  We extend the theoretical analysis of a recently proposed single subspace
learning algorithm, called Dual Principal Component Pursuit (DPCP), to the case
where the data are drawn from of a union of hyperplanes. To gain insight into
the properties of the $\ell_1$ non-convex problem associated with DPCP, we
develop a geometric analysis of a closely related continuous optimization
problem. Then transferring this analysis to the discrete problem, our results
state that as long as the hyperplanes are sufficiently separated, the dominant
hyperplane is sufficiently dominant and the points are uniformly distributed
inside the associated hyperplanes, then the non-convex DPCP problem has a
unique global solution, equal to the normal vector of the dominant hyperplane.
This suggests the correctness of a sequential hyperplane learning algorithm
based on DPCP. A thorough experimental evaluation reveals that hyperplane
learning schemes based on DPCP dramatically improve over the state-of-the-art
methods for the case of synthetic data, while are competitive to the
state-of-the-art in the case of 3D plane clustering for Kinect data.
