  We consider the problem of learning a linear subspace from data corrupted by
outliers. Classical approaches are typically designed for the case in which the
subspace dimension is small relative to the ambient dimension. Our approach
works with a dual representation of the subspace and hence aims to find its
orthogonal complement; as such, it is particularly suitable for subspaces whose
dimension is close to the ambient dimension (subspaces of high relative
dimension). We pose the problem of computing normal vectors to the inlier
subspace as a non-convex $\ell_1$ minimization problem on the sphere, which we
call Dual Principal Component Pursuit (DPCP) problem. We provide theoretical
guarantees under which every global solution to DPCP is a vector in the
orthogonal complement of the inlier subspace. Moreover, we relax the non-convex
DPCP problem to a recursion of linear programs whose solutions are shown to
converge in a finite number of steps to a vector orthogonal to the subspace. In
particular, when the inlier subspace is a hyperplane, the solutions to the
recursion of linear programs converge to the global minimum of the non-convex
DPCP problem in a finite number of steps. We also propose algorithms based on
alternating minimization and iteratively re-weighted least squares, which are
suitable for dealing with large-scale data. Experiments on synthetic data show
that the proposed methods are able to handle more outliers and higher relative
dimensions than current state-of-the-art methods, while experiments in the
context of the three-view geometry problem in computer vision suggest that the
proposed methods can be a useful or even superior alternative to traditional
RANSAC-based approaches for computer vision and other applications.
