Efficient Subspace Approximation Algorithms

We consider the problem of fitting a subspace of a specified dimension k to a set P of n points in a"e (d) . The fit of a subspace F is measured by the L (tau) norm, that is, it is defined as the tau-root of the sum of the tau th powers of the Euclidean distances of the points in P from F, for some tau a parts per thousand yen1. Our main result is a randomized algorithm that takes as input P, k, and a parameter 0 < epsilon < 1; runs in nd . 2(O(tau k2/epsilon log2 k/epsilon)) time, and returns a k-subspace that with probability at least 1/2 has a fit that is at most (1+epsilon) times that of the optimal k-subspace.