Learning Robust Subspace Clustering

We propose a low-rank transformation-learning framework to robustify subspace clustering. Many high-dimensional data, such as face images and motion sequences, lie in a union of low-dimensional subspaces. The subspace clustering problem has been extensively studied in the literature to partition such high-dimensional data into clusters corresponding to their underlying low-dimensional subspaces. However, low-dimensional intrinsic structures are often violated for real-world observations, as they can be corrupted by errors or deviate from ideal models. We propose to address this by learning a linear transformation on subspaces using matrix rank, via its convex surrogate nuclear norm, as the optimization criteria. The learned linear transformation restores a low-rank structure for data from the same subspace, and, at the same time, forces a high-rank structure for data from different subspaces. In this way, we reduce variations within the subspaces, and increase separations between the subspaces for more accurate subspace clustering. This proposed learned robust subspace clustering framework significantly enhances the performance of existing subspace clustering methods. To exploit the low-rank structures of the transformed subspaces, we further introduce a subspace clustering technique, called Robust Sparse Subspace Clustering, which efficiently combines robust PCA with sparse modeling. We also discuss the online learning of the transformation, and learning of the transformation while simultaneously reducing the data dimensionality. Extensive experiments using public datasets are presented, showing that the proposed approach significantly outperforms state-of-the-art subspace clustering methods.