Efficient Subspace Approximation Algorithms

Abstract

We consider the problem of fitting a subspace of a specified dimension k to a set P of n points in ℝd. The fit of a subspace F is measured by the Lτ norm, that is, it is defined as the τ-root of the sum of the τth powers of the Euclidean distances of the points in P from F, for some τ≥1. Our main result is a randomized algorithm that takes as input P, k, and a parameter 0 < ε < 1; runs in nd · 20(τκ2/εlog2 κ/ε) time, and returns a k-subspace that with probability at least 1/2 has a fit that is at most (1+ε) times that of the optimal k-subspace. © 2011 Springer Science+Business Media, LLC.