Discrete & Computational Geometry

, Volume 36, Issue 4, pp 593–607 | Cite as

On the Least Median Square Problem

  • Jeff EricksonEmail author
  • Sariel Har-PeledEmail author
  • David M. MountEmail author


We consider the exact and approximate computational complexity of the multivariate least median-of-squares (LMS) linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in \({\Bbb R}^d\) and a parameter k, the problem is equivalent to computing the narrowest slab bounded by two parallel hyperplanes that contains k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide nearly matching lower bounds for these problems. These lower bounds hold under the assumptions that k is Ω(n) and that deciding whether n given points in \({\Bbb R}^d\) are affinely non-degenerate requires Ω(nd) time.


Extra Point Vertical Line Segment Vertical Slab Degeneracy Problem Linear Regression Estimator 
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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of Computer Science, University of IllinoisUrbana, IL 61801USA
  2. 2.Department of Computer Science and Institute for Advanced Computer Studies, University of MarylandCollege Park, MD 20742USA

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