  We propose a novel SPARsity and Clustering (SPARC) regularizer, which is a
modified version of the previous octagonal shrinkage and clustering algorithm
for regression (OSCAR), where, the proposed regularizer consists of a
$K$-sparse constraint and a pair-wise $\ell_{\infty}$ norm restricted on the
$K$ largest components in magnitude. The proposed regularizer is able to
separably enforce $K$-sparsity and encourage the non-zeros to be equal in
magnitude. Moreover, it can accurately group the features without shrinking
their magnitude. In fact, SPARC is closely related to OSCAR, so that the
proximity operator of the former can be efficiently computed based on that of
the latter, allowing using proximal splitting algorithms to solve problems with
SPARC regularization. Experiments on synthetic data and with benchmark breast
cancer data show that SPARC is a competitive group-sparsity inducing
regularizer for regression and classification.
