  Recent methods for learning a linear subspace from data corrupted by outliers
are based on convex $\ell_1$ and nuclear norm optimization and require the
dimension of the subspace and the number of outliers to be sufficiently small.
In sharp contrast, the recently proposed Dual Principal Component Pursuit
(DPCP) method can provably handle subspaces of high dimension by solving a
non-convex $\ell_1$ optimization problem on the sphere. However, its geometric
analysis is based on quantities that are difficult to interpret and are not
amenable to statistical analysis. In this paper we provide a refined geometric
analysis and a new statistical analysis that show that DPCP can tolerate as
many outliers as the square of the number of inliers, thus improving upon other
provably correct robust PCA methods. We also propose a scalable Projected
Sub-Gradient Method method (DPCP-PSGM) for solving the DPCP problem and show it
admits linear convergence even though the underlying optimization problem is
non-convex and non-smooth. Experiments on road plane detection from 3D point
cloud data demonstrate that DPCP-PSGM can be more efficient than the
traditional RANSAC algorithm, which is one of the most popular methods for such
computer vision applications.
