  State-of-the-art subspace clustering methods are based on expressing each
data point as a linear combination of other data points while regularizing the
matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$
regularization is guaranteed to give a subspace-preserving affinity (i.e.,
there are no connections between points from different subspaces) under broad
theoretical conditions, but the clusters may not be connected. $\ell_2$ and
nuclear norm regularization often improve connectivity, but give a
subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$,
$\ell_2$ and nuclear norm regularizations offer a balance between the
subspace-preserving and connectedness properties, but this comes at the cost of
increased computational complexity. This paper studies the geometry of the
elastic net regularizer (a mixture of the $\ell_1$ and $\ell_2$ norms) and uses
it to derive a provably correct and scalable active set method for finding the
optimal coefficients. Our geometric analysis also provides a theoretical
justification and a geometric interpretation for the balance between the
connectedness (due to $\ell_2$ regularization) and subspace-preserving (due to
$\ell_1$ regularization) properties for elastic net subspace clustering. Our
experiments show that the proposed active set method not only achieves
state-of-the-art clustering performance, but also efficiently handles
large-scale datasets.
