On the least median square problem

We consider the exact and approximate computational complexity of the multivariate LMS linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in 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, under the assumption that deciding whether n given points in ℝ^d are affinely nondegenerate requires Ω(n^d) time.