Subspace clustering is the problem of clustering data points into a union of low-dimensional linear affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work (4, 19, 24, 20) provided strong theoretical guarantee for sparse subspace clustering (4), the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from different subspaces are clustered together. Such guarantee, however, is not sufficient for the clustering to be correct, due to the notorious "graph connectivity problem" (15). In this paper, we investigate the graph connectivity problem for noisy sparse subspace clustering and show that a simple post-processing procedure is capable of delivering consistent clustering under certain "general position" or "restricted eigenvalue" assumptions. We also show that our condition is almost tight with adversarial noise perturbation by constructing a counter-example. These results provide the first exact clustering guarantee of noisy SSC for subspaces of dimension greater then 3.
Nevertheless, the above-mentioned results do not rule out cases when the subgraph of each subspace is not well connected. For instance, an empty graph trivially obeys SEP. As a less trivial example, if we connect points in each subspace in disjoint pairs, then the degree of every node will be non-zero, yet the graph does not reveal much information for clustering. It is not hard to construct a problem such that Lasso-SSC will output exactly this. For the original noiseless SSC, the problem becomes trickier since the solution is more constrained. In @cite_9 it was shown that when subspace dimension is no larger than 3, SSC outputs block-wise connected similarity graph under very mild conditions; however, the graph connectivity is easily broken when subspace dimension exceeds 3. Though a simple post-processing step was remarked in [Footnote 6 in Section 5] ssc to alleviate the graph connectivity issue on noiseless data, it is unclear how to extend their method when data are corrupted by noise.
@cite_9
2054199929
Sparse Subspace Clustering (SSC) is one of the recent approaches to subspace segmentation. In SSC a graph is constructed whose nodes are the data points and whose edges are inferred from the L1-sparse representation of each point by the others. It has been proved that if the points lie on a mixture of independent subspaces, the graphical structure of each subspace is disconnected from the others. However, the problem of connectivity within each subspace is still unanswered. This is important since the subspace segmentation in SSC is based on finding the connected components of the graph. Our analysis is built upon the connection between the sparse representation through L1-norm minimization and the geometry of convex poly-topes proposed by the compressed sensing community. After introduction of some assumptions to make the problem well-defined, it is proved that the connectivity within each subspace holds for 2- and 3-dimensional subspaces. The claim of connectivity for general d-dimensional case, even for generic configurations, is proved false by giving a counterexample in dimensions greater than 3.