The subspace approximation problem with outliers, for given $n$ points in $d$ dimensions $x_{1},ldots, x_{n} in R^{d}$, an integer $1 leq k leq d$, and an outlier parameter $0 leq alpha leq 1$, is to find a $k$-dimensional linear subspace of $R^{d}$ that minimizes the sum of squared distances to its nearest $(1-alpha)n$ points. More generally, the $ell_{p}$ subspace approximation problem with outliers minimizes the sum of $p$-th powers of distances instead of the sum of squared distances. Even the case of robust PCA is non-trivial, and previous work requires additional assumptions on the input. Any multiplicative approximation algorithm for the subspace approximation problem with outliers must solve the robust subspace recovery problem, a special case in which the $(1-alpha)n$ inliers in the optimal solution are promised to lie exactly on a $k$-dimensional linear subspace. However, robust subspace recovery is Small Set Expansion (SSE)-hard. We show how to extend dimension reduction techniques and bi-criteria approximations based on sampling to the problem of subspace approximation with outliers. To get around the SSE-hardness of robust subspace recovery, we assume that the squared distance error of the optimal $k$-dimensional subspace summed over the optimal $(1-alpha)n$ inliers is at least $delta$ times its squared-error summed over all $n$ points, for some $0 < delta leq 1 - alpha$. With this assumption, we give an efficient algorithm to find a subset of $poly(k/epsilon) log(1/delta) loglog(1/delta)$ points whose span contains a $k$-dimensional subspace that gives a multiplicative $(1+epsilon)$-approximation to the optimal solution. The running time of our algorithm is linear in $n$ and $d$. Interestingly, our results hold even when the fraction of outliers $alpha$ is large, as long as the obvious condition $0 < delta leq 1 - alpha$ is satisfied.
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