  We consider the problem of subspace clustering: given points that lie on or
near the union of many low-dimensional linear subspaces, recover the subspaces.
To this end, one first identifies sets of points close to the same subspace and
uses the sets to estimate the subspaces. As the geometric structure of the
clusters (linear subspaces) forbids proper performance of general distance
based approaches such as K-means, many model-specific methods have been
proposed. In this paper, we provide new simple and efficient algorithms for
this problem. Our statistical analysis shows that the algorithms are guaranteed
exact (perfect) clustering performance under certain conditions on the number
of points and the affinity between subspaces. These conditions are weaker than
those considered in the standard statistical literature. Experimental results
on synthetic data generated from the standard unions of subspaces model
demonstrate our theory. We also show that our algorithm performs competitively
against state-of-the-art algorithms on real-world applications such as motion
segmentation and face clustering, with much simpler implementation and lower
computational cost.
