Hyperplane Clustering via Dual Principal Component Pursuit

State-of-the-art methods for clustering data drawn from a union of subspaces are based on sparse and low-rank representation theory. Existing results guaranteeing the correctness of such methods require the dimension of the subspaces to be small relative to the dimension of the ambient space. When this assumption is violated, as is, for example, in the case of hyperplanes, existing methods are either computationally too intense (e.g., algebraic methods) or lack theoretical support (e.g., K-hyperplanes or RANSAC). The main theoretical contribution of this paper is to 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 expected properties of the non-convex `1 problem associated with DPCP (discrete problem), 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 (in a deterministic sense) inside their associated hyperplanes, then the non-convex DPCP problem has a unique (up to sign) global solution, equal to the normal vector of the dominant hyperplane. This suggests a sequential hyperplane learning algorithm, which first learns the dominant hyperplane by applying DPCP to the data. In order to avoid hard thresholding of the points which is sensitive to the choice of the thresholding parameter, all points are weighted according to their distance to that hyperplane, and a second hyperplane is computed by applying DPCP to the weighted data, and so on. Experiments on corrupted synthetic data show that this DPCP-based sequential algorithm dramatically improves over similar sequential algorithms, which learn the dominant hyperplane via state-of-the-art single subspace learning methods (e.g., with RANSAC or REAPER). Finally, 3D plane clustering experiments on real 3D point clouds show that a K-Hyperplanes DPCP-based scheme, which computes the normal vector of each cluster via DPCP, instead of the classic SVD, is very competitive to state-of-the-art approaches (e.g., RANSAC or SVD-based K-Hyperplanes).