Algorithms and Hardness for Subspace Approximation

The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm instead. Most of the previous work on subspace approximation has focused on small or constant $k$ and $p$, using coresets and sampling techniques from computational geometry. In this paper, extending another line of work based on convex relaxation and rounding, we give a polynomial time algorithm, \emph{for any $k$ and any $p \geq 2$}, with the approximation guarantee roughly $\gamma_{p} \sqrt{2 - \frac{1}{n-k}}$, where $\gamma_{p}$ is the $p$-th moment of a standard normal random variable N(0,1). We show that the convex relaxation we use has an integrality gap (or "rank gap") of $\gamma_{p} (1 - \epsilon)$, for any constant $\epsilon > 0$. Finally, we show that assuming the Unique Games Conjecture, the subspace approximation problem is hard to approximate within a factor better than $\gamma_{p} (1 - \epsilon)$, for any constant $\epsilon > 0$.