  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.
