" of P. Well known instances of this problem include nding, e.g., the smallest enclosing box 10], smallest enclosing simplex 11], minimum volume ball 17, 20], (2-D) smallest enclosing ellipsoid 16, 20, 2, 14], and minimum volume annulus 1, 8, 5]. In this paper, we consider the following generic variant: Given a set of n points in R d , nd the smallest range in question that contains (at least) a certain quantile (up to 50%) of the data. Although this variant has been studied to some extent in computational geometry | instances include enclosing k points by a circle 7, 4] and nding the smallest axis parallel rectangle enclosing k points 15] | the algorithms proposed deal mainly with speciic cases and their running times are relatively high (O(n 2 log n) and O(n 3), respectively, for the minimum enclosing circle and axis parallel rectangle in the plane). stems from the growing need for eecient data analysis techniques that are robust to outlying/noisy observations. While such data can be handled successfully due to robust statistical estimators, the eecient computation of these estimators continues to pose a formidable algorithmic challenge. The basic measure of the robustness of an estima-tor is its breakdown point, that is, the fraction (up to 50%) of outlying data that can corrupt the estimator (see, e.g., 13] for an exact deenition). For example , Rousseeuw's least median-of-squares (LMS) regression estimator 12] (which is equivalent to nding the narrowest hyperstrip contanining at least 50% of the data) is among the best known 50% breakdown-point estimators. The best exact algorithm known for this problem runs in O(n d+2 log n) time 19]. Because of this high running time, a simple O(n logn) Monte Carlo algorithm (for xed d) is commonly used 13]. However, the latter provides no guarantee of accuracy (even probabilistic) unless the data set satisses certain assumptions. 1 Other robust estimators considered in this paper are the minimum volume disk (e.g., the minimum volume ellipsoid (MVE)) and the minimum volume annulus (MVA) enclosing (at least) half of the points in P. As in the case of LMS, algorithms known for these estimators have either high complexity or they provide no guarantee of accuracy due to their Monte Carlo nature 13]. In an …