Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit

Chong You, Daniel Robinson, Rene Vidal; Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016, pp. 3918-3927

Subspace clustering methods based on ell_1, l_2 or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, ell_1 regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, l_2 and nuclear norm regularization provide efficient closed form solutions, but require very strong assumptions to guarantee a subspace-preserving affinity, e.g., independent subspaces and uncorrupted data. In this paper we study a subspace clustering method based on orthogonal matching pursuit. We show that the method is both computationally efficient and guaranteed to give a subspace-preserving affinity under broad conditions. Experiments on synthetic data verify our theoretical analysis, and applications in handwritten digit and face clustering show that our approach achieves the best trade off between accuracy and efficiency. Moreover, our approach is the first one to handle 100,000 data points.