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We consider a fundamental problem in unsupervised learning called\emph{subspace recovery}: given a collection of $m$ points in $\mathbb{R}^n$,if many but not necessarily all of these points are contained in a$d$-dimensional subspace $T$ can we find it? The points contained in $T$ arecalled {\em inliers} and the remaining points are {\em outliers}. This problemhas received considerable attention in computer science and in statistics. Yetefficient algorithms from computer science are not robust to {\em adversarial}outliers, and the estimators from robust statistics are hard to compute in highdimensions.  Are there algorithms for subspace recovery that are both robust to outliersand efficient? We give an algorithm that finds $T$ when it contains more than a$\frac{d}{n}$ fraction of the points. Hence, for say $d = n/2$ this estimatoris both easy to compute and well-behaved when there are a constant fraction ofoutliers. We prove that it is Small Set Expansion hard to find $T$ when thefraction of errors is any larger, thus giving evidence that our estimator is an{\em optimal} compromise between efficiency and robustness.  As it turns out, this basic problem has a surprising number of connections toother areas including small set expansion, matroid theory and functionalanalysis that we make use of here.

Algorithms and Hardness for Robust Subspace Recovery