Computationally Efficient Algorithms for High-Dimensional Robust Estimators

Given a set of n distinct points in d-dimensional space that are hypothesized to lie on a hyperplane, robust statistical estimators have been recently proposed for the parameters of the model that best fits these points. This paper presents efficient algorithms for computing median-based robust estimators (e.g., the Theil-Sen and repeated median (RM) estimators) in high-dimensional space. We briefly review basic computational geometry techniques that were used to achieve efficient algorithms in the 2-D case. Then generalization of these techniques to higher dimensions is introduced. Geometric observations are followed by a presentation of O(n^{d − 1} log n) expected time algorithms for the d-dimensional Theil-Sen and RM estimators. Both algorithms are space optimal; i.e., they require O(n) storage, for fixed d. Finally, an extension of the methodology to nonlinear domain(s) is demonstrated.