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The subspace approximation problem Subspace(k, p) asks for a k dimensional linear subspace that fits a given set of m points in Rn optimally. The error for fitting is a generalization of the least squares fit and uses the lp norm of the distances (l2 distances) of the points from the subspace, e.g., p = ∞ means minimizing the l2 distance of the farthest point from the subspace. Previous work on subspace approximation considers either the case of small or constant k and p [27, 11, 14] or the case of p = ∞ [16, 8, 17, 7, 24, 23, 29].

Algorithms and Hardness for Subspace Approximation