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 $γ_{p} \sqrt{2 - \frac{1}{n-k}}$, where $γ_{p}$ 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 $γ_{p} (1 - ε)$, for any constant $ε> 0$. Finally, we show that assuming the Unique Games Conjecture, the subspace approximation problem is hard to approximate within a factor better than $γ_{p} (1 - ε)$, for any constant $ε> 0$.

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