Qiang Qiu, Guillermo Sapiro.
Year: 2015, Volume: 16, Issue: 7, Pages: 187−225
A low-rank transformation learning framework for subspace clustering and classification is proposed here. Many high- dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding 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. 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 nuclear norm as the modeling and optimization criteria. The learned linear transformation restores a low-rank structure for data from the same subspace, and, at the same time, forces a maximally separated structure for data from different subspaces. In this way, we reduce variations within the subspaces, and increase separation between the subspaces for a more robust subspace clustering. This proposed learned robust subspace clustering framework significantly enhances the performance of existing subspace clustering methods. Basic theoretical results presented here help to further support the underlying framework. To exploit the low-rank structures of the transformed subspaces, we further introduce a fast subspace clustering technique, which efficiently combines robust PCA with sparse modeling. When class labels are present at the training stage, we show this low-rank transformation framework also significantly enhances classification performance. Extensive experiments using public data sets are presented, showing that the proposed approach significantly outperforms state-of-the-art methods for subspace clustering and classification. The learned low cost transform is also applicable to other classification frameworks.