Clustering Consistent Sparse Subspace Clustering

Subspace clustering is the problem of clustering data points into a union of low-dimensional linear or affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and has been drawing avid attention in machine learning and statistics recently. In particular, a line of recent work (Elhamifar and Vidal, 2013; Soltanolkotabi et al., 2012; Wang and Xu, 2013; Soltanolkotabi et al., 2014) provided strong theoretical guarantee for the seminal algorithm: Sparse Subspace Clustering (SSC) (Elhamifar and Vidal, 2013) under various settings, and to some extent, justified its state-of-the-art performance in applications such as motion segmentation and face clustering. The focus of these work has been getting milder conditions under which SSC obeys "self-expressiveness property", which ensures that no two points from different subspaces can be clustered together. Such guarantee however is not sufficient for the clustering to be correct, thanks to the notorious "graph connectivity problem" (Nasihatkon and Hartley, 2011). In this paper, we show that this issue can be resolved by a very simple post-processing procedure under only a mild "general position" assumption. In addition, we show that the approach is robust to arbitrary bounded perturbation of the data whenever the "general position" assumption holds with a margin. These results provide the first exact clustering guarantee of SSC for subspaces of dimension greater than 3.