Subspace approximation with outliers

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 $0 \leq α\leq 1$, is to find a $k$-dimensional linear subspace of $R^{d}$ that minimizes the sum of squared distances to its nearest $(1-α)n$ 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 $(1-α)n$ 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 $(1-α)n$ inliers is at least $δ$ times its squared-error summed over all $n$ points, for some $0 < δ\leq 1 - α$. With this assumption, we give an efficient algorithm to find a subset of $poly(k/ε) \log(1/δ) \log\log(1/δ)$ points whose span contains a $k$-dimensional subspace that gives a multiplicative $(1+ε)$-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 $α$ is large, as long as the obvious condition $0 < δ\leq 1 - α$ is satisfied.

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