Proceedings of the 20th Annual ACM Symposium on Computational Geometry, 273-279, 2004.
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 Rd and a parameter k, the problem is equivalent to computing the slab bounded by two parallel hyperplanes of minimum separation 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 on the computational complexity of these problems, under the assumption that deciding whether n given points in Rd are affinely nondegenerate requires Ω(nd) time.