  Sparse subspace clustering (SSC) is a state-of-the-art method for segmenting
a set of data points drawn from a union of subspaces into their respective
subspaces. It is now well understood that SSC produces subspace-preserving data
affinity under broad geometric conditions but suffers from a connectivity
issue. In this paper, we develop a novel geometric analysis for a variant of
SSC, named affine SSC (ASSC), for the problem of clustering data from a union
of affine subspaces. Our contributions include a new concept called affine
independence for capturing the arrangement of a collection of affine subspaces.
Under the affine independence assumption, we show that ASSC is guaranteed to
produce subspace-preserving affinity. Moreover, inspired by the phenomenon that
the $\ell_1$ regularization no longer induces sparsity when the solution is
nonnegative, we further show that subspace-preserving recovery can be achieved
under much weaker conditions for all data points other than the extreme points
of samples from each subspace. In addition, we confirm a curious observation
that the affinity produced by ASSC may be subspace-dense---which could
guarantee the subspace-preserving affinity of ASSC to produce correct
clustering under rather weak conditions. We validate the theoretical findings
on carefully designed synthetic data and evaluate the performance of ASSC on
several real data sets.
